EXISTENCE OF HOMOCLINIC CONNECTIONS CORRESPONDING TO BILAYER
STRUCTURES IN AMPHIPHILIC POLYMER SYSTEMS
By
Li Yang

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Applied Mathematics - Doctor of Philosophy
2013

ABSTRACT
EXISTENCE OF HOMOCLINIC CONNECTIONS CORRESPONDING TO
BILAYER STRUCTURES IN AMPHIPHILIC POLYMER SYSTEMS
By
Li Yang
Bilayer structures are central to amphiphilic polymer systems which possess a phase
which wets two immiscible fluids. The amphiphilic component forms thin layers which
separate the immiscible phases. When one of the immiscible phases and the amphiphilic
material are proportional, and scarce, then the mixture can be modeled as two phase and the
bilayer structures as homoclinic connections. We prove the existence of the bilayer structures
(homoclinic solutions) for the functionalized Cahn-Hilliard equation, whose equilibriums
support these structures. We employ two methods: a functional analytical approach and a
variant of Lin’s method. The functional analytical approach is based upon a Newton type
contraction mapping and it gives the leading order description of the homoclinic connection
in terms of a homoclinic connection of a low-order problem. The contraction mapping
construction also requires a non-degeneracy condition, which we conjecture is associated to
an orbit-flip bifurcation of the homoclinic connection within the higher-order system. Lin’s
method is an implementation of the Lyapunov-Schmidt method to prove the existence of
heteroclinic chains in dynamical systems. Because of the degeneracy of the full problem, we
apply Lin’s method only to a more restricted parameter regime.

TABLE OF CONTENTS
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iv

Chapter 1 Introduction to the Functionalized Cahn-Hilliard Equation . .
1.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
9

Chapter 2 Functional Analytical Method . . . . . . . . . .
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Construction of Critical points by Contraction Mapping .
2.3 Conditioning of the Newton map . . . . . . . . . . . . .

.
.
.
.

27
27
30
44

Chapter 3 Analysis of the Second-Order System . . . . . . . . . . . . . . . .
3.1 Analysis of the second order eigenvalue problem . . . . . . . . . . . . . . . .
3.2 Heteroclinic limit of the second-order problem . . . . . . . . . . . . . . . . .

71
71
81

Chapter 4 Lin’s Method . . . . . .
4.1 Introduction . . . . . . . . . . .
4.2 Lin’s Orbit . . . . . . . . . . . .
4.3 Estimates for the Jump . . . . .
4.4 Solving the bifurcation equation
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LIST OF FIGURES

Figure 1.1

[28] Numerical evolution of (1.0.4). Reading left to right, the initial data is four circles of “water” (u = b+ ) within a background of
“polymer”(u = b− ). The boundary between two domains is given
by a front-type solution. In the early stage the high curvature circles grow at the expense of the lower curvature ones, however, the
single layer interfaces are unstable to an antipodal elongation, and
form bilayer structures, which lengthen and meander until achieving
a quasi-equlibrium at critical bilayer width. For interpretation of the
references to color in this and all other figures, the reader is referred
to the electronic version of this dissertation. . . . . . . . . . . . . . .

8

Figure 1.2

Illustration of P (left) and W (right). . . . . . . . . . . . . . . . . .

9

Figure 1.3

The tilted-shifted potential G. . . . . . . . . . . . . . . . . . . . . .

11

Figure 1.4

The homoclinic pulse profile of (1.1.6).

. . . . . . . . . . . . . . . .

12

Figure 1.5

The illustration of spectrum of Lα . The colored strip denotes the
essential spectrum with begins at −W (b) < 0. . . . . . . . . . . . .

15

Figure 1.6

Figure 1.7

Figure 1.8

Figure 1.9

Depiction of the stable manifold of the equilibrium of a homoclinic
orbit under an orbit flip bifurcation. For η = 0 the homoclinic orbit
(red line) converges to the equilibrium along the fast stable direction,
for η > 0 it converges to its equilibrium on one side, and flips the side
for η < 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

The illustration of the scaling (S)-(1.1.3) and the domain of the nondegeneracy in Theorem 1.1 in the left Figure and the illustration of
the domain of applicability (S’)-(1.1.4) of Theorem 1.2 in the right
Figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

The illustration of the scaling (S)-(1.1.3) and the domain of the nondegeneracy in Theorem 1.1 in the left Figure and the illustration of
the domain of applicability (S’)-(1.1.4) of Theorem 1.2 in the right
Figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

heteroclinic connection θ = 0.

21

iv

. . . . . . . . . . . . . . . . . . . . .

Figure 1.10

Under the assumption (S)-(1.1.3) the illustration of the eigenvalue of
Dx f (x; δ) (p ;0) for δ = 0. . . . . . . . . . . . . . . . . . . . . . . .

22

Under the assumption (S)-(1.1.3) the illustration of the eigenvalue of
Dx f (x; δ)|(p ;δ) , 0 < δ
1 for δ = 0. . . . . . . . . . . . . . . . . .

23

Figure 1.12

Perturbed heteroclinic connection for Γ1 . . . . . . . . . . . . . . . .

24

Figure 1.13

Lin’s orbit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

Figure 3.1

The homoclinic solution of (1.1.6) in red color and the heteroclinic
solution of (1.1.8) in blue color. At z = 0, φm and φh are on the cross
section Σin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

1

Figure 1.11

1

Figure 3.2

The homoclinic solution of (1.1.6) in red color and the heteroclinic
solution of (1.1.8) in blue color. At z = τm , φm and φh are on the
cross section Σin . At z = τ∗ , P (φh (τ∗ )) = 0. . . . . . . . . . . . .

v

86

Chapter 1
Introduction to the Functionalized
Cahn-Hilliard Equation
A central goal of polymer chemistry is the design of materials with novel macroscopic properties by controlling the spontaneous generation of nanoscaled, phase separated networks,
[7]. One mechanism to generate such networks is through the “functionalization” of hydrophobic polymer chains by the addition of acid terminated side-chains. In the presence of
a polar solvent the end groups interact exothermically, driving the spontaneous generation
of polymer-solvent or nanoparticle-solvent interfaces. The resulting phase separated network
structures can be exploited for charge selective conduction, and have important applications
to efficient energy conversion devices such as polymer electrolyte membranes for fuel cells,
[58, 54], dye sensitized solar cells, [42], and bulk-heterojunction solar cells, [51].
The phase separation of microemulsions is typically dominated by interfacial energies.
The Cahn-Hilliard (CH) Energy introduced in [15] is a diffuse interface model that characterizes a binary mixture by a phase field function u that maps Ω ∈ Rn into mixture
values [−1, 1]. It models the free energy as a balance between entropic effects, which seek to
homogenize the species, and the mixture potential, W , which assigns energies to blends

E(u) =

ε2
| u|2 + W (u) dx,
2
Ω
1

(1.0.1)

where ε

1 controls the width of the inner structures. The Cahn-Hilliard equation is a

mass-conserving H −1 gradient flow of the free energy functional (1.0.1); it describes the phase
separation and domain coarsening of binary mixtures of inert materials. For an appropriate
choice of double well potential W , solutions of the Cahn-Hilliard equation rapidly undergo a
spinodal decomposition into “pure states” or “phases” occupying respective minima of wells
of W , separated by transition layers of O(ε) thickness, called single-layer interfaces. These
“single layer” morphologies are competitors for the minimizer of the Cahn-Hilliard energy.
The considerable attention has been paid to the evolution of the spatial domains occupied
by the respective phases. Pego first established the motion of the inferfaces as a MullinsSekerka type flow, [52]. He introduced the chemical potential as the solution of Laplace’s
equation on each spinodal domain with the interfacial curvature as a Dirichlet condition on
the internal interfaces. For ε

1 he showed that the leading order normal velocity of the

interface of the spinodal domains can be obtained from the jump in the normal derivative of
the chemical potential defined on the complementary domains. More rigorous derivations of
Pego’s results quickly followed, particularly [2] and later [49]. The Gamma-convergence of
the Cahn-Hilliard energy to the surface area functional was rigorously established in [47, 70].
That is, for ε

1, minimizing sequences uε which converge to a limit in L1 (Ω) localize their

gradients on an interface Γ ∈ Rn while E(uε ) tends to a value which is proportional to the
interfacial surface area.
While mixtures of inert materials generically seek to minimize surface area, functionalized materials have embedded charged groups which interact exothermically with polar
solvents., spontaneously generating polymer-solvent interface. A primary example is Nafion,
a functioinalized fluorocarbon polymer frequently used as a membrane separator in polymer
electrolyte membrane fuel cells. Early small angle X-ray scattering (SAXS) experiments,
2

[34], lead Hsu and Gierke to hypothesized that a balance between the elastic energy of
the interface and the hydrophilic surface interactions among the charged functional groups
and the solvent drive Nafion to generate a water-filled, “pearled” pore network comprised of
small 4-5 nanometer balls interconnected by thin 1-2 nanometer cylindrical pores. Nanoscale
pearling has also be studied in functionalized diblock polymers, [31, 16]. There has since
been considerable controversy as numerical and experimental investigation of the microstructure of Nafion and related perfluorinated membranes have lead to the proposition of bi-layer
morphologies, [69, 39], cylindrical pores, [66], an inverted pore morphology with solvent
groups surrounding cylinders of crystallized backbone, [59, 58], spherical clusters, [25], as
well as more complex morphologies suggested by atomistic simulation [76, 38, 50]. Central
to each of these models is the recognition that the solvent and ionic groups aggregate in the
perfluorinated polymer matrix to form a connected network that allows for efficient proton
transport through the nanometer-scale clusters.
For these applications the morphologies we consider are not the minimizers of the CahnHilliard energy but come from its vast families of saddle points structures. Indeed, it is
considerably less well-known that the critical points of the Cahn-Hilliard energy encompass
large classes of network structures: domains for which the characteristic width of the minority
phase domain scales with ε, giving rise to long, thin, percolating structures: bilayers, pores,
pearled-pores, and micelle clusters, [50]. The network structures are fundamentally distinct
from the more familiar surface area minimizers, as ε → 0+ : the network morphologies grow
thinner, and longer, and do not approach a fixed limit in L1 (Ω).
The “functionalized” Cahn-Hilliard energy (FCH), [28], is a reformulation of the CahnHilliard energy, generating a new free energy which possesses the same families of critical
points as the Cahn-Hilliard but which permits a facile mechanism to select the critical
3

points with desired attributes as its quasi-minimizers. The model affords a finite interfacial
width, accommodates merging and other topological reorganization, and couples naturally to
momentum balance and other macroscopic mass transport equations. We assign a negative
value to interfacial energy via the Cahn-Hilliard energy, and balance the negative CahnHilliard energy against the square of its own variational derivative. In general, we denote
such an energy by F, and call it the functionalization of the orginal energy, E,
F(u) =

1
Ω2

δE 2
ε2
| u|2 + W (u)
−η
˜
δu
2

dx.

(1.0.2)

Here E represents a free energy functional of Cahn-Hilliard type, see (1.0.1), where W is
a smooth, double-well potential with equal global minima at states u = b± , with b− < b+
and µ± := W (b± ) > 0. The FCH remaps this paradigm, balancing the square of the first
variation of the CH energy against a small multiple of itself. Viewing the square of the first
variation of the CH energy as the bending energy of the interface, physical considerations
suggests the constant η be a small parameter that represents the strength of functionalization
˜
and u is a phase function that differentiates between the solvent and polymer backbone
regions represented by two minima of a double-well potential W .
The term functionalization is borrowed from synthetic chemistry where it refers to the
addition of hydrophilic (functional) groups to a hydrophobic polymer to modify its solubility.
Mathematically, “functionalization” is a systematic reformulation of the original energy.
Indeed for the η = 0 problem, all critical points of E, that is the solutions of δE = 0,
˜
δu
render F(u) = 0 and hence are global minimizers of F. The parameter η breaks this highly
˜
degenerate situation: crucially, for η > 0, the perturbation term favors the critical points
˜
of E with more surface area. It has been demonstrated for a broad class of energies that
4

their associated functionalized form is bounded below and possesses global minimizers over
natural function spaces, [56]. An energy similar to the FCH, called the Φ6 model, has been
proposed for amphiphilic systems, in which two immiscible fluids are mixed with a surfactant
forming a microemulsion at the interface, [30]. The Φ6 model was motivated by SAXS data
which can be related to the reciprocal of the Fourier transform of the second variation of the
energy evaluated at a constant background state.
Higher order energies, which resemble the FCH with η < 0 and an untilted well W ,
˜
have been proposed, [45, 71]. Indeed a De Giorgi conjecture concerning the Γ limit of
the FCH energy for η < 0 with an untilted well has been established, [57]. Extensions
˜
of these models to address deformations of elastic vesicles subject to volume constraints,
[23, 24], and multicomponent models which incorporate a variable intrinsic curvature have
been investigated, [73]. However, the single-layer interface forms the essential underpinning
of each of these models.
It is instructive to view the FCH energy as a diffuse-interface regularization of a CanhamHelfrich, [17, 32] sharp interface energy of the form

ECH (Γ) :=

Γ

a1 H 2 − a2 dS,

(1.0.3)

however this identification is potentially misleading as it is predicated on the assumption
that the underlying structures are of co-dimention 1 and free of defects, such as end-caps
and junctions. Over R3 the FCH free energy supports co-dimension one bilayer interfaces, as
well as and a wide range of stable co-dimension 2 and co-dimension 3 morphologies, [21, 28],
in addition to many locally stable defect structures. The structure of the problem, and the
physically motivating examples, change fundamentally and dramatically with the sign of η .
˜

5

For these reasons the FCH merits a disctinct name, see [54], which evokes the amphiphilic
nature of functionalized polymers.
It is crucial to emphasize the distinction between single-layer interfaces, which separate
two dissimilar phases across a co-dimension one interface, and bilayers which separate two
identical phases by a thin region of a second phase. Significantly, the single-layer framework
can not support perforation of the interface. In many biological processes it is essential
to understand the opening and closing of pores within a vesicle, or the roll-up of a bicelle
into a closed vesicle, [68]. Single layer models treat the inside and outside of a vesicle as
distinct phases: they can not be merged. In contrast, the η > 0 perturbation supports stable,
˜
strongly incompressible bilayers which admit not only the opening of perforations, but the
roll-up of the bilayer into a solid filament or its break-up into a collection of solid micelle,
in a manner which naturally accounts for the competition between these morphologically
distinct structures for a scarce surfactant phase. We do not address this competition within
the current work, rather we prove the existence of the bilayer structures of the FCH energy.
The Cahn-Hilliard equation, the H −1 mass preserving gradient flow on the energy E,
drives the system to minimize interface area while preserving the volume fraction, which
describes the coarsening processes of binary alloys [9, 14, 15]. Network formation in functionalized polymers is a fundamentally different process, and the FCH energy takes this into
consideration by balancing the solvation energy released by formation of water-acid interface
against the elastic energy required to bend the interface, and associated polymer backbones.
The functionalized Cahn-Hilliard equation is a mass preserving gradient flow of the FCH
energy
ut = −G

δF
= −G (ε2 ∆ − W (u) + η )(ε2 ∆u − W (u)) ,
˜
δu

6

(1.0.4)

where G is any positive, self-adjoint operator whose only kernel is the constant factor 1.
Examples include the zero-mass projection Π0 , which subtracts the average value,

Π0 f := f −

1
f (x)dx,
|Ω| Ω

(1.0.5)

as well as the negative Laplacian −∆.
We are interested in the critical points of (1.0.4) which are also homoclinic solutions,

˜
G (ε2 ∆ − W (u) + η )(ε2 ∆u − W (u)) = 0.

(1.0.6)

Inverting the gradient operator, (1.0.6) can be written as

(ε2 ∆ − W (u) + η )(ε2 ∆u − W (u)) = θ,
˜

(1.0.7)

where the constant θ can be viewed as a Lagrangian multiplier associated to the mass
conservation. We look for the flat-interface co-dimension one bi-layer solutions Φm (z), after
rescaling in transverse dimension and neglecting the tangential variation, Φm is the solution
of
2
˜
∂z − W (Φm ) + η

2
∂z u − W (Φm ) = θ,

(1.0.8)

which is homoclinic to the back-ground state b. It follows from (1.0.8) that the constants b
and θ are connected via the relation

θ = (W (b) − η )W (b).
˜

(1.0.9)

For θ = 0, (1.0.8) supports single-layer heteroclinic structures which dominate the gradient
7

Consider the “simplest” mass preserving gradient flow
ut = −Π0

δFCH (u)
δu

F (u)

￿

￿￿￿ ￿
￿
= −Π0 ￿2∆ − W ￿￿(u) + η ￿2∆u − W ￿(u) ,

Figure 1.1: [28] Numerical evolution of (1.0.4). Reading left to right, the initial data is four
8
circles of “water” (u = b+ ) within a background of “polymer”(u = b− ). The boundary
between two domains is given by a front-type solution. In the early stage the high curvature
circles grow at the expense of the lower curvature ones, however, the single layer interfaces
are unstable to an antipodal elongation, and form bilayer structures, which lengthen and
meander until achieving a quasi-equlibrium at critical bilayer width. For interpretation of
the references to color in this and all other figures, the reader is referred to the electronic
version of this dissertation.

flow of the Cahn-Hilliard energy. For 0 < |θ|

1, we will show that the system has a

family of homoclinic bi-layer solutions parameterized by bi-layer width. We demonstrate
this using two distinct approaches: a functional analytical approach based upon a Newtontype contraction mapping and a dynamical systems approach based upon Lin’s method.

In the next section, we will introduce the definitions and notations which will be used
in the rest of paper and state our main theorems. The existence of homoclinic solution by
functional analytical contraction mapping method will be illustrated in Chapter 2 and 3. In
addition, Lin’s method to prove the existence of homoclinic orbits in a less degenerated case
is presented in Chapter 4.
8

1.1

Main Results

Before we state the main theorems, we introduce some definitions, assumptions and scaling
which will be used throughout the paper.
(H) The well potential W is a smooth double well W = P 2 where P is a convex function
with transverse zeros at b± with b− < b+ , W (b± ) = W (b± ) = 0 and µ± := W (b± ) >
0, see Fig 1.2.

b−

b+

b−

b+

Figure 1.2: Illustration of P (left) and W (right).

The first parameter we introduce is the background state b of the homoclinic pulse profile.
The goal of the functional analytical method is to characterize the leading order profile of the
9

homoclinic solution of (1.0.8) in terms of a homoclinic solution of a second order equation.
If θ = 0, it is clear that the equilibrium state b for the homoclinic solution of (1.0.8) can
not equal the minimum b− of the potential W . To this end in Chapter 2 we introduce a
parameter which shifts the minimum of the potential W to b, as well as a parameter which
tilts the well potential G, yielding a new double well with one minimum at b and the other
one near b+ . The tilted potential G is generically not an equal-depth well, indeed

G(b) = G (b) = 0,

(1.1.1)

while G (b+ ) = 0 and G (b+ ) < 0. Consequently there exists another point φmax < b such
that
G(φmax ) = 0,

G (φmax ) < 0,

(1.1.2)

as is depicted in Fig 1.3.
The functionalization parameter, η > 0, reflects the balance of the bending energy of
˜
the interface against the surface energy. To study the breaking we scale it as η = δ 2 η. Our
˜
analysis is performed in a neighborhood of η = 0, b = b− which we summarize as our scaling
assumption
(S) Fix η ∈ R and β < 0. Then our standard scaling is
η = ηδ 2 ,
˜

b = b− +δ 2 β,

for 0 < δ

1.

(S)− (1.1.3)

We prove the existence of the homoclinic solution under the scaling assumption (S)-(1.1.3)
using the contraction mapping in Chapter 2. The dynamical system method presented in
Chapter 4 yields a less concise characterization of the homoclinic solution and is presented
10

W
G

Tilt
δα

b+
b

b−
Sh ift
δ 2β

Figure 1.3: The tilted-shifted potential G.

under a less degenerate scaling assumption (S’)
(S’) Fix η , β such that − min {µ± } < η < 0 and β < 0. Our scaling in order to apply Lin’s
˜
˜
method is

b = b− + δ 2 β,

for 0 < δ

1.

(S’)− (1.1.4)

11

φm
δ 1 /2

b+

b

Figure 1.4: The homoclinic pulse profile of (1.1.6).

We construct the homoclinic solution in the space defined as

Xb := u u − b ∈ H 4 (R) .

(1.1.5)

The contraction mapping approach seeks to construct the homoclinic solution Φm of (1.0.8)
in the neighborhood of φm , the homoclinic solution of the second-order differential equation

φm = G (φm ),

12

(1.1.6)

which resides in Xb . Under the scaling (S)-(1.1.3) G = G(u; δ) with G(u; δ = 0) = W (u).
The difficulty in setting up the contraction mapping argument is that the linearization of
the system (1.0.8) about φm is degenerate. To understand this degeneracy we first consider
θ = 0 in (1.0.8)
2
∂z − W (u) + η
˜

2
∂z u − W (u) = 0.

(1.1.7)

We denote by φh the heteroclinic solution of

φh = W (φh ),

(1.1.8)

which connects the two minima b± of W and satisfies
lim φh (z) = b− ,

z→−∞

lim φ (z)
z→∞ h

= b+ ,

(1.1.9)

and φh (0) = 0. Linearizing the differential equation (1.1.7) around φh we obtain

Lh := (Lh + η ) Lh ,
˜

(1.1.10)

2
Lh := ∂z − W (φh ).

(1.1.11)

where

From (1.1.8) we can see that the operator Lh has a kernel spanned by φh and hence Lh also
has it as its kernel.
The saddle-saddle connection is broken when the well depths are unequal, W (b− ) =
W (b+ ), breaking the heteroclinic chain connecting between b− and b+ into a homoclinic
orbit and introducing a second small eigenvalue in the associated linearization. This hidden13

symmetry eigenvalue makes the bifurcation of the homoclinic orbit from the heteroclinic
chain a degenerate problem. Removing this degeneracy is the main effort of the contraction
mapping construction. In particular we show that this eigenvalue + eigenvector pair has
√
a connection to the Modica-Mortola W function which plays a fundamental role in Γ√
convergence analysis of the Cahn-Hilliard energy [8, 43]. Indeed we show that the W is
the optimal shape of the perturbed potential well to detune the degeneracy associated with
the bifurcation of the homoclinic orbit from the heteroclinic one.
A key step of removing the degeneracy is the introduction of the tilted potential well

G(u; α, b) = G0 (u; b) − δαg(u; b),

(1.1.12)

where, as motivated in Chapter 2,

η
˜
G0 := W (u) − W (b) − W (b)(u − b) − (u − b)2 ,
4

(1.1.13)

corresponds to the horizontal shift of the potential to absorb the mass constraint. The tilting
function

u

g(u; b) :=

Ws (t; b) dt,

(1.1.14)

b

removes the degeneracy of the heteroclinic to homoclinic bifurcation. Here Ws (u; b) = W (u−
δ 2 β) is a shifting of the double well.
A reasonable choice for g follows from an analysis of the linearized operator Lα obtained
by linearizing (1.1.6) about φm ,

2
Lα = ∂z − G (φm ; α).

14

(1.1.15)

The point spectrum and associated eigenfunctions of Lα play an important role in contraction
mapping argument. In particular Lα has a translated eigenvalue at the origin and an O(δ)
positive ground-state eigenvalue λ0 . An appropriate choice of g will yield a second order
equation (1.1.6) whose homoclinic solution is the correct bi-layer Ansatz to O(δ 2 ). Removing
√
the O(δ) term from the perturbation expansion require Lα g (φm ) = O(δ). The W choice
for tilting perturbation in (1.1.14) is optimal since it renders

W (φm ) = ψ0 + O(δ),

(1.1.16)

where ψ0 is the ground state eigenfunction of operator Lα corresponding to the eigenvalue
λ0 , see Fig 1.5.

σ(Lα )

λ0 = O(δ)
ψ0 = W (φm) + O(δ)

λ1 = 0
ψ1 = φm

Figure 1.5: The illustration of spectrum of Lα . The colored strip denotes the essential
spectrum with begins at −W (b) < 0.

15

We also demonstrate that Lin’s method applies to construct the homoclinic orbit. We
adapt Lin’s method [44, 61], and rewrite (1.0.8) as a one-parameter vector field,

x = f (x, θ),
˙

(1.1.17)

where x = (u, u , u , u )T and f : R4 × R → R4 is smooth and
θ = W (b)(W (b) − η ).
˜

(1.1.18)

As in the case of the contraction mapping construction, the application of Lin’s method
requires unfolding a degeneracy. The classical application of Lin’s method requires that
the leading eigenvalues of Dx f (x, θ)|(p

1,2 ,0)

, where p1,2 are the equilibria connected by the

heteroclinic orbit, be semi-simple. In the case at hand the matrix eigenvalues form a Jordan
block for the unperturbed, δ = 0, case; moreover for δ = 0 the Jordan block unfolds smoothly
in δ forming real eigenvalues which perturb at O(δ 2 ). This co-dimension two situation
greatly complicates our primary goal: to determine when the unfolding of the heteroclinic
orbit coincides with an orbit-flip bifurcation. The contraction mapping construction realizes
the leading order structure of the homoclinic as the solution of an associated second-order
system. In an orbit-flip bifurcation the homoclinic solution of a fourth-order system converges
to its equilibria along the fast stable or fast unstable directions within the stable or unstable
manifolds respectively, see Fig 1.6. The degeneracy in our system makes the analysis of the
orbit flip bifurcation particularly technical – at δ = 0 the stable and unstable manifolds have a
single eigenvector which perturbs into two distinct eigenvectors for δ = 0. We must determine
the projection of the homoclinic onto the fast and slow eigenvectors of the stable/unstable

16

manifolds as they bifurcation away from each other.
δ = 0, η > 0

δ=0

δ = 0, η < 0

Figure 1.6: Depiction of the stable manifold of the equilibrium of a homoclinic orbit under an
orbit flip bifurcation. For η = 0 the homoclinic orbit (red line) converges to the equilibrium
along the fast stable direction, for η > 0 it converges to its equilibrium on one side, and flips
the side for η < 0.

Our primary results, stated below, show the existence of a homoclinic orbit that bifurcates
out of the heteroclinic orbit. In the contraction mapping approach we show that, up to a codimension one condition, characterized in terms of a Melnikov-type integral, the bifurcating
homoclinic orbit can be well described by a homoclinic solution of an associated secondorder dynamical system. Our primary conjecture is that this co-dimension one condition is
equivalent to the orbit-flip bifurcation in the dynamical systems construction. Indeed, it is at
an orbit-flip bifurcation, when the decay rate of the homoclinic switches from the fast-stable
to the slow-stable rate, that a fourth dynamical system is most unlike a second order one.
The full analysis of the orbit-flip bifurcation is outside the scope of this thesis, instead we
use Lin’s method to construct the homoclinic orbit in the non-degenerate case (S’)-(1.1.4)
for which η is fixed at a non-zero value, yielding a heteroclinic connection at δ = 0 that has
˜
simple matrix eigenvalues.
17

The main theorem of this paper is
Theorem 1.1. Let the equal-depth double well W satisfying (H1) be given and let φh denote
the heteroclinic solution of (1.1.8). Let η , β be given by the scaling (S)-(1.1.3) and η, β
˜
satisfy
(H2 )

Ah β + Ah η > ν δ ω ,
2
1

(1.1.19)

for some ν > 0 independent of δ where ω > 0 defined in (3.2.42) only depends on W . The
constants Ah and Ah depend only upon the heteroclinic orbit φh ,
2
1
9 5
2
Ah := − µ+ (b+ − b− ) + 3 W (φh )(φh − b− ), (φh )2 ,
1
2
2

(1.1.20)

Ah :=
2

(1.1.21)

W (φh )(φh − b− ), (φh )2

2

.

Then there exists a solution Φm ∈ Xb of (1.0.8) which is homoclinic to b, admits the following
expansion
Φm = φm (z; α∗ (δ)) + O(δ 2 ),

(1.1.22)

in H 4 where φm ∈ Xb is the corresponding solution of the second-order differential equation
(1.1.6), with α∗ given by the smooth function α∗ = α∗ (δ; β, η) which satisfies

α∗ (δ; β, η) :=

−

3
2
µ+ (b+

√

− b− )β

2 g(b+ )

√
+ O( δ).

(1.1.23)

Remark 1.1. In the neighborhood to the left of b− depicted in Fig 1.8 there is a “bad” ray
which is precisely excluded by condition (H2). Theorem 1.1 holds within the boxed domain
except on this ray.

18
















‘bad’ ray
δ2















η
˜

δ2

b−

b

Figure 1.7: The illustration of the scaling (S)-(1.1.3) and the domain of the nondegeneracy in
Theorem 1.1 in the left Figure and the illustration of the domain of applicability (S’)-(1.1.4)
of Theorem 1.2 in the right Figure.

We also apply Lin’s method to construct the homoclinic orbit under the scaling (S’)(1.1.4). Since b = b− + βδ 2 , W (b− ) = W (b− ) = 0 and µ− = W (b− ) > 0 we observe
that

θ = W (b− ) W (b− ) − η βδ 2 + O(δ 3 ),
˜
= µ− (µ− − η )βδ 2 + O(δ 3 ).
˜

(1.1.24)

For the case θ = 0, the vector system (1.1.17) has two hyperbolic equilibria p1 := (b+ , 0, 0, 0)T
and p2 := (b− , 0, 0, 0)T satisfying f (pi ; θ = 0) = 0. For θ = 0, (1.1.17) has two symmetric
heteroclinic connections q1 (z), q2 (z) between the bi-saddle hyperbolic equilibrium points p1

19

η
˜
b−
δ2





























− min{µ± }

b

O(1)

Figure 1.8: The illustration of the scaling (S)-(1.1.3) and the domain of the nondegeneracy in
Theorem 1.1 in the left Figure and the illustration of the domain of applicability (S’)-(1.1.4)
of Theorem 1.2 in the right Figure.

and p2 , see Figure 1.9

lim q1 (z) = p1 ,

lim q (z)
z→∞ 1

= p2 ,

(1.1.25)

lim q2 (z) = p2 ,

lim q (z)
z→∞ 2

= p1 .

(1.1.26)

z→−∞

z→−∞

The system (1.1.17) is reversible, that is symmetric under the transformation z → −z
and has two hyperbolic equilibria p1 and p2 at θ = 0.
For δ = 0, then θ = 0 and the spectrum of Dx f (x; δ) (p ;0) i = 1, 2 is given by
i
√
σ(Dx f (x; δ) (p ;0) ) = {± µ− },
1
√
σ(Dx f (x; δ) (p ;0) ) = {± µ+ }.
2

20

(1.1.27)
(1.1.28)

q1(z)

p2

p1

q2(z)
Figure 1.9: heteroclinic connection θ = 0.

with µ± > 0. Moreover, they are not the simple eigenvalues, having a 2-dimensional Jordan
block structure, which does not satisfies the hypothesis of Lin’s method, [44, 61], that the
leading eigenvalues be semisimple. For the Jordan block case, the δ = 0 structure does not
readily contain the full information. In this thesis we apply Lin’s method to construct the
homoclinic orbit in the non-degenerate case (S’)-(1.1.4) for which η is fixed at a non-zero
˜
value, yielding a heteroclinic connection at δ = 0 that has simple matrix eigenvalues. For
δ = 0, then θ = 0 and the spectrum of Dx f (x; δ) (p ;0) is given by
i
√
σ(Dx f (x; δ) (p ;0) ) = {± µ− , ±
1
√
σ(Dx f (x; δ) (p ;0) ) = {± µ+ , ±
2

µ− − η },
˜

(1.1.29)

µ+ − η }.
˜

(1.1.30)

which satisfies the hypothesis of Lin’s method.
We construct the section planes Σi , i = 1, 2, which are transverse to qi (z) at some
point, say(without loss of generality) at qi (0), see Fig 1.13. For θ = 0, the stable and
unstable manifolds W s (pi ) and W u (pi ), i = 1, 2 for our system (1.1.17) are two-dimensional.
21

σ(Dxf (p1; 0))

√
− µ−

√

µ−

Figure 1.10: Under the assumption (S)-(1.1.3) the illustration of the eigenvalue of
Dx f (x; δ) (p ;0) for δ = 0.
1

Moreover, our system (1.1.17) admits that the intersection between of the stable and unstable
manifolds along the heteroclinic connection is of dimension one,

Tq (0) W u (p1 ) ∩ Tq (0) W s (p2 ) = span{q1 (0)},
˙
1
2

(1.1.31)

Tq (0) W u (p2 ) ∩ Tq (0) W s (p1 ) = span{q2 (0)}.
˙
2
1

(1.1.32)

where Tq M denotes the tangent space of the manifold M at q. We introduce the subspace
Zi , i = 1, 2 such that

R4 = Z1 ⊕ Tq (0) W u (p1 ) + Tq (0) W s (p2 ) ,
1
1

(1.1.33)

R4 = Z2 ⊕ Tq (0) W u (p2 ) + Tq (0) W s (p1 ) .
2
2

(1.1.34)

Under the assumption (S’)-(1.1.4) we show that the heteroclinic connecting orbit Γ1 :=
{q1 (z)|z ∈ R} does not lie in the strong stable manifold of p2 or the strong unstable manifold
of p1 . This condition excludes the orbit flip for Γ1 , [61]. Due to the reversibility property of
our system (1.1.17) it also excludes the orbit flip for Γ2 := {q2 (z)|z ∈ R}.
22

σ(Dxf (p1; δ)), 0 < δ

O(δ 2)

√
− µ− −√µ− − η
˜

1

O(δ 2)

√

µ− − η
˜

√

µ−

Figure 1.11: Under the assumption (S)-(1.1.3) the illustration of the eigenvalue of
Dx f (x; δ)|(p ;δ) , 0 < δ
1 for δ = 0.
1

Now we are ready to apply Lin’s method [44, 61, 37] to demonstrate the bifurcation of
the homoclinic solution from the heteroclinic connection for θ = 0. To start we construct
±
the perturbed heteroclinic connection qi (z), which converges to p1,2 as z → ±∞ and has

a possible discontinuity jump in Z1,2 , see Figure 1.12. The discontinuity, called the jump
∞
∞
ξi = ξi (θ), depends on the parameter θ and gives a bifurcation equation for the existence of

such perturbed connections for θ = 0. The solvability condition for the bifurcation equation
is
ψi (s) Dθ f (qi (s), 0) ds = 0,

i = 1, 2,

(1.1.35)

R

where ψi spans the subspace Zi . We show in Lemma 4.1 that this condition (1.1.35) is
satisfied for our system. The second step of Lin’s method is to construct the ‘Lin’s orbits’.
These piecewise continuous orbits x± (z) are solutions to the system (1.1.17) and lie in a
i
±
neighborhood of the perturbed heteroclinic orbits qi (z) and these orbits have the prescribed

flying times 2ω2 from Σ1 to Σ2 . Moreover, they also satisfy

23

Σ1

−
q1 (z)

Z1

+
q1 (z)

q1 (z)

p2

p1

Figure 1.12: Perturbed heteroclinic connection for Γ1 .

• x± (0) ∈ Σi , i = 1, 2,
i
• x+ (0) − x− (0) ∈ Zi , i = 1, 2,
i
i
• x− (−∞) = x+ (∞) and x+ (ω2 ) = x− (−ω2 ),
1
2
1
2

24

see Fig 1.13. We prove the existence and uniqueness of those ‘Lin’s orbits’ x± (θ, ω2 )(z) and
i
derive an expression for the jump

ξi (θ, ω2 ) := < ψi , x+ (θ, ω2 )(0) − x− (θ, ω2 )(0) >,
i
i
ω

∞
= ξi (θ) + ξi 2 (θ),

i = 1, 2.

(1.1.36)

To obtain the homoclinic orbit, we require the jumps to be zero, i.e., ξ1 (θ, ω2 ) = 0 which by
the symmetry property of the system (1.1.17) also implies ξ2 = 0. We also derive the leading
order term of ξ1 (ω2 , θ)
u

u

ξ1 (ω2 , θ) = M1 θ + cu (θ)e−2ω2 λ2 (θ) + o(e−2ω2 λ2 (θ) ),

where λu (θ) =
2

√

(1.1.37)

µ+ and the function cu (·) is smooth and cu (0) = 0. The following theorem,

which is the main result of our application of Lin’s method, is a consequence of the implicit
function theorem.
Theorem 1.2. Let η , δ and double well W be given and satisfy (H1) and (S’)-(1.1.4). Then
˜
there exists δ0 > 0 such that for all δ ∈ (0, δ0 ) there exists a homoclinic solution of (1.0.8)
denoted by Φm which is homoclinic to b where b satisfies (1.0.9).
Remark 1.2. Both Theorems 1.1 and 1.2 give the existence of a homoclinic solution Φm
of (1.0.8). Theorem 1.1 provides for a sharp characterization of the homoclinic solution
Φm in terms of a homoclinic solution of a corresponding second-order problem (1.1.6). On
the other hand, Theorem 1.2 gives a perturbation expansion of the homoclinic orbit but does
not classify the leading order expression in terms of a lower-order problem. The homoclinic
solution is only shown to exist in the neighborhood of the heteroclinic solution that connects
25

Σ1
q1

x−
1

x+
1
ω2
p1

p2
ω2

Σ2
x−
2
q2

x+
2
Figure 1.13: Lin’s orbit.

the two minimums of W . While Theorem 1.1 requires the additional condition (H2) and the
restriction b > b− , Theorem 1.2 does not permit η to scale with δ.
˜

26

Chapter 2
Functional Analytical Method
In this chapter, we use a contraction mapping argument to construct a homoclinic solution
Φm of (1.0.7). A key issue is that the natural contraction mapping arising from Newton’s
method is poorly scaled; we identify a tuning parameter corresponding to a tilting of the
double well, see (1.1.14) and (2.2.3), which conditions the mapping.

2.1

Introduction

We are interested in the critical point of (1.0.4) which is also a homoclinic solution and
satisfies
Π0

δF
= 0,
δu

(2.1.1)

where Π0 is the zero mass projection. Observing that the kernel of Π0 is comprised of spatial
constants, so if the shifted energy H satifies
δH
δF
=
+ c,
δu
δu

27

(2.1.2)

for some constant c, then u will satisfy (2.1.1) exactly when it satisfies (2.1.2). In the sequel
we look for the homoclinic solution of the reformulated problem

Π0

δH
δF
=
= 0.
δu
δu

(2.1.3)

As a first step we rewrite the energy (1.0.2). Integrating by parts we find

1 4
ε (∆u)2 + ε2 f1 (u)| u|2 + f2 (u) dx,
2

F(u) =

(2.1.4)

Ω

where we have introduced the functions

η
˜
f1 (u) = W (u) − ,
2
1
f2 (u) =
(W (u))2 − η W (u).
˜
2

(2.1.5)
(2.1.6)

We modify the potential f2 to have a double zero at u = b,

f3 (u) = f2 (u) − f2 (b)(u − b) − f2 (b).

(2.1.7)

From (2.1.7) it is easy to see that for our case c = −f2 (b). Define the shifted energy
H(u) =

1 4
ε (∆u)2 + ε2 f1 (u)| u|2 + f3 (u) dx,
2
Ω

(2.1.8)

whose variational derivative differs from that of F by a constant. We rewrite the shifted

28

energy in terms of the potential
s2

u

G0 (u) ≡

η
˜
f1 (s1 ) ds1 ds2 = WT (u; b) − (u − b)2 ,
4

(2.1.9)

b s1 =b

where WT (u; b) = W (u) − W (b) − W (b)(u − b) is the result of shifting the double zero of W
at b− to b by subtracting the Taylor polynomial. Integrating by parts on f1 in (2.1.8) and
completing the square we obtain

H(u) =

2
1 2
ε ∆u − G0 (u) + p(u) dx,
Ω2

(2.1.10)

2
where p(u) = f3 (u) − 1 G0 (u) . Analysis of the potential p performed in [28] yields the
2

following expression
Lemma 2.1. The potential p takes the form

p(u) =

WT (u) − W (b)(u − b) W (b) − η WT (u) +
˜
+

η
˜
η
˜
WT (u) − (u − b) (u − b),
2
4

(2.1.11)

with a double zero at u = b. Under the scaling (S)-(1.1.3) then p(u) = δ 2 p2 (u) where
p20 (u)

p2 (u) = µ− (W (u) − W (b) − µ− (u − b)) β +
p21 (u)

+

1
W (u)(u − b) − W (u) + W (b) +O(δ),
2

(2.1.12)

in L∞ ([b, −b]), and µ− = W (b− ). Moreover, if uW (u) > 0, then both p20 and p21 are
non-positive on [b, −b], and zero at z = b.
29

2.2

Construction of Critical points by Contraction Mapping

Restricting to one space dimension and rescaling the transverse direction by ε the EulerLagrange equation associated to the shifted energy becomes

δH
2
(u) = (∂z − G0 (u))(uzz − G0 (u)) + p (u) = 0.
δu

(2.2.1)

Recalling the scaling (S)-(1.1.3), i.e., η = δ 2 η and b = b− + δ 2 β, (2.2.1) becomes
˜
2
F (u) := ∂z − G (u) − αδg (u)

uzz − G (u) − αδg (u) + p (u) = 0,

(2.2.2)

where we have introduced the tilted well

G(u) := G0 (u) − δαg(u),

(2.2.3)

g is defined in (1.1.14), and α is the δ-scaled parameter that tunes the shape of the potential.
We will show that for δ small enough, we can generate a solution of the full EulerLagrange equation in a small neighborhood of φm (z; α, b) via a modified Newton’s method.
Here φm (z; α, b) is the solution of the second order equation (1.1.6) associated to G. To this
end we define the Newton map

N (u) = u − L−1 (F (u)) ,
α

30

(2.2.4)

where the self-adjoint linear operator

Lα = Lα − δαg (φm )

2

+ αδG (φm )g (φm ) + δ 2 α2 g (φm )g (φm ) + p2 (φm ) , (2.2.5)

is the linearization of F about φm . Here Lα is defined in (1.1.15). Expanding Lα we may
write it as

Lα = L2 + αδ G (φm )g (φm ) − g (φm )Lα − Lα g (φm ) +
α
+δ 2 α2 g (φm )g (φm ) + α2 g (φm )2 + p2 (φm ) .

(2.2.6)

Spectrum analysis of Lα is performed in [28].
Lemma 2.2. Let g be as in (1.1.14) with δ

1. Then there exists ν0 > 0, independent of

δ, such that the spectrum of the linear operator Lα given in (1.1.15) consists of two point
eigenvalues
σp (Lα ) = {λ1 = 0, λ0 (α, b)},

(2.2.7)

and a remainder contained in (−∞, −ν0 ]. The ground state eigenvalue is given by the formula
ˆ
λ0 = δ λ0 + O δ 2

> 0,

(2.2.8)

where
4αγ0
ˆ
λ0 :=
.
φm 2
2

31

(2.2.9)

Here
√
1
γ0 := − g (φm )g(φm ), g (φm ) 2 = − 2g (φmax )g(φmax ) + O δ 2
=

1
√
µ+ g(b+ ) + O δ 2

> 0.

,
(2.2.10)

The corresponding normalized eigenfunctions take the form

ψ1 =
ψ0 =

φm
,
φm 2
g (φm )
+ O(δ),
g (φm ) 2

(2.2.11)
(2.2.12)

with the equality on ψ0 holding in L2 . Moreover we may relate g (φm ) to the translational
eigenvalue
1
1
g (φm ) = √ |φm | + O δ 2
2

,

(2.2.13)

in the L1 norm.
Proof. The operator Lα is a self-adjoint, 2nd-order, Sturn-Liouville operator and its spectrum is real. Taking the z derivative of (1.1.6) shows, from the translational symmetry, that
φm is in the kernel of Lα . This kernel has one node so the Sturn-Liouville theory implies
that the ground state eigenvalue, λ0 , must be positive. To develop an expansion for λ0 and
the ground state ψ0 > 0, from definition (2.2.3) of G we may rewrite the potential G as

G(s) = g (s)

2

− δαg(s) + δ 2 h(s),

32

(2.2.14)

where the higher order term

h(s) =
=

η
W (s) − W (b) − W (b)(s − b) − W (s − βδ 2 ) − (u − b)2
4
η
W (s) − µ− (s − b) β − (u − b)2 + O(δ),
4

(2.2.15)

has a double zero at s = b, is O(1) over the range of φm and is smooth. The definition
(1.1.14) of g implies that g is only C 1 with discontinuities in its second derivative at s = b,
b+ +βδ 2 . However g is piece-wise C ∞ , and is C ∞ on the range of φm , since φmax < b+ +βδ 2 .
Applying Lα to g (φm ) and using (1.1.6) and its first integral
1
2
φ
= G(φm ),
2 m

(2.2.16)

to eliminate derivative of φm , we obtain

Lα g (φm ) = 2g (φm )G(φm ) + g (φm )G (φm ) − G (φm )g (φm ).

(2.2.17)

Taking derivative of (2.2.14) with respect to s we may eliminate G for g and h and their
derivatives. The leading order terms cancel, yielding

Lα g (φm ) = −2δαg (φm )g(φm ) + δ 2 r(φm ),

(2.2.18)

where the second order term

r(s) = 2g h + g h − g h ,

33

(2.2.19)

is zero at s = b and hence r(φm ) is O(1) in

· L2 . Since the RHS of (2.2.18) is even, it is

orthogonal to the kernel of Lα and we may invert, yielding

g (φm ) = −

2δα g (φm )g(φm ) + δr(φm ), ψ0 2
ψ0 + δg ⊥ ,
λ0

(2.2.20)

in L2 , where g ⊥ ⊥ ψ0 is O(1). Taking the L2 norm of both sides of (2.2.20), we deduce that
g (φm ) 2 = O(1) since φm ranges from b to φmax with O(1) derivatives. From this we may
infer that λ0 = O(δ), which further yields the asymptotic expression

λ0 =

2δα g (φm )g(φm ), ψ0 2
+ O(δ 2 ).
g (φm ) 2

(2.2.21)

Dividing both sides of (2.2.20) by their L2 norms yields (2.2.12), and using this to substitute
for ψ0 in (2.2.21) yields (2.2.8). Using (2.2.14) and (2.2.16) we can obtain

g (φm )

2

1
= |φm |2 + δαg(φm ) − δ 2 h(s).
2

(2.2.22)

1
|φm |2 + δαg(φm ) − δ 2 h(s).
2

(2.2.23)

We may isolate g (φm ),

g (φm ) =

Expanding this expression yields (2.2.13), except for a neighborhood of z = 0 where φm (z) =
√
√
φ (0)z + O(z 3 ) where φm (0) = G (φmax ) = O( δ). These estimates yield the O( δ) error

34

bound in (2.2.13). To simplify the expression for γ0 we use (2.2.13) to write
1
1
γ0 = − √
g (φm )g(φm )|φm |dz + O δ 2 ,
2 R
∞
√
1
=
2
∂z g (φm ) g(φm )dz + O δ 2 ,
0

1
= −√
2

g (φm )

2

− 2g (φm )g(φm )

∞

.

(2.2.24)

0

√
However g (b) = 0 and g (φm (0)) = g (φmax ) = O( δ) so the first term is O(δ). Similarly,
g(b) = 0, and the expression reduces to
√
1
γ0 = − 2g (φm )g(b+ ) + O δ 2

.

(2.2.25)

Finally, from definition (1.1.14) of g we obtain

g (φmax ) =

W (φmax − βδ 2 )

W (φmax − βδ 2 )
1
µ+
= −
+ O δ2 .
2
2

(2.2.26)

From Lemma 2.2 we see that Lα has two small eigenvalues λ0 > 0 and λ1 = 0 with
positive, even groundstate ψ0 > 0 and ψ1 = φm / φm 2 . Since Lα is an O(δ), relatively
compact perturbation of the operator L2 , we deduce that it has two small eigenvalues,
α
which we denote by Λ0 = λ2 + O(δ) and Λ1 = O(δ). It is important to indentify the leading
0
order behavior in Λ0 , which comes from the perturbations, since λ2 = O(δ 2 ) is formally
0
lower order. We introduce the spectral projections used in the rest of the paper. For i = 0, 1

35

we define the spectral projections πi associated to the eigenvector ψi of Lα

πi f := (f, ψi )2 ψi ,

(2.2.27)

and their sum π = π0 + π1 and their complement πi = I − πi .
˜
Lemma 2.3. The operator Lα is self-adjoint, with real spectrum. Moreover there exists
ν > 0, independent of δ, such that

σp (Lα ) ∩ {|λ| < ν} = {Λ0 , Λ1 },

(2.2.28)

with associated eigenfunctions Ψ0 and Ψ1 ,

Ψ0 = ψ0 + O(δ),

(2.2.29)

Ψ1 = ψ1 + O(δ),

(2.2.30)

in L2 . Here ψ0 and ψ1 are the eigenfunctions of Lα corresponding to two small eigenvalues
λ0 and λ1 . Moreover

Λ0 = δ 2

√
p2 (φm ), (φm )2 2 + 6 2α2 g(φmax )(g (φmax ))2
ˆ
λ2 +
0
φm 2
2

5

+ O δ2

,

(2.2.31)

where α is the tuning parameter introduced in (2.2.3), p2 is defined in (2.1.12) and the scaling
ˆ
λ0 of λ0 is defined in (2.2.9).
Proof. The operator Lα is a relatively compact, O(δ) perturbation of L2 , and as such the
α
point spectrum of Lα is the square of that of Lα , up to O(δ). From lemma (2.2), L2 has
α
two small eigenvalues {0, λ2 }, with the rest of its spectrum an O(1) distance from the origin,
0
36

hence we may find a ν > 0 which verifies (2.2.28). We obtain formulas for the eigenvalues via
a regular perturbation expansion. We expand the operator Lα = L2 + δL1 + δ 2 L2 + O(δ 3 )
α
where

L1 = α G (φm )g (φm ) − g (φm )L − Lg (φm ) ,
L2 =

α2 g (φm )g (φm ) + α2 g (φm )2 + p2 (φm ) .

(2.2.32)
(2.2.33)

Similarly the eigenfunction and and eigenvalue have expansions

Ψ0 = ψ0 + δΨ0,1 + δ 2 Ψ0,2 + O(δ 3 ),

(2.2.34)

Λ0 = λ2 + δΛ0,1 + δ 2 Λ0,2 + O(δ 3 ).
0

(2.2.35)

In light of these expansions, the eigenvalue equation

Lα Ψ0 = Λ0 Ψ0 ,

(2.2.36)

becomes

(L2 + δL1 + δ 2 L2 )(ψ0 + δΨ0,1 + δ 2 Ψ0,2 )
= (λ2 + δΛ0,1 + δ 2 Λ0,2 )(ψ0 + δΨ0,1 + δ 2 Ψ0,2 ) + O(δ 3 ).
0
(2.2.37)

From the relation L2 ψ0 = λ2 ψ0 , the O(1) terms cancel and at O(δ) we find,
0

(L2 − λ2 )Ψ0,1 = Λ0,1 ψ0 − L1 ψ0 .
0
37

(2.2.38)

Focusing on the second term, L1 ψ0 , in the right hand side of (2.2.38), by the definition of
L1 (2.2.32) we have
L1 ψ0 = α G (φm )g (φm )ψ0 − Lg (φm )ψ0 + O(δ).

(2.2.39)

From (2.2.12) and (2.2.22) we have the expansion

1
2
G (φm ) g (φm ) + O(δ),
g (φm ) 2
1
2
G (φm ) φm + O(δ).
=
2 g (φm ) 2

G (φm )g (φm )ψ0 =

(2.2.40)

Applying (3.1.4) from lemma 3.2 we further simplify this expression

G (φm )g (φm )ψ0 =

1
Lφ + O(δ).
2 g (φm ) 2 m

(2.2.41)

This identity permits us to factor the operator L out of L1 ψ0 , yielding the expression
L1 ψ0 = αL

1
φ − g (φm )ψ0
2 g (φm ) 2 m

+ O(δ).

(2.2.42)

However from (2.2.12) and (3.1.25) we find a leading-order cancelation yielding the result
L1 ψ0 = O(δ), and hence (2.2.38) reduces to
(L2 − λ2 )Ψ0,1 = Λ0,1 ψ0 .
0

(2.2.43)

The operator L2 − λ2 has kernel spanned by ψ0 and an O(δ 2 ) eigenspace spanned by ψ1 .
0
However by parity considerations the right-hand side of (2.2.43) is orthogonal to ψ1 and the

38

solvability of (2.2.38) requires that,
Λ0,1 = 0.

(2.2.44)

Since we require Ψ0,1 ⊥ ψ0 = 0, we deduce that Ψ0,1 = 0 and (2.2.37) simplifies to
(L2 + δL1 + δ 2 L2 )(ψ0 + δ 2 Ψ0,2 ) = (λ2 + δ 2 Λ0,2 )(ψ0 + δ 2 Ψ0,2 ) + O(δ 3 ).
0

(2.2.45)

Collecting the O(δ 2 ) terms in (2.2.45), and recalling L1 ψ0 = O(δ) we obtain
1
(L2 − λ2 )Ψ0,2 = − L1 ψ0 − L2 ψ0 + Λ0,2 ψ0 .
0
δ

(2.2.46)

Again, the right-hand side is even about z = 0 and hence orthogonal to ψ1 , we choose Λ0,2
to make it orthogonal to ψ0 , obtaining

Λ0,2 =

1L ψ
δ 1 0

+ L2 ψ0 , ψ0

||ψ0 ||2
2

2

1
= (L1 ψ0 , ψ0 )2 + (L2 ψ0 , ψ0 )2 ,
δ

(2.2.47)

where the second equality follows since ψ0 2 = 1.
We analyze the first term of (2.2.47), substituting the expression (2.2.32) for L1 obtaining
the equality

(L1 ψ0 , ψ0 )2 = α (G (φm )g (φm )ψ0 , ψ0 )2 − (g (φm )Lψ0 , ψ0 )2 − (Lg (φm )ψ0 , ψ0 )2 .
(2.2.48)
Since L is self-adjoint and Lψ0 = λ0 ψ0 , we have the reduction

(L1 ψ0 , ψ0 )2 = α G (φm )g (φm )ψ0 , ψ0 2 − 2αλ0 (g (φm )ψ0 , ψ0 )2 .

39

(2.2.49)

From (3.1.22) we obtain an expression for ψ0 which we substitute into (2.2.49), yielding

(L1 ψ0 , ψ0 )2 =

α
ρ2
g



T1

 1

G (φm )g (φm )(φm )2 dz +

 R2
T2

√

+
R

2G (φm )g (φm )|φm |

√

T3

−

δ(h1 − h2 ) − δh3 dz +



3

ˆ
δ λ0 g (φm )(φm )2 dz  + O δ 2

R

.

(2.2.50)

We examine the terms T1 , T2 and T3 in the integral above one-by-one. Using (3.1.4) to
eliminate (φm )2 in T1 we obtain

T1 =

1
Lφm , g (φm ) 2 .
2

(2.2.51)

Using (3.1.9) and (2.2.9) to eliminate g (φm ) we obtain the reduction,

T1 =

δ
ˆ
ρg λ0 (ψ0 , φm )2 − 2α(g (φm )g(φm ), π φm )2 + O(δ 2 ).
˜
2

(2.2.52)

From (3.1.6) we see that the first inner product in (2.2.52) is O(δ 2 ). For the second inner
product we observe that π φm = φm + O(δ) and hence
˜

T1 = −δα g (φm )g(φm ), φm 2 + O(δ 2 ).

40

Applying (3.1.28) from the appendix we obtain,

T1 = δα

√
1
||φm ||2 − 2g(φmax )(g (φmax ))2 + O(δ 2 ).
2
4

(2.2.53)

Addressing the second term on the right-hand side of equality (2.2.50), we split it into three
parts
T21

√

T2 =
R

2δG (φm )g (φm )|φm |h1 dz +
T22

−

√
R

2δG (φm )g (φm )|φm |h2 dz +

R

√
δ 2G (φm )g (φm )|φm |h3 dz .
(2.2.54)

Addressing T21 term of (2.2.50), we substitute the expression (2.2.13) for g (φm ) to obtain

T21 =

√

G (φm )(φm )2 h1 dz + O(δ 2 ).

δ

(2.2.55)

R

From (3.1.13) we eliminate |φm |h1 to obtain
δα
T21 = √
G (φm )|φm |g(φm ) dz + O(δ 2 ).
2 R

(2.2.56)

Since φm is even and φm is odd, we may break the integral into twice the half-line value
√
T21 = −δα 2

∞
0

G (φm )φm g(φm ) dz + O(δ).

41

(2.2.57)

From (1.1.6) we deduce that φm = G (φm )φm , , integrating by parts we find
∞
√
1
T21 = δα 2 G (φm (0))g(φm (0)) − √
φm φm dz + O(δ 2 ),
2 0
√
1
= δα 2 G (φm (0))g(φm (0)) + √ ||φm ||2 + O(δ 2 ).
2
2 2

(2.2.58)

Using (3.1.29) we can re-write the equality as
√
1
T21 = δα(2 2g(φm (0))(g (φm (0)))2 + ||φm ||2 ) + O(δ 2 ).
2
2

(2.2.59)

Turning to T22 and using (3.1.14) on T22 we find

G (φm )(φm )2

T22 =

√

δh2 + δh3 dz + O(δ 2 ).

(2.2.60)

R

However from (3.1.4) and the self-adjointness of L we obtain

T22 =

φm L

√
δh2 + δh3 dz + O(δ 2 ).

(2.2.61)

R

From (3.1.11) we obtain that

3

δφm Lh3 dz + O(δ 2 ).

T22 =

(2.2.62)

R

From (3.1.12) we deduce that

Lh3 = −2α˜ g (φm )g(φm ) + O(δ),
π

42

(2.2.63)

and hence

T22 = −2δα
= −2δα

3

˜
φm π g (φm )g(φm ) dz + O(δ 2 ),
R

3

φm g (φm )g(φm ) dz + O(δ 2 )

(2.2.64)

R

where we again used that π φm = φm + O(δ). Finally, applying (3.1.28) yields,
˜

T22 = δα

1
||φm ||2 − 2c3
2
2

3

+ O(δ 2 ).

(2.2.65)

Finally, returning to (2.2.50) we consider the third term and using (3.1.8) show that

ˆ
T3 = δ λ0

g (φm )φm dz
R

=

ˆ
3
δ λ0
√
|φm |φm dz + O δ 2
2 R

3

= O δ2

.

(2.2.66)

Combining the expressions (2.2.53), (2.2.66), (2.2.59) and (2.2.65) for T1 , T21 , T22 , T3 we
obtain
1
α2
(L1 ψ0 , ψ0 )2 =
δ
φm 2
2

√
1
1
φm 2 + 6 2g(φmax )(g (φmax ))2 + O δ 2
2
2

.

(2.2.67)

Turning our attention to the second term of Λ0,2 in (2.2.47) we use the definition (2.2.33) of
L2 to write this term as
2
α2 g (φm )g (φm ) + α2 (g (φm ))2 + p2 (φm ) ψ0 dz,

(L2 ψ0 , ψ0 ) =
R

=

1
φm 2 R
2

1

α2 g (φm )g (φm ) + α2 (g (φm ))2 + p2 (φm ) (φm )2 dz + O δ 2

,

(2.2.68)
43

where the second equality comes from the relation ψ0 =

|φm |
φm 2

1

+ O(δ 2 ) which is a conse-

quence of (2.2.12) and (2.2.13). Integrating by parts on the second term in the integrand we
obtain

(L2 ψ0 , ψ0 ) =

1
φm 2 R
2

1

−α2 g (φm )g (φm )φm + p2 (φm )(φm )2 dz + O δ 2

. (2.2.69)

Finally, applying (3.1.25) to the first term in the integrand we obtain,
1
1
p2 (φm ), (φm )2 2 − 2 α2 φm 2
2
(L2 ψ0 , ψ0 ) =
+ O δ2
φm 2
2

.

(2.2.70)

Combining the equalities (2.2.67) and (2.2.70) we simplify (2.2.47) to
√
1
p2 (φm ), (φm )2 2 + 6 2α2 g(φmax )(g (φmax ))2
Λ0,2 =
+ O δ2
2
φm 2

.

(2.2.71)

Inserting this expression into (2.2.35) and recalling the scaling (2.2.9) of λ0 , we obtain
(2.2.31).

2.3

Conditioning of the Newton map

As a first step towards understanding the conditioning of the Newton map, we Introduce the
spectral projections Π0 (α), Π1 (α) associated to the operator Lα
Π0 (α) u := (u, Ψ0 (·, α))2 Ψ0 ,

(2.3.1)

Π1 (α) u := (u, Ψ1 (·, α))2 Ψ1 ,

(2.3.2)

44

where Ψ0 , Ψ1 are the two small eigenvalues associated to the operator Lα . Their sum is
defined
Π(α) := Π1 (α) + Π2 (α).

(2.3.3)

˜
Π0 (α) := I − Π0 (α),

(2.3.4)

˜
Π1 (α) := I − Π1 (α).

(2.3.5)

˜
˜
˜
Π(α) := Π0 (α) + Π1 (α).

(2.3.6)

Their complements are

Their sum is defined

Because of the small eigenvalues Λ0 and Λ1 of the linearization Lα the Newton map (2.2.4)
˜
is ill-conditioned. We also observe that Π(α)L−1 : L2 → H 4 is uniformly bounded. We may
α
“remove” Λ1 by restricting Lα to act upon even functions. For the second small eigenvalue
Λ0 in order to condition the Newton map we may tune α = α(δ; β, η) to eliminate Λ0 for
fixed β and η. Here we recall the scaling b = b− + δ 2 β, η = δ 2 η, and p = δ 2 p2 and determine
˜
α = α(δ; β, η) for which the homoclinic solution φ∗ = φ∗ (z; α, β, η) of (1.1.6) satisfies
m
m

(F (φ∗ ), Ψ0 (·, α))2 = 0.
m

(2.3.7)

Lemma 2.4. For b, η given in (S)-(1.1.3) with β < 0, there exists a C 1 function α∗ =
˜
α∗ (δ; β, η) such that φ∗ := φm (·, α∗ ), the homoclinic solution of (1.1.6) satisfies
m
(F (φ∗ ), Ψ0 (·, α∗ ))2 = 0,
m

45

(2.3.8)

where Ψ0 is the eigenfunction corresponding to the smallest eigenvalue of Lα . Moreover, the
function α∗ enjoys the asymptotic expansion
α∗ =

√
µ2 (b+ − b− )β
− + √
+ O( δ).
2γ0

(2.3.9)

Remark 2.1. The form of the leading order term of α∗ requires that β be negative. Although
α∗ depends on β and η, it follows from (2.3.9) that the leading order expression of α∗ depends
only on β, indeed for fix β > 0
√
∂α∗
= O( δ)
∂η

1.

(2.3.10)

In the sequel, quantities denoted with “*” are evaluated at α = α∗ (δ; β, η). We will subsequently have a separate function, α = α(u), which will be unadorned.
Proof. We fix β and η and introduce the function

h(α, δ; β, η) := (F (φm (·, α), Ψ0 (·, α)))2 .

(2.3.11)

For a given value of β and η we wish to find α∗ = α∗ (δ) such that
h(α∗ , δ) = 0.

(2.3.12)

Substituting u = φm into the expression (2.2.2) for F , recalling the definition of Lα and
using (1.1.6) we can rewrite h as

h(α, δ) = (L − δαg (φm ))(−δαg (φm )) + δ 2 p0 (φm ), Ψ0 (α)

46

2

.

(2.3.13)

We apply (2.2.18) and recall that Ψ0 (α) = ψ0 (α) + O(δ 2 ) to obtain

h(α, δ) =

δ 2 α2 g (φm )g (φm ) − δαLg (φm ) + δ 2 p2 (φm ), ψ0 (α)

2

+ O(δ 3 ),

= δ 2 α2 g (φm )g (φm ) + 2α2 g (φm )g(φm ) + p2 (φm ), ψ0 (α)

2

+ O(δ 3 ).
(2.3.14)

We apply (3.1.22) to rewrite ψ0
5
δ2
h(α, δ) = √
α2 g (φm )g (φm ) + 2α2 g (φm )g(φm ) + p2 (φm ), |φm | + O δ 2
2
2ρg
5
2δ 2 ˜
h(α) + O δ 2
= √
2ρg

,

,

(2.3.15)

where, since φm is even and φm < 0 for z > 0, we have introduced
˜
h1

˜
h(α, δ) = −α2

∞
0

˜
h2

g (φm )g (φm )φm dz −2α2

∞
0

g (φm )g(φm )φm dz +

˜
h3

−

∞
0

p2 (φm )φm dz .

(2.3.16)

Recalling φm (0) = φmax we obtain
1
1
1
˜
h1 =
(g (φm (∞)))2 − (g (φmax ))2 = O δ 2
2
2

.
(2.3.17)

47

˜
Integrating by parts on the second term h2 and recalling that φm (0) = φmax we have

˜
h2 = −4

∞
0

g (φm )g(φm )φm dz,

= 4g (φmax )g(φmax ) + 4

∞
0

g (φm )g (φm )φm dz.

(2.3.18)

˜
˜
The second integral of h2 equals to 4h1 and hence
1

˜
h2 = 4g (φmax )g(φmax ) + O δ 2

.

(2.3.19)

˜
Integrating by parts on h3 and recalling φm (0) = φmax yields
1

˜
h(α, δ) = 2p2 (φmax ) + 4α2 g (φmax )g(φmax ) + O δ 2

.

(2.3.20)

Therefore we can choose

α=

−

1
p2 (φmax )
+ O δ2
2g (φmax )g(φmax )

,

(2.3.21)

such that
˜
h(α, δ = 0) = 0.

(2.3.22)

By (3.1.3), (2.2.10) from Lemma 2.1 and 3.1, we can simplify the expression (2.3.21),

α=

1
µ2 (b+ − b− )β
− + √
+ O δ2
2γ0

48

.

(2.3.23)

Since γ0 > 0 and b+ − b− > 0, we choose β < 0 to render α is real. If we establish
˜
∂h
(α, δ = 0) =
∂α

µ2 (b+ − b− )β
− + √
> 0,
2γ0

(2.3.24)

and
µ2 (b+ − b− )β
− + √
= O(1),
2γ0

(2.3.25)

then by the implicit function theorem, there exists a C 1 function α∗ = α(δ) such that
˜
h(α∗ , δ) = 0, thereby completing the proof for this lemma. We first observe that
∞
∞
˜
∂h
(α, δ) = ∂α −2α2
g (φm )g (φm )φm dz − 4α2
g (φm )g(φm )dz +
∂α
0
0

−2

∞

0

p2 (φm )φm dz

1

.

+ O δ2

(2.3.26)

The first term of (2.3.26)

∂α −2α2

∞
0

g (φm )g (φm )φm dz

= −4α
−2α2

∞
0

g (φm )g (φm )φm dz +

∞
0

−2α2

0

−2α2

0

∞
∞

g (φm )g (φm )φm ∂α φm dz +
(g (φm ))2 φm ∂α φm dz +
g (φm )g (φm )∂α φm dz,
(2.3.27)

49

by (3.1.24) from appendix we observe that

∂α φm = − √

1

1

φm + O δ 2

ˆ
2λ0
1
1
∂α φm = − √
φm + O δ 2
ˆ
2λ0
in

,

(2.3.28)

,

(2.3.29)

· L2 norm so that (2.3.27) can be simplified
∂α

∞

2α2

0
∞

g (φm )g (φm )φm dz ,
√

2α2 ∞
g (φm )g (φm )φm dz +
g (φm )g (φm )φm
ˆ
λ0
0
0
√ 2
∞
∞
1
2α
g (φm )g (φm )φm + O δ 2 ,
= −2α(g (φm ))2 +
ˆ
λ0
0
0

= −4α

1

dz + O δ 2

1

= O(δ 2 ).

,

(2.3.30)

Addressing the second term of (2.3.26), we expand it as
∞

∂α −4α2
∞

= −8α
−4α2

0
∞
0

0

g (φm )g(φm )φm dz

g (φm )g(φm )φm dz − 4α2

∞
0

g (φm )g (φm )φm ∂α φm dz − 4α2

g (φm )g(φm )φm ∂α φm dz +
∞
0

g (φm )g(φm )∂α φm dz.
(2.3.31)

Integrating the first term of this expansion by parts and applying (2.3.29) to the remaining

50

three terms, we obtain the reduction

∂α −4α2

∞
0

g (φm )g(φm )φm dz
∞

= 8αg (φm (0))g(φm (0)) + 8α
g (φm )g (φm )φm dz +
0
√
1
2 2α2 ∞
+
g (φm )g(φm )φm dz + O δ 2
ˆ
λ0
0
1

= 8αg (φmax )g(φmax ) + O δ 2

.

(2.3.32)

The third term of (2.3.26) enjoys the expansion

∂α −

∞
0

p2 (φm )φm dz

= −2

∞
0

p2 (φm )φm ∂α φm dz − 2

∞
0

p2 (φm )∂α φm dz, (2.3.33)

and using (2.3.29) we find

∂α −

∞
0

p2 (φm )φm dz

=

√

∞

2

ˆ
λ0

= O

0
1
δ2

1

p2 (φm )φm dz + O δ 2
.

,
(2.3.34)

Combining the results of (2.3.30), (2.3.32) and (2.3.34), the expression (2.3.26) reduces to
˜
∂h
(α, δ) = 8αg (φmax )g(φmax ).
∂α

(2.3.35)

The relation (2.2.10) allows us to rewrite (2.3.35) as
˜
√
∂h
(α, δ = 0) = −4 2γ0 .
∂α
Since γ0 is O(1) and non-zero, the lemma follows.
51

(2.3.36)

We have established the existence of a C 1 function α∗ = α∗ (z; β, η) = O(1) such that
the homoclinic solution φ∗ (z) = φm (z; α∗ ) satisfies
m
(F (φ∗ ), Ψ0 (.; α∗ ))2 = 0.
m

(2.3.37)

From the definition (2.2.2) of F (u) we may expand F (φ∗ )
m

2
F (φ∗ ) = ∂z − G (φ∗ ) − αδg (φ∗ )
m
m
m

(φ∗ ) − G (φ∗ ) − αδg (φ∗ ) + p (φ∗ ). (2.3.38)
m
m
m
m

Recalling the definition (1.1.15) of Lα∗ and that φ∗ is the homoclinic solution of (1.1.6) we
m
may further deduce

2
F (φ∗ ) = α∗ δ 2 g (φ∗ )g (φ∗ ) − α∗ δLα∗ g (φ∗ ) + δ 2 p2 (φ∗ ).
m
m
m
m
m

(2.3.39)

From (2.2.18) of we see that

Lα∗ g (φ∗ ) = −2δα∗ g (φ∗ )g(φ∗ ) + δ 2 r(φ∗ ),
m
m
m
m
where r(φ∗ ) = O(1) in
m

(2.3.40)

· L2 . So the residual of F at φ∗ takes the form
m

2
F (φ∗ ) = δ 2 α∗ g (φ∗ )g (φ∗ ) + 2g (φ∗ )g(φ∗ ) + p2 (φ∗ ) + δ 3 r(φ∗ ).
m
m
m
m
m
m
m

(2.3.41)

Since φ∗ is even, and hence to Ψ1 we may introduce the function ξ∗ defined by
m
˜
ξ∗ := Π(α∗ )L−1 F (φ∗ ) = O(δ 2 ).
α∗
m

52

(2.3.42)

In particular φ∗ → ξ∗ is the first iteration of Newton map. We fix δ, η and β, then define
m
k
He (R) to be the subspace of H k comprised of even functions. And for ρ > 0 introduce

∗
Bρ =

5

4
u ∈ He (R) u − (φ∗ − ξ∗ ) H 4 ≤ ρδ 2
m

.

(2.3.43)

∗
For u ∈ Bρ we may decompose u as

u = φ∗ − ξ∗ + v0
m

(2.3.44)

u = φm (z; α) − ξ∗ + v,

(2.3.45)

and also as

where in the second decomposition α, and hence v, are to be determined. Using the second
decomposition we expand

F (u) = F (φm ) + Lα (−ξ∗ + v) + N (−ξ∗ + v),

(2.3.46)

where N (v) H 4 ≤ c v 2 4 represents nonlinear terms. The Newton map can be expressed
H
as

N (u) = u − L−1 (F (φm ) + Lα (−ξ∗ + v) + N (−ξ∗ + v)) ,
α
˜
= φm − L−1 F (u),
α

(2.3.47)
(2.3.48)

where we have introduced

˜
F (u) := F (φm (α)) + N (−ξ∗ + v).
53

(2.3.49)

The function F (φ∗ ) admits the expansion
m

F (φ∗ ) = F (φm ) + Lα (φ∗ − φm ) + N (φ∗ − φm ),
m
m
m

(2.3.50)

which, when re-arranged takes the form

F (φm ) = F (φ∗ ) + Lα (φm − φ∗ ) − N (φ∗ − φm ).
m
m
m

(2.3.51)

˜
Substituting (2.3.51) into the definition (2.3.49) of F we obtain

˜
F (u) = F (φ∗ ) + Lα (φ∗ − φm ) − N (φ∗ − φm ) + N (−ξ∗ + v).
m
m
m

(2.3.52)

In fact for a small v we can further decompose the nonlinear term as

N (v) = Q(v, v) + R(v),

(2.3.53)

where the term

1
Q(v, w) := − Lα G (φm ) vw − G (φm )vLα (w) + O(δ v
2

w ),

(2.3.54)

is quadratic in v, while R represents terms that are cubic or higher. Recalling the two
decompositions of u (2.3.44), (2.3.45), the v-terms are related by

v = v0 + (φ∗ − φm ),
m

54

(2.3.55)

so that

Q(−ξ∗ + v) = Q(−ξ∗ + v0 + φ∗ − φm ),
m
= Q(φ∗ − φm ) + Qs (−ξ∗ + v0 , φ∗ − φm ) + Q(−ξ∗ + v0 ), (2.3.56)
m
m
˜
where Qs (v, w) := Q(v, w) + Q(w, v). With these manipulations we may expand F (u) as

˜
F (u) = F (φ∗ ) + Lα (φ∗ − φm ) + Qs (−ξ∗ + v0 , φ∗ − φm ) +
m
m
m
+Q(−ξ∗ + v0 ) − R(φ∗ − φm ) + R(−ξ∗ + v).
m

(2.3.57)

Lemma 2.5. Let α∗ and φ∗ be as defined in lemma 2.4. Under the assumption (H2) of
m
∗
Theorem 1.1, there exist ρ1 , ρ2 > 0 such that for any u ∈ Bρ1 , there is a unique α = α(u; β, η)

satisfying
|α − α∗ | < ρ2 δ 2 ,

(2.3.58)

˜
F (u, α(u)) ⊥ Ψ0 (·; α(u)).

(2.3.59)

for which

˜
Here Ψ0 is the ground state eigenfunction of Lα given in (1.1.15) and F is as given in
(2.3.49). In particular, u may be written in the form

u = φm (z; α) − ξ∗ + v,

(2.3.60)

5

where v H 4 = O δ 2 .
∗
Proof. We want to show from the implicit function theorem that given u ∈ Bρ1 there is a

55

unique α = α(u) such that

˜
H(α, u; β, η, δ) := F (u), Ψ0 (α)

2

= 0,

(2.3.61)

and that α is smooth in u in H 4 . We first remark that Lemma 2.4 establishes the fact that

H(α∗ , φ∗ − ξ∗ ; β, η, δ) = 0.
m

(2.3.62)

˜
and α is smooth in u. By the expansion of F (u) given in (2.3.57), we may write H(α, u; β, η, δ)
as

H(α, u; β, η, δ) = (F (φ∗ ) + Lα (φ∗ − φm ) + Qs (−ξ∗ + v0 , φ∗ − φm ) + Q(−ξ∗ + v0 )+
m
m
m
−R(φ∗ − φm ) + R(−ξ∗ + v), Ψ0 (α))2 .
m

(2.3.63)

The first three terms on the right hand side of (2.3.63) are linear in α − α∗ . The remaining
terms are quadratic in α − α∗ or higher order terms. In addition the leading order inhomogeneous term is (Q(ξ∗ , ξ∗ ), Ψ0 (α∗ ))2 . We conclude that H(α, u; β, η, δ) can be rewritten
as

H(α, u; b, η, δ) = (α − α∗ )B − (Q(ξ∗ , ξ∗ ), Ψ0 (α∗ ))2 + O |α − α∗ |2 v H 4 ,

(2.3.64)

where the coefficient B comes from the first three terms. The remaining of the proof requires
that we compute B explicitly to determine conditions under which it is non-zero at leading

56

order. We focus on the first term on the RHS of (2.3.63), using (2.3.8) we may rewrite it as

(F (φ∗ ), Ψ0 (α))2 = (F (φ∗ ), Ψ0 (α) − Ψ0 (α∗ ))2 ,
m
m
=

F (φ∗ ),
m

(2.3.65)

∂ψ0
(α∗ ) (α − α∗ ) + O(|α∗ − α|2 ).
∂α
2

(2.3.66)

The first contribution to B arises from

B1 :=

F (φ∗ ),
m

∂ψ0
(α∗ ) .
∂α
2

(2.3.67)

From (2.3.41) we may expand B1

2
B1 = δ 2 α∗ ((g (φ∗ )g (φ∗ ) + 2g (φ∗ )g(φ∗ )) + p2 (φ∗ ),
m
m
m
m
m

∂ψ0
(α∗ ) + O(δ 3 ).
∂α
2

(2.3.68)

Applying (3.1.24) and (3.1.25) we further simplify B1
δ2
B1 = − √
ˆ
2 (φ∗ ) 2 λ∗
m
0

2
α∗ ∗
(φ ) + 2α∗ g (φ∗ )g(φ∗ ) + p2 (φ∗ ), (φ∗ )
m
m
m
m
2 m

+ O(δ 3 ),
2

(2.3.69)
ˆ
ˆ
where λ∗ := λ0 (α∗ ). By (3.1.28) we may combine the first two terms,
0
B1 = − √

δ2

√ 2
2 2α∗ g(φ∗ )(g (φ∗ ))2 + p2 (φ∗ ), (φ∗ )
max
max
m
m
ˆ
2 (φ∗ ) 2 λ∗
m
0

+ O(δ 3 ), (2.3.70)

Addressing the second term of (2.3.63) we obtain

(Lα (φ∗ − φm ), Ψ0 (α))2 = Λ∗ (α∗ − α)B2 + O(|α∗ − α|2 ),
m
0

57

(2.3.71)

where Λ∗ := Λ0 (α∗ ) and
0

∂φm
(α∗ ), Ψ0 (α∗ ) .
∂α
2

B2 := Λ∗
0

(2.3.72)

Applying (3.1.24) and (2.2.29) we further simplify
Λ∗
B2 = √ 0 (φ∗ ) 2 + O(δ 3 ).
m
ˆ
2λ∗

(2.3.73)

0

The remaining terms that are linear in α−α∗ are (Qs (φ∗ −φm , v0 ), Ψ0 (α))2 and (Qs (−ξ∗ , φ∗ −
m
m
φm ), Ψ0 (α))2 . The first of these terms is

(Qs (φ∗ − φm , v0 ), Ψ0 (α))2 = (α∗ − α)B3 + O(|α∗ − α|2 ),
m

(2.3.74)

where the coefficient
B3 :=

Qs

∂φm
(α∗ ), v0 , Ψ0 (α) .
∂α
2

(2.3.75)

Using the definition of Qs and Q, (2.3.54), this term has the expansion

Qs (

∂φm
∂φm
∂φm
(α∗ ), v0 ) = Q(
(α∗ ), v0 ) + Q(v0 ,
(α∗ )),
∂α
∂α
∂α
1
∂φm
∂φm
= − Lα G (φ∗ )
(α∗ )v0 − G (φ∗ )
(α∗ )Lα v0 +
m
m
2
∂α
∂α
1
∂φm
∂φm
− Lα G (φ∗ )v0
(α∗ ) − G (φ∗ )v0 Lα (
(α∗ )) +
m
m
2
∂α
∂α
+O δ||v0 ||H 4 .

(2.3.76)

58

Substituting this term into B3 we obtain
∂φm
∂φm
Lα G (φ∗ )(v0
(α∗ ) +
(α∗ )v0 ) , Ψ0 (α) +
m
∂α
∂α
2
∂φm
− G (φ∗ )v0 Lα
(α∗ ), Ψ0 (α) +
m
∂α
2
∂φm
− G (φ∗ )
(α∗ ) Lα v0 , Ψ0 (α) +
m
∂α
2

B3 = −

1
2

+O δ v0 H 4 .

(2.3.77)

Using the self-adjointness of Lα and (2.2.29) to replace Ψ0 (α) by ψ0 (α) yields the form
λ
B3 = − 0
2

G (φ∗ )(v0
m

∂φm
∂φm
(α∗ ) +
(α∗ )v0 ), ψ0 (α) +
∂α
∂α
2

−λ0 G (φ∗ )v0 ψ0 (α∗ ), ψ0 (α) 2 +
m
∂φm
(α∗ ) Lα v0 , ψ0 (α) +
− G (φ∗ )
m
∂α
2
+O δ||v0 ||H 4 .

(2.3.78)

5

Recalling that λ0 = O(δ) and v0 = O δ 2

B3 = − G (φ∗ )
m

the leading order term is

7
∂φm
(α∗ ) Lα v0 , ψ0 (α) + O δ 2
∂α
2

.

(2.3.79)

Using (2.2.12) and (2.2.13) we obtain

ψ0 =

1
|(φ∗ ) |
m
+ O δ2
(φ∗ ) 2
m

59

,

(2.3.80)

in L2 norm. Combining (2.3.80), (3.1.24) and (3.1.4) we may further simplify B3

B3 = − √

7

1
ˆ
2λ0 (φ∗ ) 2
m

Lα (φ∗ ) , Lα v0 2 + O δ 2
m

.

(2.3.81)

ˆ
From the definition (2.2.8) of λ0 , the form (2.3.9) of α∗ from Lemma 2.4 and the definition
(2.2.10) of γ0 and identity (3.1.26), the coefficient of B3 may be reduced to
5
√
2
2
ˆ
2 (φ∗ ) 2 (λ∗ )2 = 4µ+ α∗ (φ∗ ) 2 = −8µ+ (b+ − b− )β + O( δ).
m 2
m 2 0

(2.3.82)

Substituting this term into B3 we obtain

B3 =

1
5
2
8µ+ (b+
5

Using the fact that v0 = O δ 2

Lα (φ∗ ) , Lα v0 2 + O(δ 3 ).
m

(2.3.83)

− b− )β

we conclude

5

B3 = O δ 2

.

(2.3.84)

The final term that is linear in α − α∗ takes the form
(Qs (−ξ∗ , φ∗ − φm ), Ψ0 (α))2 = (α∗ − α)B4 + O(|α∗ − α|2 ),
m

(2.3.85)

where the coefficient
B4 :=

Qs (−ξ∗ ,

∂φm
(α∗ )), Ψ0 (α∗ ) .
∂α
2

60

(2.3.86)

From the definition of Qs and Q (2.3.54), we have

Qs (−ξ∗ ,

∂φm
∂φm
∂φm
(α∗ )) = Q(−ξ∗ ,
(α∗ )) + Q(
(α∗ ), −ξ∗ ),
∂α
∂α
∂α
∂φm
∂φm
1
=
Lα G (φ∗ )ξ∗
(α∗ ) + G (φ∗ )ξ∗ Lα
(α∗ ) +
m
m
2
∂α
∂α
1
∂φm
∂φm
+ Lα G (φ∗ )
(α∗ )ξ∗ + G (φ∗ )
(α∗ )Lα ξ∗ +
m
m
2
∂α
∂α
+O δ||ξ∗ ||L2 .

(2.3.87)

Substituting back into B4 we obtain

B4 =

∂φm
∂φm
(α∗ ) +
(α∗ )ξ∗ ) , Ψ0 (α∗ ) +
∂α
∂α
2
∂φm
+ G (φ∗ )ξ∗ Lα
(α∗ ), Ψ0 (α∗ ) +
m
∂α
2
∂φm
+ G (φ∗ )
(α∗ )Lα ξ∗ , Ψ0 (α∗ ) +
m
∂α
2
1
2

Lα G (φ∗ )(ξ∗
m

+O δ||ξ∗ ||L2 .

(2.3.88)

Using (3.1.24) to replace ∂φm /∂α by ψ0 and using the self-adjointness of Lα and (2.2.29),
we obtain the simplified form

B4 =

λ0
2

G (φ∗ )(ξ∗
m

∂φm
∂φm
(α∗ ) +
(α∗ )ξ∗ ), ψ0 (α∗ ) +
∂α
∂α
2

+λ0 G (φ∗ )ξ∗ ψ0 (α∗ ), ψ0 (α∗ ) 2 +
m
∂φm
+ G (φ∗ )
(α∗ )Lα ξ∗ , ψ0 (α) +
m
∂α
2
+O δ||ξ∗ ||L2 .

(2.3.89)

61

Recalling that ξ∗ = O(δ 2 ) and λ0 = O(δ), the leading order term is
B4 =

G (φ∗ )
m

∂φm
(α∗ )Lα ξ∗ , ψ0 (α∗ ) + O(δ 3 ).
∂α
2

(2.3.90)

Using the definition (2.3.43) of ξ∗ and (3.1.24), we obtain
B4 =

∂φm
˜
G (φ∗ )
(α∗ ) ψ0 (α∗ ) Lα Π0 (α∗ )L−1 F (φ∗ )dz + O(δ 3 ).
m
α∗
m
∂α
R

(2.3.91)

Using (3.1.24), (3.1.22), we obtain

B4 = √

1
2

ˆ
(φ∗ ) 2 λ∗
m
0

R

˜
G (φ∗ )((φ∗ ) )2 Lα Π0 (α∗ )L−1 F (φ∗ )dz + O(δ 3 ),
m
m
α∗
m

(2.3.92)

Applying the identity (3.1.4) and recalling from (2.2.6) that Lα∗ = L2 ∗ + O(δ), we obtain
α
B4 = √

5

1
ˆ
2 (φ∗ ) 2 λ∗ R
m
0

˜
Π0 (α∗ )F (φ∗ ) (φ∗ ) dz + O δ 2
m
m

.

(2.3.93)

Substituting (2.3.41) into the expression for F (φ∗ ) and using (3.1.25) and (3.1.29) to simplify
m
the resulting expression we obtain
√ 2
5
δ2
B4 = √
2 2α∗ g(φ∗ )(g (φ∗ ))2 + (p2 (φ∗ ), (φ∗ ) )2 + O δ 2
max
m
m
max
ˆ
2 (φ∗ ) 2 λ∗
m
0

.
(2.3.94)

We have decomposed the quantity B in (2.3.64) as

B = B1 − B2 − B3 − B4 .

62

(2.3.95)

Combining the expression (2.3.70), (2.3.73), (2.3.84), (2.3.94) for B1 , B2 , B3 and B4 respectively, we obtain the expression

B = −

√ 2
2δ

√ 2
2 2α∗ g(φ∗ )(g (φ∗ ))2 + (p2 (φ∗ ), (φ∗ ) )2 +
max
max
m
m

ˆ
(φ∗ ) 2 λ∗
m
0
∗
5
Λ
− √ 0 (φ∗ ) 2 + O δ 2
m
ˆ
2λ∗

.

(2.3.96)

0

Using the expression (2.2.31) of Lemma 2.3 to replace Λ∗ we obtain
0

B = −

√ 2
2δ

√ 2
1
ˆ
(φ∗ ) 2 (λ∗ )2 + 5 2α∗ g(φ∗ )(g (φ∗ ))2 +
max
max
m 2 0
2

ˆ
λ∗ (φ∗ ) 2
m
0
3
+ (p2 (φ∗ ), (φ∗ ) )2
m
m
2

5

+ O δ2

.

(2.3.97)

Our goal is to establish conditions under which the leading order term of B is non-zero.
Using the form (2.3.9) of α∗ , the definition (2.2.10) of γ0 , and identities (3.1.26) and (3.1.27)
the second term in B reduces to
√
√ 2
5 5
2
5 2α∗ g(φ∗ )(g (φ∗ ))2 = − µ+ (b+ − b− )β + O( δ).
max
max
2

(2.3.98)

Finally the expression (2.1.12) for p2 allows us to reduce the third term of B to the explicit
form

(p2 (φ∗ ), ((φ∗ ) )2 )2 = βµ− W (φ∗ ), ((φ∗ ) )2
m
m
m
m
+

2

+

η
W (φ∗ )(φ∗ − b), ((φ∗ ) )2 + O(δ).
m
m
m
2
2

(2.3.99)

Combining these expressions and collecting terms in β and η, we express B = B(β, η, δ) in

63

the form
B=−

√ 2
2δ
ˆ
λ∗ (φ∗ ) 2
m
0

5

(A1 β + A2 η) + O δ 2

,

(2.3.100)

where we have introduced
9 5
3
2
A1 := − µ+ (b+ − b− ) + µ− W (φ∗ ), ((φ∗ ) )2 ,
m
m
2
2
2
3
W (φ∗ )(φ∗ − b), ((φ∗ ) )2 .
A2 :=
m
m
m
4
2

(2.3.101)
(2.3.102)

√
So long as the leading order term (A1 β + A2 η) in B is non-zero (at least to O( δ)) we may
solve (2.3.64) to obtain

α = α∗ −

5
(Q(ξ∗ , ξ∗ ), Ψ0 (α∗ ))2
+ O δ2
B

.

(2.3.103)

In particular we remark that |α − α∗ | = O(δ 2 ) which follows from ξ∗ H 2 = O(δ 2 ) and
Q(ξ∗ , ξ∗ ) 2 = O(δ 4 ). Note that A1 , A2 depend on η and β. To examine this dependence
we define
F (η, β, δ) := A1 (η, β, δ)β + A2 (η, β, δ)η.

(2.3.104)

As δ → 0 the homoclinic solution φ∗ bifurcates out the heteroclinic solution φh ; this bim
furcation is not smooth in

· 2 . However the inner products in A1 and A2 are smooth as

δ → 0, as we establish in Lemma 3.7. Consequently applying (3.2.33) and (3.2.34) of Lemma
3.7 we may conclude

F (η, β, δ) = F (η, β, 0) + O(δ ω )
= Ah β + Ah η + O(δ ω ),
1
2

64

(2.3.105)

where ω = min{C, 1 } and C is some positive constant depending upon the exponential
2
dichotomies of the second order differential equation(1.1.8) and hence is independent of η, β
and δ. Here we have introduced
9 5
2
Ah := − µ+ (b+ − b− ) + 3µ− W (φh ), (φh )2 ,
1
2
2
3
W (φh )(φh − b− ), (φh )2 .
Ah :=
2
2
2

(2.3.106)
(2.3.107)

In particular Ah and Ah do not depend on η, β or δ. They only depend on the heteroclinic
1
2
solution φh of the second order differential equation (1.1.8) which is fully determined by the
double well potential W .
As long as β and η satisfies the condition (H2 )

Ah β + Ah η > ν δ ω ,
1
2

(2.3.108)

for some ν > 0 independent of δ, then F (η, β, δ) is bounded away from zero by an O(δ ω )
and hence (2.3.103) holds.

Lemma 2.6. Let α = α(u; η, β, δ) be given in Lemma 2.5. Then for every ρ > 0 there exists
δ0 > 0 such that the Newton map

˜
N (u) = φm − L−1 F (u),
α

(2.3.109)

∗
∗
defined in (2.2.4) maps Bρ into Bρ .

Remark 2.2. In (2.3.109) α in the subscript where we emphasize that α = α(u, δ) is a
65

function of u as given in Lemma 2.5. However, α∗ = α(δ; β, η) is as defined in (2.3.9).
∗
∗
Proof. We show that the Newton map takes Bρ into Bρ . With α = α(u) the Newton map

becomes
−1

˜
N (u) = φm − Πα Lα

F (φ∗ ) + Lα (φ∗ − φm ) + Qs (ξ∗ + v0 , φ∗ − φm ) +
m
m
m

+Q(ξ∗ + v0 ) − R(φ∗ − φm ) + R(ξ∗ + v) ,
m

(2.3.110)

˜
where Πα Lα maps L2 into H 4 with an O(1) norm. We perform a Neumann expansion on
˜
˜
˜
(Πα Lα )−1 in terms of Π∗ L∗ := Πα∗ Lα∗ , and obtain
˜
N (u) = φ∗ − Π∗ L∗
m
−1

˜
− Π∗ L∗

−1

F (φ∗ ) +
m

[Qs (ξ∗ + v0 , φ∗ − φm ) + Q(ξ∗ + v0 )] + O(δ 4 ). (2.3.111)
m

Observe that F (φ∗ ) = O(δ 2 ) and |α − α∗ | = O(δ 2 ). While ξ∗ H 4 = O(δ 2 ) implies that
m
˜
Π∗ L∗

−1

˜
= Π∗ L∗

(Qs (ξ∗ + v0 , φ∗ − φm ) + Q(ξ∗ + v0 ))
m

−1

((α∗ − α)Qs (ξ∗ + v0 , ∂α φ) + Q(ξ∗ + v0 )) ,

= O(δ 4 ).

(2.3.112)

˜
In particular, since N (φ∗ ) = φ∗ − Π∗ L∗
m
m

−1

F (φ∗ ), we have
m

N (u) − N (φ∗ ) H 4 = O(δ 4 ).
m
∗
Since ρ is fixed, taking δ sufficiently small implies that N (u) ∈ Bρ .

66

(2.3.113)

Lemma 2.7. Let α = α(u; β, η, δ) be as defined in Lemma 2.5. Then for every ρ > 0 there
∗
exists δ0 > 0 such that the operator N is an asymptotically strong contraction on Bρ for all

0 < δ < δ0 . That is, there exists γ = O(1) > 0 such that
1

||N (u1 ) − N (u2 )||H 4 ≤ γδ 2 ||u1 − u2 ||H 4 ,

(2.3.114)

∗
for all u1 , u2 ∈ Bρ .

Proof. Recall the two expansions of u defined in (2.3.44) and (2.3.45)

i
ui = φ∗ + ξ∗ + v0
m

(2.3.115)

ui = φi (z; αi ) + ξ∗ + v i , i = 1, 2.
m

(2.3.116)

Correspondingly from (2.3.111) we have

1
2
˜
N (u1 ) − N (u2 ) = (Π∗ L∗ )−1 Qs (ξ∗ + v0 , φ∗ − φ1 ) − Qs (ξ∗ + v0 , φ∗ − φ2 )+
m
m
m
m
1
2
+Q(ξ∗ + v0 ) − Q(ξ∗ + v0 ) + h.o.t. .

(2.3.117)

From the definition (2.3.54) of Q we expand

1
i
i
i
Q(ξ∗ + v0 ) = − Lαi G (φi )(ξ∗ + v0 )2 − G (φi )(ξ∗ + v0 )Lα∗ (ξ0 + v0 ) +
m
m
2
+O(δ||ξ∗ ||L2 ||v0 ||H 4 ),
i
i
= Q(ξ∗ ) + Q(v0 ) − Lαi G (φi )ξ∗ v0 − G (φi )ξ∗ Lαi v0 +
m
m
i
i
−G (φi )v0 Lαi ξ∗ + O(δ||ξ∗ ||L2 ||v0 ||H 4 ),
m

(2.3.118)
67

from which we deduce that

2
1
˜ ∗
Π∗ L−1 Q(ξ∗ + v0 ) − Q(ξ∗ + v0 )

L2

˜
Π∗ L−1
∗

≤

2
1
Q(v0 ) − Q(v0 )

L2

+

2
1
+ Lα1 G (φ1 )ξ∗ v0 − Lα2 G (φ2 )ξ∗ v0 2 +
m
m
L
2
1
+ G (φ1 )ξ∗ Lα1 v0 − G (φ2 )ξ∗ Lα2 v0 ) 2 +
m
m
L
2
1
+ (G (φ1 )v0 Lα1 − G (φ1 )v0 Lα2 )ξ∗ 2 .
m
m
L

(2.3.119)

5

i
Recalling that ξ∗ H 4 = O(δ 2 ) and v0 H 4 = O(δ 2 ), we deduce that

1
2
˜ −1
Π∗ L∗ Q(ξ∗ + v0 ) − Q(ξ∗ + v0 )

L2

˜
Π∗ L−1
∗

≤

1
2
Q(v0 ) − Q(v0 )

L2

1
2
+γ1 δ v0 − v0 2 .
L

+
(2.3.120)

From the definition (2.3.54) of Q we expand

1
2
Q(v0 ) − Q(v0 )

1
1
2
L G (φ1 )(v0 )2 − Lα2 G (φ2 )(v0 )2 ) 2 +
≤
m
m
2 α1
L
L2
1
1
2
2
+ G (φ1 )v0 Lα1 v0 − G (φ2 )v0 Lα2 v0 2 ,
m
m
L
1
2
≤ γ2 δ v0 − v0

H4

.

(2.3.121)

In particular the first term from RHS of (2.3.117) satisfies the estimate

1
2
˜
Π∗ L−1 Q(ξ∗ + v0 ) − Q(ξ∗ + v0 )
∗

68

L2

1
2
≤ γ3 δ v0 − v0

H4

,

(2.3.122)

where we have introduced the constant

˜ ∗
γ3 := Π∗ L−1 2
(γ1 + γ2 ).
L →H 4

(2.3.123)

Addressing the second term on the RHS of (2.3.117), we recall that Qs (v, w) = Q(v, w) +
Q(w, v) and definition (2.3.54) of Q to obtain

2
1
Qs (ξ∗ + v0 , φ∗ − φ1 ) − Qs (ξ∗ + v0 , φ∗ − φ2 )
m
m
m
m

L2

T1
1
2
≤ Lα1 G (φ1 )(ξ∗ + v0 )(φ∗ − φ1 ) − Lα2 G (φ2 )(ξ∗ + v0 )(φ∗ − φ2 ) 2 +
m
m
m
m
m
m
L
T2
1
2
+ G (φ1 )(ξ∗ + v0 )Lα1 (φ∗ − φ1 ) − G (φ2 )(ξ∗ + v0 )Lα2 (φ∗ − φ2 )
m
m
m
m
m
m

L2

+

T3
1
2
+ G (φ1 )(φ∗ − φ1 )Lα1 (ξ∗ + v0 ) − G (φ2 )(φ∗ − φ2 )Lα2 (ξ∗ + v0 )
m
m
m
m
m
m

L2

.

(2.3.124)

The first term T1 on the RHS of (2.3.124) satisfies

T1 =

1
Lα1 G (φ1 )(ξ∗ + v0 )∂α φm (α∗ − α1 )+
m
2
−Lα2 G (φ2 )(ξ∗ + v0 )∂α φm (α∗ − α2 )
m

L2

.

(2.3.125)

However

|α∗ − αi | = O(δ 2 ),
i
ξ0 + v0

H4

i = 1, 2

= O(δ 2 ),

69

i = 1, 2

(2.3.126)
(2.3.127)

and we deduce that
1

2
1
T1 ≤ γ4 δ 2 v0 − v0 H 4 ,

(2.3.128)

for some γ4 > 0, independent of δ. Similar estimates apply to T2 , T3 and we conclude that

1
2
Qs (ξ∗ + v0 , φ∗ − φ1 ) − Qs (ξ∗ + v0 , φ∗ − φ2 )
m
m
m
m

1

2
1
≤ γ5 δ 2 v0 − v0 4 ,
L2
H

(2.3.129)

for a constant γ5 > 0, independent of δ < δ0 . Combining the inequalities (2.3.122) and
(2.3.129) with (2.3.117) we deduce that

N (u1 ) − N (u2 ) H 4 ≤

1
2
˜
Π∗ L−1 (Qs (ξ∗ + v0 , φ∗ − φ1 ) − Qs (ξ∗ + v0 , φ∗ − φ2 ))
∗
m
m
m
m
1
2
˜ ∗
+ Π∗ L−1 (Q(ξ∗ + v0 ) − Q(ξ∗ + v0 ))
1

H4

H4

+

,

1

1
2
≤ γδ 2 v0 − v0 2 = γδ 2 u1 − u2 H 4 ,
L

(2.3.130)

for some constant γ > 0, independent of δ < δ0 .
The proof of Theorem 1.1 follows by applying Lemma 2.6, Lemma 2.7 and the contraction
∗
mapping principle to derive the existence of a unique fixed point Φm of N in Bρ0 .

70

Chapter 3
Analysis of the Second-Order System
In chapter 2 we use the contraction mapping argument to prove the existence of the homoclinic solution of full system (1.0.8), describing it as a perturbation of the homoclinic solution
of the associated second-order differential equation (1.1.6),

φm = G (φm ),

(3.0.1)

where G is defined in (2.2.3).

3.1

Analysis of the second order eigenvalue problem

Here we establish some of the properties of spectrum of the second order system used in
Chapter 2. Recall that φm (z; α, b) is the homoclinic solution of the second order equation
(1.1.6).
Lemma 3.1. Under the scaling (S)-(1.1.3), where b = b− + δ 2 β and η = δ 2 η there exists a
˜
smooth function

α∗ (b) = −

δ
η
µ− β(b+ − b− ) + (b+ − b− )2 + O(δ 2 ),
g(b+ )
4

(3.1.1)

for which G(·; α∗ (b), b) has a double zero at u = φ∗ (b). For α > α∗ , this double zero breaks
max
71

into two zeros, the smaller of which takes the form

φmax (α, b) = b+ −

3
2δαg(b+ )
+ O δ2
µ+

.

(3.1.2)

In particular, the value φmax (α) is the maximum of φm over z ∈ R, and
p2 (φmax ) = −µ2 (b+ − b− )β −
+

1
2αg(b+ )
µ+
µ− (µ+ − µ− )β +
(b+ − b− )η δ 2 + O(δ),
µ+
2

(3.1.3)
which is positive so long as β < 0.
Proof. The expression (3.1.2) follows from a Taylor expression of G near b+ , and (3.1.3)
results from substitution of (3.1.2) into (2.1.12).
Lemma 3.2. Let φm (z; α, b) is the homoclinic solution of the second order equation (1.1.6),
then the following equalities hold

Lα φm = G (φm )(φm )2 ,

(3.1.4)

Lα ψ0 = G (φm )φm ψ0 + λ0 ψ0 ,

(3.1.5)

Lα (zφm ) = 2φm ,

(3.1.6)

Lα (zψ0 ) = 2ψ0 + λ0 ψ0 ,

(3.1.7)

2
where Lα = ∂z − G (φm ) is the linearization of (1.1.6) about φm and λ0 is the ground-state

eigenvalue of Lα with eigenfunction ψ0 .
Proof. The first equality follows from taking partial derivatives of (1.1.6) with respect to z
while the second equality arises from taking partial derivative of the eigenvalue equation for
ψ0 with respect to z. The last two equalities follow from distributing the action of Lα .
72

Lemma 3.3. The function g (φm ) admits the following expansions
√
1
g (φm ) = √ |φm | + δh1 + O(δ 2 ),
2

(3.1.8)

where h1 L1 + h1 L2 = O(1). In addition we have

g (φm ) =

√
φm 2
√ ψ0 + δh2 + δh3 + O(δ 2 ),
2

(3.1.9)

in L2 norm, where the correction terms take the form

1
h1 = √
δ

1
1
|φm |2 + δαg(φm ) − √ |φm | ,
2
2

h2 = Cψ0 ,

(3.1.10)
(3.1.11)

h3 = −2α(˜ L)−1 g (φm )g(φm ) +
π

(r(φm ), ψ0 )2
ψ0 .
ˆ
λ0

(3.1.12)

Moreover
|φm |h1 =

δ
αg(φm ) + O(δ),
2

(3.1.13)

and
1
|φm |g (φm ) = √ |φm |2 + δαg(φm ) + O(δ 2 ).
2

(3.1.14)

Proof. From (1.1.6) we observe its first integral form

1
(φm )2 = G(φm ).
2

73

(3.1.15)

Applying the identity (3.1.15) to (2.2.14) we obtain

1
(φ )2 = (g (φm ))2 − δαg(φm ) + δ 2 h(φm ).
2 m

(3.1.16)

Solving for g (φm ) we obtain

g (φm ) =

1
|φ |2 + δαg(φm ) + O(δ 2 )
2 m

1
= √ |φm | +
2

1
1
|φm |2 + δαg(φm ) − √ |φm |
2
2

+ O(δ 2 ).

(3.1.17)

Expanding the expression (3.1.17) yields (3.1.8), except for a neighborhood of z = 0 with
√
√
φm = φm (0)z +O(z 2 ) where φm (0) = G (φm (0)) = O( δ). These estimates yield the O( δ)
error bound in L2 and L1 , which recovers the equality (3.1.8). From (3.1.10) it is easy to
derive (3.1.13) and plug it into (3.1.8) yielding (3.1.14). Recalling (2.2.18)

Lg (φm ) = −2δαg (φm )g(φm ) + δ 2 r(φm ),

(3.1.18)

where r(s) = 2g h + g h − g h and r(φm ) is O(1) in L2 . Since the right hand side of
(3.1.18) is even, it is orthogonal to the kernel of L and we can invert L, yielding

g (φm ) = −

δ 2 (r(φm ), ψ0 )2
2δα(g (φm )g(φm ), ψ0 )2
ψ0 +
ψ0 +,
λ0
λ0
H1
˜
−2δαL−1 Πg

(φm )g(φm ) + O(δ 2 ),

(3.1.19)

in L2 . Combining (2.2.8), (2.2.10), (2.2.12) and (2.2.13) we may further simplify the leading

74

coefficient of ψ0 in (3.1.19) yielding
2δα(g (φm )g(φm ), ψ0 )2
,
λ0
1
||φm ||2
√
+ O(δ 2 ).
2

H1 := −
=

(3.1.20)

Therefore

1
φm 2
√ ψ0 + δ 2 Cψ0 +
2

g (φm ) =

:=h2

(r(φm ), ψ0 )2
˜
ψ0 − 2αL−1 Πg (φm )g(φm ) +O(δ 2 ),
ˆ0
λ

+δ

(3.1.21)

:=h3

where C is some constant.

Remark 3.1. From the results of Lemma 3.3 we may refine (3.1.8) to obtain the equality

ψ0 =

√

2

φm 2

√
1
√ |φm | + δ(h1 − h2 ) − δh3 + O(δ 2 ),
2

(3.1.22)

in L2 norm, where h1 , h2 and h3 are defined in (3.1.10), (3.1.11) and (3.1.12).
Lemma 3.4. Let φm (z; β, η, δ) be the homoclinic solution of (1.1.6). Then the following
important identities hold

1
∂ψ0
φm
= −√
+ O δ2
ˆ
∂α
2λ0 ||φm ||2

75

,

(3.1.23)

in L2 norm, while
∂α φm = √

1
1
|φm | + O δ 2
ˆ
2λ0

,

(3.1.24)

in L2 norm.
1
g (φm )g (φm ) = φm + O(δ),
2

(3.1.25)

in L∞ norm. In addition
1
1
1
g(φm (0)) = g(b+ ) + O(δ 2 ) = √ φm 2 + O δ 2 ,
2
2 2
1
µ+
g (φm (0)) = −
+ O δ2 ,
2
√
(φm , g(φm )g (φm ))2 = 2g(φm (0))(g (φm (0)))2 +
1
1
− ||φm ||2 + O δ 2
2
4

(3.1.26)
(3.1.27)

,

(3.1.28)

where
1
1
g(φm (0))(g (φm (0)))2 = g(φm (0))G (φm (0)) + O δ 2
2

.

(3.1.29)

Proof. From the definition (1.1.14) of g(φm ) we can obtain

g (φm ) =

Ws (φm ),

g (φm ) =

1 Ws (φm )
,
2 Ws (φm )

76

(3.1.30)

from which we deduce

1
W (φm ),
2 s
1
=
G (φm ) + O(δ),
2
1
=
φ + O(δ),
2 m

g (φm )g (φm ) =

(3.1.31)

which recovers (3.1.27). From (1.1.6) and the definition of G(φ) (2.2.3) we have

φm = G (φm ),
= G0 (φm ) − δαg (φm ).

(3.1.32)

Then taking derivative with respect to α we obtain

(∂α φm )

= G0 (φm )∂α φm − δαg (φm )∂α φm − δg (φm ),
= G (φm )∂α φm − δg (φm ),

(3.1.33)

so from the definition of L (1.1.15) we obtain

L∂α φm = −δg (φm ),
= −δ(g (φm ), ψ0 )2 ψ0 + δ π0 g (φm ).
˜

(3.1.34)

δ(g (φm ), ψ0 )2
ψ0 − δL−1 π0 g (φm ),
˜
λ0

(3.1.35)

Inverting L yields
∂α φm = −

77

in L2 norm. From (2.2.12) we observe that π0 g (φm ) = O(δ). It yields
˜

∂α φm = −

(ψ0 , g (φm ))2
ψ0 + O(δ 2 ).
ˆ0
λ

(3.1.36)

Combining the results of (3.1.9), (2.2.12) and (2.2.13) we may further simplify
||φ ||
∂α φm = − √m 2 ψ0 + O
ˆ
2λ0
1
= √
|φ | + O
ˆ m
2λ0

1

δ2
1

δ2

,
,

(3.1.37)

in L2 which covers (3.1.24). In order to prove (3.1.23) it follows from (2.2.12)
g (φm ) ∂α φm ||g (φm )||2 − g (φm ) ∂α ||g (φm )||2
∂ψ0
+ O(δ),
=
∂α
||g (φm )||2
2

(3.1.38)

in L2 norm. Observing that

∂α ||g (φm )||2

= ∂α

(g (φm

))2 dz

1/2

,

R

=

Rg

(φm )g (φm )∂α φm dz
,
2||g (φm )||2

(3.1.39)

from (3.1.24) and (3.1.25) we have

∂α ||g (φm )||2 =

(φ
R g √ m )g

(φm )|φm |dz

1

,
ˆ
2 2λ0 ||g (φm )||2
∞
1
g (φm )g (φm ), φm dz
+ O δ2 ,
= − 0 √
ˆ
2λ0 ||g (φm )||2
+ O δ2

∞
1
φ φ dz
= − √0 m m
+ O δ2
ˆ
2 2λ0 ||g (φm )||2
1

= O δ2

.
78

,
(3.1.40)

So (3.1.38) becomes

1
∂ψ0
g (φm ) ∂α φm
=
+ O δ2
∂α
||g (φm )||2

.

(3.1.41)

Combining the results of (3.1.24), (3.1.8) and (3.1.25) we may further simplify

1
∂ψ0
g (φm )g (φm )
+ O δ2
= −
ˆ
∂α
λ0 ||g (φm )||2
1
φm
= −
+ O δ2
ˆ
2λ0 ||g (φm )||2

= −√

1

φm

ˆ
2λ0 ||φm ||2

+ O δ2

,
,
,

(3.1.42)

in L2 norm which covers (3.1.23). To prove (3.1.27) we use (3.1.25) and (3.1.8) to obtain

1

2g (φm )g (φm )g (φm )g(φm )dz + O δ 2

(φm , g(φm )g (φm ))2 =
R

=

,

√

, (3.1.43)

R

1

2g (φm )g (φm )|φm |g(φm )dz + O δ 2

by even-odd symmetry of φm and φm and integration by parts we obtain

(φm , g(φm )g (φm ))2 = −
=

√

∞
0

√
1
2 2g (φm )g (φm )φm g(φm )dz + O δ 2

,(3.1.44)

2g(φm (0))(g (φm (0)))2 +
∞√

+

1

2(g (φm ))2 g (φm )φm dz + O δ 2

0

79

. (3.1.45)

Applying (3.1.8) and (3.1.25) again we obtain

(φm , g(φm )g (φm ))2 =

√
2g(φm (0))(g (φm (0)))2 +
−

=

∞
0

1

2(g (φm ))2 (g (φm ))2 dz + O δ 2

,

√
2g(φm (0))(g (φm (0)))2 +
−

∞
0

1
1
(φm )2 dz + O δ 2
2

,

(3.1.46)

which recovers (3.1.28). In order to prove (3.1.26) we use (3.1.16) and the fact that φm (0) = 0
to obtain
g (φm (0)) =

1

δαg(φm (0)) + O(δ) = O δ 2

.

(3.1.47)

From (2.2.14) and the equality (3.1.47) we obtain

G (φm (0)) = 2 (g (φm (0)))2 + g (φm (0))g (φm (0)) + O(δ),
1

= 2(g (φm (0)))2 + O δ 2

,

(3.1.48)

so
1

G (φm (0))g(φm (0)) = 2(g (φm (0)))2 g(φm (0)) + O δ 2
which finishes the proof of (3.1.29).

80

,

(3.1.49)

3.2

Heteroclinic limit of the second-order problem

The differential equation (1.1.6) reduces to (1.1.8) in the limit as δ approaches to 0. In this
section we study the convergence of the homoclinic φm of (1.1.6) to the heteroclinic φh of
(1.1.8).
We perform the analysis in the dynamical systems framework. We first write (1.1.6) as
a system, introducing
u = φ, v = φ .

(3.2.1)


 ˙  
 u   v 
,
 =
G (u)
v

(3.2.2)

Then we can rewrite (1.1.6) as

here · means ∂/∂z. Define





(3.2.3)

x = F (x, δ),
˙

(3.2.4)

 u 
x =  ,
v
then (3.2.2) becomes

where F (x, δ) is defined by the right hand side of (3.2.2). It is also easy to see that p+ :=
(b+ , 0)T is a saddle point equilibrium point of (3.2.2) in case δ = 0. The spectrum of
Dx F (x, δ) (p ,0) consists of two matrix eigenvalues
+
σ(Dx F (x, δ) (p ,0) ) = λs ∪ λu ,
+

81

(3.2.5)

√
√
where λs = − µ+ < 0 and λu = µ+ > 0. Since µ+ = W (b+ ) > 0 it follows that p+ is
saddle equilibrium point. For δ = 0 the matrix eigenvalues of

σ(Dx h(x, δ) (p ,δ) ) = λs (δ) ∪ λu (δ),
+

(3.2.6)

where λs (δ) < 0 and λu (δ) > 0 perturb smoothly from λs and λu upon changing δ, respecu
s
tively. We denote the stable and unstable eigenspaces of Dx h(x, δ) (p ,δ) by E0 and E0
+
u
s
respectively and choose local coordinates x = (u, v) ∈ E0 ⊕ E0 . By an appropriate smooth

coordinate change near b+ seen in Lemma 6.1 in [33], (3.2.4) takes the normal form

u = λs (δ)u + f s (u, v),
˙

(3.2.7)

v = λu (δ)v,
˙

(3.2.8)

where f s (u, v) satisfies

f s (u, v) = O(|u|2 ),

(3.2.9)

as |u|, |v| → 0. We also define the local section Σin as
Σ+ = (u, v) u = ν, |v| ≤ ν

(3.2.10)

where ν > 0 is a small constant. It is easy to check that (3.2.4) is a reversible system with
a symmetry plane. We can also define the symmetry plane in (u, v) coordinates as

Σsym = (u, v) u = K(v) ,

82

(3.2.11)

where K is a smooth function satisfying that K(v) → 0 as v → 0 and K (0) = 0.

Σin

Σsym

Figure 3.1: The homoclinic solution of (1.1.6) in red color and the heteroclinic solution of
(1.1.8) in blue color. At z = 0, φm and φh are on the cross section Σin .

Lemma 3.5. For all sufficiently small ν and ξ such that 0 < ξ < ν then there exists
τ > 0 and a unique corresponding solution of (3.2.8) (u, v)(z) = (u, v)(z; ν, ξ; δ) satisfying
the conditions
•
u(0) = ν,

|v(0)| ≤ ν

(3.2.12)

•
u(τ ) = K(v(τ )) = ξ.

(3.2.13)

Moreover
|v(0)| ≤ C ξ K −1 (ξ),

τ∼

where C is a positive constant independent of ξ and ν.
83

ln(ξ)
.
|λs |

(3.2.14)

Proof. By rescaling we may assume ν = 1 and ξ < 1, then the normal form (3.2.8) with
f s (u, v) = νO(|u|2 ). Then from (3.2.7) and (3.2.13) we have

v(z) = K −1 (ξ) eλu (z−τ ) ,

(3.2.15)

and
z

u(z) = νeλs z +

eλs (z−t) f s (u, v)dt.

(3.2.16)

0

We define the weighted norm u := supz∈[0,∞) e−λs z |u(z)| and denote by U the Banach
space which it forms. A ball in U is defined

BR = u u ≤ R .

(3.2.17)

We define Γ : U → U as the map given by the right hand side of (3.2.16). We claim that
there is a ball BR ⊂ U for which Γ(BR ) ⊂ BR . Let u ∈ BR then by the definition of BR it
follows that |u| ≤ R eλs z . Consequently, from the definition of Γ(u) in (3.2.16) we have the
point-wise estimate

|Γ(u)| =

z

νeλs z +

≤ νeλs z + C

0

eλs (z−t) f s (u(t), v(t))dt ,
z

0

eλs (z−t) |u(t)|2 dt,

(3.2.18)

where C > 0 is a constant independent of u ∈ BR , and the last inequality results from

84

(3.2.9). Using the fact that |u| ≤ R eλs z we obtain
Γ(u) ≤ νeλs z + CR2 eλs z
≤

eλs z

ν

+ CR2

z

eλs t dt,

0

.

(3.2.19)

We deduce that for ν small enough, there exists R > 0 such that Γ(u)

≤ R for all

u ∈ BR . Applying the contraction mapping theorem to BR we deduce the existence of a
unique solution u of (3.2.8)-(3.2.7) subject to u(0) = ν. Moreover, since u(0) = ν > ξ and
u(z) approaches 0 as z approaches infinity, it follows from the intermediate value theorem
that there exists τ > 0 such that u(τ ) = ξ. From (3.2.15) and λu > 0 we deduce that
|v(0)| < K −1 (ξ) < ν. Since u ∈ BR we have the point-wise estimate |u(z)| ≤ R eλs z from
which we obtain the bound
τ
0

eλs (τ −t) f s (u, v)dt ∼

1 2λs τ
e
,
|λs |

(3.2.20)

Since u(τ ) = ξ, we may use (3.2.16) to solve for τ which satisfies the asymptotic relation

τ∼

ln(b+ − φmax )
ln(ξ)
=
.
λs
λs

(3.2.21)

From (3.2.15) it follows
|v(0)| ≤ C ξ K −1 (ξ).

(3.2.22)

Remark 3.2. ν can be chosen to be O(1) but small enough that Γ(u) ≤ R which implies
that ν < 1/4C.
85

Remark 3.3. The above lemma is very similar to Shil’nikov variable method which deals
well with local flow near the saddle equilibrium point.
Under the assumption (H) we know that the double well potential W actually has the
following form W = P 2 and P is a parabola that opens upward with two zeros b± , see
Fig 1.2. We translate the homoclinic and heteroclinic orbits so that φm (0) = φmax and
φh (0) = 0. We choose τm and τh are chosen so that φm (τm ) and φh (τh ) are on the cross
section Σ+ and note that τm < 0 and τh > 0. From lemma 3.5 we can see that τm depends
on δ since ξ = b+ − φmax and φmax depends on δ. However, τh is independent of δ.

τ∗
P (φh ) > 0
P (φh ) < 0

Σ+
τm

b−

b+

Σsym

Figure 3.2: The homoclinic solution of (1.1.6) in red color and the heteroclinic solution of
(1.1.8) in blue color. At z = τm , φm and φh are on the cross section Σin . At z = τ∗ ,
P (φh (τ∗ )) = 0.

Lemma 3.6. The homoclinic and heteroclinic solutions, translated so that φm (0) = φmax
and φh (0) = 0 are proximal in the following sense,

φm − φh (· − τm + τh ) − δ 2 β

86

1

H 1 (−∞,0]

≤ C δ2,

(3.2.23)

where the translations τm and τh are chosen such that φm (τm ) and φh (τh ) are on the cross
section Σ+ .
Proof. From the local coordinates used in Lemma 3.5 and Lemma 3.1 it follows ξ =
√
b+ − φmax = O( δ). From remark 3.2 we can choose ν in Lemma 3.5 independent of δ. So
from (3.2.14) we have

φm − φh (· − τm + τh ) − δ 2 β

1

H 1 [τm ,0]

≤ C δ2.

(3.2.24)

The time τ∗ is chosen so that P (φh (τ∗ − τm + τh )) = 0. From (3.1.15) and using the fact
that G(φm ) > 0 for −∞ ≤ z ≤ τm we have
2G(φm ).

(3.2.25)

√
2W (φh ) = − 2P (φh ).

(3.2.26)

φm =

Similarly we can also obtain

φh =

Let y = φm − φh (· − τm + τh ) − δ 2 β. From (3.2.25) and (3.2.26) we have
φm − φh (· − τm + τh ) − δ 2 β

=

2G(φm ) −

2W (φh (· − τm + τh )),

=

2W (φm ) −

2W (φh (· − τm + τh )) +

−

=

√

2W (φm ) +

2G(φm ),

2(P (φh (· − τm + τh )) − P (φm )) +

−
87

2W (φm ) +

2G(φm ).

(3.2.27)

2W (φm ) −

From the definition of G (2.2.3) we observe that

y =

√

√
2G(φm ) = O( δ). So

√
2P (φh (· − τm + τh ))y + O( δ, y 2 ).

(3.2.28)

Note that P (φh (· − τm + τh )) < 0 for z ∈ (τ∗ , τm ] and P (φh (· − τm + τh )) > 0 for
√
z ∈ (−∞, τ∗ ). At the time τm y = O( δ) which follows from (3.2.24). Since ν is O(1), then
it is easy to see that the flight time τ∗ − τm is uniformly bounded, independent of δ here.
By using (3.2.28) we have

1

y = O δ2

,

τ∗ − 1 ≤ z ≤ τm .

(3.2.29)

Using the fact that τ∗ − τm + 1 is bounded independent of δ and (3.2.29) it yields
1

φm − φh (· − τm + τh ) − δ 2 β

H 1 [τ∗ −1,τm ]

≤ C δ2.

(3.2.30)

On the interval (−∞, τ∗ − 1) we have P (φh ) < −C0 < 0 for some C0 > 0 independent of δ.
From (3.2.28) we have

1

y ≤ eC0 (z−τ∗ +1) y(τ∗ − 1) + O δ 2

,

−∞ ≤ z ≤ τ∗ − 1,

(3.2.31)

Moreover, since y tends to zero at an exponential rate as z → −∞ we obtain the bound
φm − φh (· − τm + τh ) − δ 2 β

1

H 1 [−∞,τ∗ −1]

≤ C δ2.

Combining the results of (3.2.24), (3.2.30), and (3.2.32) we recovers (3.2.23).

88

(3.2.32)

Remark 3.4. Lemma 3.6 is applied in the proof of Lemma 2.5. In Lemma 2.5 the implicit
function theorem is applied which requires that φm approaches φh as δ goes to zero.

Lemma 3.7. There exists ω > 0 for which the quantities A1 and A2 which are defined by
(2.3.101) and (2.3.102) in Lemma 2.5 admit the the asymptotic expansion

A1 (η, β; δ) = Ah (η, β) + O(δ ω ),
1

(3.2.33)

A2 (η, β; δ) = Ah (η, β) + O(δ ω ),
2

(3.2.34)

point-wise, where we have introduced the heteroclinic Melnikov integrals
5
9 2
Ah := − µ+ (b+ − b− ) + 3µ−
W (φh ) (φh )2 dz
1
2
R
3
W (φh )(φh − b− ), (φh )2 .
Ah :=
2
2
2

(3.2.35)
(3.2.36)

Proof. Since the terms involving β and η contains factor δ, then β, η → 0 precisely when
δ → 0. Recalling the definition, (2.3.101), of A1 we have
5
3
9 2
A1 = − µ+ (b+ − b− ) + µ−
W (φ∗ ) ((φ∗ ) )2 dz
m
m
2
2
R
5
0
9 2
W (φ∗ ) ((φ∗ ) )2 dz,
= − µ+ (b+ − b− ) + 3µ−
m
m
2
−∞

(3.2.37)

where the second equality follows from the fact that φ∗ is even. Using (3.2.23) from lemma
m
3.6 and remarking that W (φh ) is uniformly bounded in

· L∞ (R) yields

−τm +τh
√
9 5
2
A1 = − µ+ (b+ − b− ) + 3µ−
W (φh ) (φh )2 dz + O( δ).
2
−∞

89

(3.2.38)

Using the definition (3.2.35) of Ah we may further simplified A1
1

A1 = Ah − 3µ−
1

∞
−τm +τh

√
W (φh ) (φh )2 dz + O( δ).

(3.2.39)

√
Choosing ξ = O( δ) in (3.2.14) we have the estimate

−τm + τh = −C1 ln δ,

(3.2.40)

where C1 is a positive constant of order 1. Substituting (3.2.40) into A1 we obtain

A1 = Ah − 3µ−
1

∞
−C1 ln(δ)

√
W (φh ) (φh )2 dz + O( δ).

(3.2.41)

Since the heteroclinic solution φh (z) of (1.1.8) exponentially approaches the equilibrium b+
as z → ∞ and W (φh ) is bounded in
A 1 = A h − C2
1
=

· L∞ (R) we obtain
∞

−C1 ln δ
Ah + O(δ ω ),
1

√
e−C3 z dz + O( δ),

1
where ω = min{C1 C3 , 2 }. The derivation of (3.2.34) is similar and omitted here.

90

(3.2.42)

Chapter 4
Lin’s Method

4.1

Introduction

Homoclinic and heteroclinic orbits play an important role in applications. Lin’s method is
regarded as of great importance in proving existence of heteroclinic and homoclinic solutions
of dynamical systems. This method was first proposed by X. Lin, [44], to construct solutions
that stay close to a finite or infinite chain of heteroclinic connections. Lin’s method is based
upon Melnikov’s method, an implementation of the Lyapunov-Schmidt method. In [61]
Sandstede developed Lin’s method and improved it by giving an better representation of the
bifurcation equation. In this section, we follow Sandstede’s way and adopt Lin’s method to
prove the existence of homoclinic solutions of (1.0.8). Reconsider the differential equation
(1.0.8)
2
∂z − W (u) + η
˜

2
∂z u − W (u) = θ.

(4.1.1)

Let Lh be the linearization of (4.1.1) around the heteroclinic solution φh for θ = 0, then
Lh = (Lh + η )Lh ,
˜

(4.1.2)

2
Lh = ∂z − W (φh ).

(4.1.3)

where

91

It is easy to see that φh is the eigenvector of Lh corresponding to eigenvalue 0. From SturnLiouville Theory, φh is the lowest eigenstate. There is a basic fact based on assumption
(S’)-(1.1.4) and we will use it through this Chapter:
Lemma 4.1. 0 is a simple eigenvalue of operator Lh , i.e., N ull(Lh ) = span{φh }.
Proof. Assume that there exists ϕ ∈ span{φh } such that
/
Lh ϕ = (Lh + η )Lh ϕ = 0
˜

(4.1.4)

By Sturn-Liouville theory, 0 is the simple eigenvalue of Lh associated with eigenvector φh
and the other eigenvalues are O(1) away from the 0. By using spectrum mapping theorem
for self-adjoint operator and the assumption (S’)-(1.1.4) η + λh = 0, we conclude that Lh + η
˜
˜
2
doesn’t have kernel, which implies that

Lh ϕ = 0.

(4.1.5)

Therefore ϕ = C φh , which contradict to the definition of ϕ.
Let
u1 = u, u2 = u˙1 , u3 = u˙2 , u4 = u˙3 ,

92

(4.1.6)

where ˙ denotes ∂/∂z, we can rewrite (4.1.1) obtaining
 ˙
 u1


 u2



 u3


u4





 
 
 
 
 
=
 
 
 
 

u2
u3
u4
˜
θ + η W (u1 ) − W (u1 )W (u1 ) + W (u1 )u2 + (2W (u1 ) − η )u3
˜
2








 . (4.1.7)





We further define
x = (u1 , u2 , u3 , u4 )T ,

(4.1.8)

and write (4.1.7) in the compact form

x = f (x; θ),
˙

(4.1.9)

where f (x; θ) represents the right-hand side of (4.1.7). Recall that (1.0.9)

θ = W (b) W (b) − η ,
˜

(4.1.10)

and under the scaling (S’)-(1.1.4) b = b− + βδ 2 and since W (b− ) = W (b− ) = 0, µ− =
W (b− )
θ = W (b− ) W (b− ) − η βδ 2 + O(δ 3 ),
˜
= µ− (µ− − η )βδ 2 + O(δ 3 ).
˜

(4.1.11)

So θ is very small θ = O(δ 2 ).
For the case θ = 0, the system (4.1.9) has equilibriums p1 := (b− , 0, 0, 0)T and p2 :=
93

(b+ , 0, 0, 0)T satisfying f (pi ; θ = 0) = 0. For θ = 0 the equilibriums are described by the
following lemma.
Lemma 4.2. There exist θ0 > 0 and a smooth function pi (θ) such that for all |θ| < θ0
the points p1 (θ) = (b− (θ), 0, 0, 0)T and p2 (θ) = (b+ (θ), 0, 0, 0)T are the equilibriums of the
system (4.1.9).
Proof. Note that f (pi ; 0) = 0 and f is smooth in x and θ. Moreover the Jacobian matrix







Dx f (x; θ) (p ;0) = 

i




0

1

0

0

0

0

1

0

0

0

0



1

µ± η − µ2
˜
±

0

2µ± − η
˜

0






,





(4.1.12)

is nonsingular. Here µ± = W (b± ). The result follows from the implicit function theorem.

Remark 4.1. From Lemma 4.2 we know that for each small |θ| < θ0 there exists the corresponding equilibriums pi (θ) satisfying f (pi (θ), θ) = 0. Without lose of generality we can
assume that f (pi , θ) = 0 for small |θ| < θ0 .
It is easy to see that the spectrum of Dx f (x; θ) (p ;0) is comprised of four eigenvalues,
i
√
σ(Dx f (x; θ) (p ;0) ) = {± µ± , ±
i

µ± − η }.
˜

(4.1.13)

√
√
√
√
˜ 1
˜
Then for p1 λs := − µ+ , λss := − µ+ − η , λu := µ+ , and λuu := µ+ − η . Under the
1
1
1

94

assumption (S’)-(1.1.4) we have η < 0 so
˜

λss < λs < 0 < λu < λuu .
1
1
1
1

(4.1.14)

√
√
√
√
Similarly for p2 λs := − µ− , λss := − µ− − η , λu := µ− , and λuu := µ− − η .
˜ 2
˜
2
2
2
λss < λs < 0 < λu < λuu .
2
2
2
2

(4.1.15)

Remark 4.2. By the Hypotheses (H) and (S’)-(1.1.4) µ± = W (b± ) > 0 and |˜| < µ± so
η
µ± , µ± − η are strictly positive and (pi , 0)T are saddle points. Furthermore, θ = 0 but small,
˜
the eigenvalues for the fixed point p1,2 be the same as (4.1.13).
The system (4.1.9) has some symmetry property in geometric sense which is very important for the later analysis. We call the dynamical system (4.1.9) S-reversible if there exists
a 4 × 4 matrix S with S 2 = I satisfying
f (S x, θ) = −S f (x, θ),

∀x ∈ R4 .

(4.1.16)

It is easy to see that if x(z) is a solution of S-reversible system (4.1.9) then S x(−z) is also
a solution. Define F ix S = {x ∈ R4 | Sx = x} and F ix(−S) = {x ∈ R4 | Sx = −x}. Since
1
x = 2 (I − S)x + 1 (I + S) then x = x1 + x2 where x1 ∈ F ix S and x2 ∈ F ix(−S).
2

Lemma 4.3. The system (4.1.9) is reversible under the map S : R4 → R4 defined by
S(u1 , u2 , u3 , u4 ) = (u1 , −u2 , u3 , −u4 ).
Proof. It is easy to verify that S f (x; θ) = −f (S x; θ).
95

(4.1.17)

Remark 4.3. for our system it is easily checked S q1 (0) = q2 (0) where S is defined in
(4.1.17).

4.2

Lin’s Orbit

We will construct Lin’s orbit in this section. The first step of construction of Lin’s orbit is
to study the splitting of the manifolds with respect to the parameter θ. Recall that φh is the
heteroclinic solution of (1.1.8). Since φh is the heteroclinic connection of (1.0.8) between b−
and b+ it follows that
q1 (z) = (φh (z), φh (z), φh (z), φh (z))T ,

(4.2.1)

is the heteroclinic connection of (4.1.9) for θ = 0 between p1 = (b− , 0, 0, 0)T and p2 =
(b+ , 0, 0, 0)T , that is
lim q1 (z) = p1 ,

z→−∞

lim q (z)
z→∞ 1

= p2 .

(4.2.2)

Correspondingly there is another heteroclinic solution q2 of (1.0.8) and

q2 (z) = (φh (−z), −φh (−z), φh (−z), −φh (−z))T ,

(4.2.3)

which connects b+ and b− satisfying
lim q2 (z) = p2 ,

z→−∞

lim q (z)
z→∞ 2

= p1 .

(4.2.4)

s
By the invariant manifold theorem, there is an unstable manifold W− and a stable manu
s
ifold W− generated by the saddle point p2 . Similarly there are stable manifold W+ and
u
unstable manifold W+ associated to p1 . Consider the differential equation obtained by the

96

linearization of (4.1.9) about Φh
v = A(z)v,
˙

(4.2.5)

where A(z) := Dx f (Φh (z), 0). We denote by T (t, s) the fundamental transition operator
for (4.2.5). It has semigroup properties: T (t, t) = I; T (t, w) = T (t, s)T (s, w) for t ≥ s ≥
w. Since limz→±∞ A(z) = Dx f (x; θ) (p ;0) is hyperbolic, it has exponential dichotomies
1,2
Q− (z), Pi− (z) for z ≤ 0 and Pi+ (z), Q+ (z) for z ≥ 0 [18]. This means that there exist
i
i
constants C ≥ 1 and ν > 0, and continuous mappings Q− (·) : R− → L(R4 ) and Pi+ (·)
i
: R+ → L(R4 ) where L(R4 ) represents the bounded linear operator on R4 . The following
properties hold:
• Q− (z) are projections on R− 4 , and Pi+ (z) are projections on R+ 4 ,
i
(Pi+ )2 = Pi+ ,

(4.2.6)

(Q− )2 = Q− .
i
i

(4.2.7)

•
Pi+ (t)Ti (t, s) = Ti (t, s)Pi+ (s),

t, s ∈ R+ ,

(4.2.8)

Q− (t)Ti (t, s) = Ti (t, s)Q− (s),
i
i

t, s ∈ R− .

(4.2.9)

97

•
Ti (t, s) Pi+ (s)

≤ C e−ν(t−s) ,

0 ≤ s ≤ t,

(4.2.10)

Ti (t, s) Q+ (s)
i

≤ C e−ν(s−t) ,

0 ≤ t ≤ s,

(4.2.11)

Ti (t, s) Pi− (s)

≤ C e−ν(t−s) ,

s ≤ t ≤ 0,

(4.2.12)

Ti (t, s) Q− (s)
i

≤ C e−ν(s−t) ,

t ≤ s ≤ 0,

(4.2.13)

where Qi ± := I − Pi± . In fact
R Q− (z) = Tq (z) W u (p2 ).
1
1

+
R P1 (z) = Tq (z) W s (p1 ),
1

(4.2.14)

+
Similarly for P2 and Q− . Let Ti∗ (t, s) be the adjoint operator of Ti (t, s), whose definition
2

is given by
< ψ, Ti (t, s)v >=< Ti∗ (t, s)ψ, v >,

v ∈ R4 , ψ ∈ R4∗ .

(4.2.15)

The operator Ti∗ (t, s) has exponential dichotomies on R+ and R− with the projections
±,∗

Pi

±,∗

(z), Qi

(z) being the adjoint operators of the projections for Ti (t, s).

• Q− (z) are projections on R− 4 , and Pi+ (z) are projections on R+ 4 ,
i
+,∗ 2
)

= Pi

−,∗ 2
)

= Qi

(Pi

(Qi

+,∗

,

(4.2.16)

−,∗

.

(4.2.17)

•
+,∗

∗
(t)Ti∗ (t, s) = Tu (t, s)Pi

−,∗

(t)Ti∗ (t, s) = Ti∗ (t, s)Qi

Pi

Qi

+,∗

t, s ∈ R+ ,

(4.2.18)

−,∗

98

(s),
(s),

t, s ∈ R− .

(4.2.19)

•
+,∗

(s)

≤ C e−ν(t−s) ,

0 ≤ s ≤ t,

(4.2.20)

+,∗

(s)

≤ C e−ν(s−t) ,

0 ≤ t ≤ s,

(4.2.21)

−,∗

(s)

≤ C e−ν(t−s) ,

s ≤ t ≤ 0,

(4.2.22)

−,∗

(s)

≤ C e−ν(s−t) ,

t ≤ s ≤ 0,

(4.2.23)

Ti∗ (t, s) Pi

Ti∗ (t, s) Qi
Ti∗ (t, s) Pi

Ti∗ (t, s) Qi
+,∗

where Qi

+,∗

:= I − Pi

−,∗

and Pi

−,∗

:= I − Qi

.

Let R denote by the range of the operator.
Corollary 1. Under the assumptions (H) and (S’)-(1.1.4) we have the following nondegeneracy condition
R Q− (0) ∩ R Pi+ (0) = span q˙i (0).
i

(4.2.24)

Proof. By theorem 4.1 in [64] and lemma (4.1) we know that the only bounded solution of
(4.2.5) is given by q˙i (z) = (φh , φh , φh , φh )T , up to constant scalar multiples, which proves
the corollary.

From corollary 1 we have

˙
R Q− (0) = span qi (0) ⊕ Yiu ,
i

R Pi+ (0) = span qi (0) ⊕ Yis ,
˙

(4.2.25)

where dim Yiu = dim Yis = 1. Define Yi := Yiu + Yis . Then

R4 = span qi (0) ⊕ Yi ⊕ Zi
˙
99

(4.2.26)

where dim Zi = 1. Finally we are ready to define a cross section Σ1 of the heteroclinic orbit
q1 by
Σi := qi (0) + Yi .

(4.2.27)

+
+
In particular we remark that R P1 (0) = R Q− (0) and R Q− (0) = R P2 (0). Then we
2
1

conclude

q2 (0) = S q1 (0),
˙
˙
Y2s = S Y1u ,

Z2 = S Z1 ,

Y2u = S Y1s .

(4.2.28)
(4.2.29)

+
−
In order to study of the splitting manifolds we look for the solutions qi (z; θ) and qi (z; θ)

defined on R+ and R− respectively satisfying the conditions:
±
• (Q1) qi (z; θ) are close to qi (z).
+
−
• (Q2) limz→∞ q1 (z; θ) = p2 , limz→−∞ q1 (z; θ) = p1 .
+
−
• (Q3) limz→∞ q2 (z; θ) = p1 , limz→−∞ q2 (z; θ) = p2 .
±
• (Q4) qi (0; θ) ∈ Σi .
+
−
∞
• (Q5) ξi ψi := qi (0; θ) − qi (0; θ) ∈ Zi .
−
−
+
+
We are looking for the solutions qi (z) = qi (z) + ri (z) and qi (z) = qi (z) + ri (z) defined
±
on R+ and R− respectively such that qi (z; θ) satisfies conditions (Q1)-(Q4) with the norm
+
−
0
ri ∞ = max{supz∈R+ |ri (z)|, supz∈R− |ri (z)|}. Define Cb+ := {r ∈ C 0 (R+ ) | r ∞ <
0
∞} and Cb− := {r ∈ C 0 (R− ) | r ∞ < ∞}. If ri is finite in norm

· ∞ and close to zero,

±
±
then by the theory of stable and unstable manifolds, qi (z) = qi (z) + ri (z) is on the desired

stable or unstable manifold.
100

Lemma 4.4. (Perturbed Heteroclinic Orbit) Given θ be small enough there is a unique
+
−
pair (qi (z; θ), qi (z; θ)) solutions of (4.1.9) satisfying (Q1)-(Q5). Moreover, the mappings
+
−
0
0
(qi (z; ·), qi (z; ·)) : R → Cb+ × Cb− are smooth.
±
±
Proof. Plug qi (z) = qi (z) + ri (z) into (4.1.9) we obtain

±
±
ri = A± (z)ri + h± (z, ri , θ),
˙±
i
i

(4.2.30)

±
where A± (z) := Dx f (qi (z), 0) and
i

±
±
h± (z, ri , θ) = f (x, θ) − f (qi , 0) − A± (z)ri .
i
i

(4.2.31)

Let’s focus on case +. From the variation of constants formula for (4.2.30) and exponential
dichotomy (4.2.20) we have

+
ri (z) = Ti (t, 0)νi +

z
0

+
Ti (z, s)Pi+ (s)hi (s, ri , θ)ds −

∞
z

+
Ti (z, s)Q+ (s)hi (s, ri , θ)ds,
i

(4.2.32)
+
:= S(ri , νi , θ).

(4.2.33)

+
where νi = Pi+ (0)zi (0). From the definition (4.2.31) of hi and the fact that the orbits of
+
Γi := {qi (t)|t ∈ R} are bounded one can easily show that the RHS of (4.2.32) S(ri , νi , θ)
0
defined for each (νi , θ) ∈ R Pi+ (0) × R is a smooth mapping from Cb+ into itself. So we can
+
0
consider equation (4.2.30) as an equation in Cb+ . Note that for (νi , θ) = (0, 0) ri = 0 is the

101

solution of (4.2.30). From the definition (4.2.31) of hi it is easy to check that

+
Dr+ S(ri , νi , θ) +
= 0.
(ri ,νi ,θ)=(0,0,0)
i

(4.2.34)

Hence by the implicit function theorem we can solve the equation (4.2.32) for (νi , θ) close
enough to (0, 0) which means
+
+
ri (z) = ri (νi , θ)(z),

(4.2.35)

for small νi ∈ R Pi+ (0). We go through the similar proof for −. From the variation of
constants formula and exponential dichotomy (4.2.22) equation (4.2.30) becomes

−
ri (z) = Ti (t, 0)ζi −

0
z

−
Ti (z, s)Q− (s)hi (s, ri (s), θ)ds +
i

z
−∞

−
Ti (z, s)Pi− (s)hi (s, ri , θ)ds,

(4.2.36)

−
where ζi = Q− (0)zi (0). Similarly we can solve (4.2.36) for (ζi , θ) close to (0, 0) which means
i

−
−
ri (z) = ri (ζi , θ)(z),

(4.2.37)

for small ζi ∈ R Q− (0). From (4.2.32) and (4.2.36) we can obtain
i
+
ri (0) = νi −

∞
0

+
Q+ (0)Ti (0, s)hi (s, ri (νi , θ)(s), θ)ds,
i

(4.2.38)

T + (νi ,θ)
−
ri (0) = ζi +

0
−∞

−
Pi− (0)Ti (0, s)hi (s, ri (ζi , θ)(s), θ)ds .
T − (ζi ,θ)

102

(4.2.39)

±
Condition (Q4) indicates that ri (0) ∈ Yi ⊕ Zi . Then it follows
=T + (νi ,θ)
+,y

+
ri (0) =

νi − Ti

∈Yis

+,z

(νi , θ) − Ti

∈Yiu

(νi , θ),

(4.2.40)

∈Z

=T − (ζi ,θ)

−,y

−
ri (0) =

ζi + Ti
∈Yiu

−,z

(ζi , θ) + Ti

∈Yis

(ζi , θ) .

(4.2.41)

∈Z

From (4.2.40), (4.2.41) and Condition (Q5) we obtain

−,y

νi = −Ti

+,y

ζi = Ti

(ξi , θ),

(νi , θ).

(4.2.42)
(4.2.43)

Note that (νi , ζi , θ) = (0, 0, 0) solve (4.2.42) and (4.2.43). From the definition (4.2.31) of hi
it is easy to check that

+,y

Dνi Ti

(νi , θ) (ν ,θ)=(0,0) = 0,
i

−,y

Dζi Ti

(ζi , θ) (ζ ,θ)=(0,0) = 0.
i

(4.2.44)
(4.2.45)

By the implicit function theorem we can solve (4.2.42)-(4.2.43) near (ζi , νi , θ) = (0, 0, 0) and

ζi = ζi (θ), νi = νi (θ).

(4.2.46)

. Combining the results of (4.2.35), (4.2.37), (4.2.46) we obtain

±
±
ri (z) = ri (θ)(z).

103

(4.2.47)

±
±
Hence qi (z) := qi (z) + ri (z) are the unique heteroclinic connections close to qi (z) satisfying

the conditions (Q1)-(Q5).
Remark 4.4. Lemma 4.4 indicates that we can construct the unique perturbed heteroclinic
±
±
orbits qi (z; θ) = qi (z) + ri (z) which comprising the stable and unstable manifolds of pi . In
±
particular the proof of Lemma 4.4 also shows that qi are S symmetric with respect to each
±
±
other, i.e., q2 = S q1 .
+
−
∞
Remark 4.5. From definition we know that ξi (θ) =< ψi , ri (0) − ri (0) >. From (4.2.32)

and (4.2.36) we have

∞
ξi (θ) = < ψi ,

= < ψi ,
=

∞
−∞

∞
0

∞

−∞

+
Q+ (0)Ti (0, s)hi (s, ri , θ)ds +
i

0
−∞

−
Pi− (0)Ti (0, s)hi (s, ri , θ)ds >,

Ti (0, s)hi (s, ri , θ) > ds,

< ψi (s), hi (s, ri , θ) > ds.

(4.2.48)

where
ψi (s) := Ti (0, s)∗ ψi ,

(4.2.49)

∞
and Ti (0, s)∗ denotes by the transpose of Ti (0, s). Note that ξi (θ = 0) = 0 and

∞
Dθ ξi θ=0 =

=

∞
−∞
∞
−∞

< ψi (s), Dθ hi (s, 0, 0) > ds,
< ψi (s), Dθ f (qi (s), 0) > ds.

(4.2.50)

By Lemma 4.1 we know that 0 is the simple eigenvalue of Lh with corresponding eigenvector
ˆ
ˆ
φh . It follows that there doesn’t exist φ such that Lh φ = φh , which implies that φh ⊥

104

†

N ull(Lh ). So
< φh , ψ † >= 0,

(4.2.51)

†

where ψ † spans the null space of Lh . By rewriting in vector forms, it is easy to check that
(4.2.51) is equivalent to
∞
−∞

< ψi (s), Dθ f (qi (s), 0) > ds = 0.

(4.2.52)

By the implicit function theorem

∞
ξi (θ) = Mi θ + O(θ2 ),

(4.2.53)

where
Mi :=

∞
−∞

< ψi (s), Dθ f (qi (s), 0) > ds.

(4.2.54)

±
Now we are ready to construct the Lin’s orbit x± (z) which are the perturbations of qi (z)
i

and could be discontinuous on the cross sections Σi . To be precise, we look for the solutions
of (4.1.9) which have the form

±
±
x± (z) = qi (z; θ) + vi (z).
i

(4.2.55)

Then plug into (4.1.9) and we have

±
±
vi = A± (z; θ)vi + h± (z, vi , θ),
˙±
i
i

105

(4.2.56)

±
where A± (z; θ) := Dx f (qi (z; θ), θ) and
i

±
±
±
h± (z, vi , θ) = f (x, θ) − f (qi , θ) − A± (z; θ)vi .
i
i

(4.2.57)

Moreover we also want the solutions x± (0) be on the cross sections Σi which implies that
i
±
±
vi (0) ∈ Σi since Lemma 4.4 indicates qi (0) ∈ Σi .

Let ω1 , ω2 ∈ R+ . Define
Xω :=

−
+
− + − +
v = (v1 , v1 , v2 , v2 ) v1 ∈ C 0 ([−ω1 , 0], R4 ), v1 ∈ C 0 ([0, ω2 ], R4 ),
−
+
v2 ∈ C 0 ([−ω2 , 0], R4 ), v1 ∈ C 0 ([0, ω1 ], R4 ) .

(4.2.58)

Here is the main theorem in this section:
Theorem 4.1. There exists constants ω0 > 0 and c0 > 0. Then for any ωi > ω0 , |θ| < c0 ,
there exists a unique solution v = v(ω, θ) ∈ Xω of (4.2.56) satisfying
±
• (L1) vi (0) ∈ Yi ⊕ Zi ,
+
−
• (L2) vi (0) − vi (0) ∈ Zi ,
+
−
+
−
• (L3) v2 (ω1 ) − v1 (−ω1 ) = d1 and v1 (ω2 ) − v2 (−ω2 ) = d2 where

−
+
d1 := q1 (θ)(−ω1 ) − q2 (θ)(ω1 ),

(4.2.59)

−
+
d2 := q2 (θ)(−ω2 ) − q1 (θ)(ω2 ).

(4.2.60)

From Lemma 4.4 and Theorem 4.1 we can construct Lin’s orbit.

106

Corollary 2. (Lin’s Orbit) There exists constants ω0 > 0 and c0 > 0. Then for any
ωi > ω0 , |θ| < c0 , there exists a unique solution x = x(ω, θ) ∈ Xω of (4.1.9) satisfying
• x± (0) ∈ Yi ⊕ Zi ,
i
• x+ (0) − x− (0) ∈ Zi ,
i
i
• x− (−ω1 ) = x+ (ω1 ) and x+ (ω2 ) = x− (−ω2 ).
1
2
1
2
Remark 4.6. Since the equation (4.1.9) is S-reversible, x(z) := S x(ω, θ)(−z) is also a
¯
solution of (4.1.9) satisfying
• x± (0) ∈ S Yi ⊕ S Zi ,
¯i
• x+ (0) − x− (0) ∈ S Zi ,
¯i
¯i
• x− (−ω1 ) = x+ (ω1 ) and x+ (ω2 ) = x− (−ω2 ).
¯1
¯2
¯1
¯2
Note that S Y1 = Y2 and S Z1 = Z2 . So x(ω, θ) is another Lin’s orbit corresponding to the
¯
same θ and ω. By the uniqueness of Lin’s orbit from Corollary 2 we conclude that the orbit
of x(ω, θ) is the same with that of x(ω, θ). So
¯

x− (z) = S x+ (−z),
2
1

(4.2.61)

x+ (z) = S x− (−z).
1
2

(4.2.62)

From the definition of symmetry of S-reversible system we obtain that Lin’s orbit x(ω, θ) is
symmetric. So we have the following lemma:
Lemma 4.5. (Symmetry) Lin’s orbit x(ω, θ) in Corollary 2 is symmetric and

ξ1 = S ξ2 ,
107

(4.2.63)

where

ξ1 (0) := x− (0) − x+ (0),
1
1

(4.2.64)

ξ2 (0) := x− (0) − x+ (0).
2
2

(4.2.65)

Remark 4.7. ξi are the jumps of Lin’s orbit. In order to construct the homoclinic orbit we
need to merge the jumps ξi = 0. From (4.2.63) it is sufficient to make ξ1 to be zero.
Now we will illustrate the main steps of proof for Theorem 4.1. Note that (4.2.56) is a
nonlinear differential equation. We will prove this theorem in several steps. Before dealing
with this nonlinear equation, we consider the linear version first,

±
±
±
v˙ = A± (z; θ)vi + gi (z),
i
i

(4.2.66)

− + − +
where g := (g1 , g1 , g2 , g2 ) ∈ Xω . Let T (·, ·; θ) be the fundamental transition operator of

the homogeneous equation (4.2.66)

±
±
v˙ = A± (z; θ)vi .
i
i

108

(4.2.67)

As before these equations also have exponential dichotomies on R± with projections Pi± (·; θ).
This means that there exists constants C > 0 and ν > 0 such that

Ti (t, s; θ) Pi+ (s; θ)

≤ C e−ν(t−s) ,

0 ≤ s ≤ t,

(4.2.68)

Ti (t, s; θ) Q+ (s; θ)
i

≤ C e−ν(s−t) ,

0 ≤ t ≤ s,

(4.2.69)

Ti (t, s; θ) Pi− (s; θ)

≤ C e−ν(t−s) ,

t ≤ s ≤ 0,

(4.2.70)

Ti (t, s; θ) Q− (s; θ)
i

≤ C e−ν(s−t) ,

t ≤ s ≤ 0,

(4.2.71)

where Qi ± := I − Pi± .
The first step is to construct the ‘Lin’s orbit’ for the linear inhomogeneous equation
(4.2.66). Consider the case i = 1 first. For the brevity of the notation we omit θ in T1 (z, s; θ)
±
and P1 (z; θ) and Q± (z; θ) in the following Lemma.
1
−
Lemma 4.6. For any ωi ∈ R+ , a1 = (a− , a+ ) ∈ R P1 (−ω1 ) × R Q+ (ω2 ) and g1 =
1 1
1
− +
±
(g1 , g1 ) ∈ C 0 ([−ω1 , 0], R4 )×C 0 ([0, ω2 ], R4 ), there exists a solution v1 = v1 (z; ω1 , ω2 , a± , g1 , θ)
1
−
+
with v1 ∈ C 0 ([−ω1 , 0], R4 ) and v1 ∈ C 0 ([0, ω2 ], R4 ) satisfying (4.2.66)

±
±
v˙1 ± = A± (z; θ)v1 + g1 (z),
1

(4.2.72)

±
v1 (0) ∈ Z1 ⊕ Y1 ,

(4.2.73)

with the jump conditions

+
−
v1 (0) − v1 (0) ∈ Z1 .

109

(4.2.74)

Moreover

≤ C(|a− | + |a+ | + g1 ),
1
1

v1

(4.2.75)

−
+
+
|Q− (−ω1 )v1 (−ω1 )| + |P1 (ω2 )v1 (ω2 )| ≤ C( g1 +,
1

+e−2ν(ω1 +ω2 ) (|a− | + |a+ |)). (4.2.76)
1
1

where

·

denotes the sup norm.

Proof. From the variation of constant formula we can obtain

−
v1 (z) =

+
v1 (z) =

z
−ω1
z
0

0

−
−
T1 (z, s)P1 (s)g1 (s)ds −

z

−
T1 (z, s)Q− (s)g1 (s)ds +
1

+T (z, 0)b− + T1 (z, −ω1 )a− ,
1
1

+
T1 (z, s)Pi+ (s)g1 (s)ds −

+T (z, 0)b+
1

ω2

z

−ω1 < z ≤ 0;(4.2.77)

+
T1 (z, s)Q+ (s)g1 (s)ds +
1

+ T1 (z, ω2 )a+ ,
1

0 ≤ z ≤ ω2 ,

(4.2.78)

+
−
where b− = R Q− (0), b+ = R P1 (0), a− = R P1 (−ω1 ) and a+ = R Q+ (ω2 ). Then
1
1
1
1
1
1

−
v1 (0)

=

:=

+
v1 (0) =

:=

0
−
− + T (0, −ω )a− + P − (0)
T1 (0, s)g1 (s)ds,
b1
1
1 1
1
−ω1
s
u
∈Y1 ⊕Z1
∈Y1
s
∈Y1 ⊕Z1
−
−
b− + Hs (ω1 , a− , g1 , θ) + Hz (ω1 , a− , g1 , θ) .
1
1
1
u
s
∈Z1
Y1
∈Y1
ω2
+
b+ + T1 (0, ω2 )a+ − Q+ (0)
T1 (0, s)g1 (s)ds,
1
1
1
0
s
u
∈Y1
∈Y1 ⊕Z1
u
∈Y1 ⊕Z1
+
+
b+ + Gs (ω2 , a+ , g1 , θ) + Gz (ω2 , a+ , g1 , θ) .
1
1
1
s
u
∈Z1
Y1
∈Y1

110

(4.2.79)

(4.2.80)

From Condition (4.2.74) we can solve that

+
b− = Gs (ω2 , a− , g1 , θ),
1
1

(4.2.81)

−
b+ = Hs (ω1 , a+ , g1 , θ).
1
1

(4.2.82)

So combining with (4.2.77) and (4.2.78) we conclude that

±
±
− +
v1 (z) = v1 (ω1 , ω2 , g1 , g1 , θ)(z),

(4.2.83)

−
+
and it is easy to see that v1 ∈ C 0 ([−ω1 , 0], R4 ) and v1 ∈ C 0 ([0, ω2 ], R4 ). From (4.2.81) and

the definition of Gs we obtain

+
|b− | = |Gs (ω2 , a− , g1 , θ)|,
1
1

≤ C( T1 (0, ω2 )a+ +
1

ω2
0

+
Q+ (0)T1 (0, s)g1 (s) ds).
1

(4.2.84)

Using the exponential dichotomies properties (4.2.68), (4.2.69) it yields

+
|b− | ≤ Ce−νω2 |a+ | + C g1
1
1

ω2
0

e−νs ds,

+
≤ C(e−νω2 |a+ | + g1 ).
1

(4.2.85)

From (4.2.77) and using the exponential dichotomies properties (4.2.70), (4.2.71) we obtain

−
v1

−
≤ C g1

z
−ω1

−
e−ν(z−s) ds + g1

0

eν(z−s) ds+,

z

+C|b− | + C|a− |,
1
1

−
≤ C( g1 + |b− | + |a− |).
1
1

111

(4.2.86)

Combining (4.2.85) and (4.2.86) it yields

−
−
+
v1 ≤ C(e−νω2 |a+ | + |a− | + g1 + g1 ).
1
1

(4.2.87)

Similarly we can also prove

+
−
+
v1 ≤ C(e−νω1 |a− | + |a+ | + g1 + g1 ).
1
1

(4.2.88)

Combining (4.2.87) and (4.2.88) we recover the estimate (4.2.95). From (4.2.77) we have

−
Q− (−ω1 )v1 (−ω1 ) = −
1

0
−ω1

−
T1 (0, s)Q− (s)g1 (s)ds + b− .
1
1

(4.2.89)

From (4.2.68) and (4.2.85) we obtain

−
−
|Q− (−ω1 )v1 (−ω1 )| ≤ C g1
1

≤

0

eνs ds + e−νω1 |b− |,
1

−ω1
C(e−ν(ω1 +ω2 ) |a+ | +
1

−
+
g1 + g1 )

(4.2.90)

Similarly we can also prove that

−
+
+
+
|P1 (ω2 )v1 (ω2 )| ≤ C(e−ν(ω1 +ω2 ) |a− | + g1 + g1 ).
1

(4.2.91)

(4.2.90) and (4.2.91) recovers the estimate (4.2.76).

±
− +
Remark 4.8. During the proof of Lemma 4.6 it is easy to see that v1 (ω1 , ω2 , a− , a+ , g1 , g1 , θ)(z)
1 1

depends on (a− , a+ , h− , h+ ) linearly.
1 1
1
1
112

Similarly we can also prove it for the case i = 2.
−
Lemma 4.7. For any ωi ∈ R+ , a2 = (a− , a+ ) ∈ R P2 (−ω2 ) × R Q+ (ω1 ) and g2 =
2
2 2
− +
±
(g2 , g2 ) ∈ C 0 ([−ω2 , 0], R4 )×C 0 ([0, ω1 ], R4 ), there exists a solution v2 = v2 (z; ω1 , ω2 , a± , g2 , θ)
2
−
+
with v2 ∈ C 0 ([−ω2 , 0], R4 ) and v2 ∈ C 0 ([0, ω1 ], R4 ) satisfying (4.2.66)
±
±
v˙ = A± (z; θ)v2 + g2 (z),
2
2

(4.2.92)

with the jump conditions

±
v2 (0) ∈ Z2 ⊕ Y2 ,
+
−
v2 (0) − v2 (0) ∈ Z2 .

(4.2.93)
(4.2.94)

Moreover

v2

≤ C(|a− | + |a+ | + g2 ),
2
2

(4.2.95)

−
+
+
|Q− (−ω2 )v2 (−ω2 )| + |P2 (ω1 )v2 (ω1 )| ≤ C( g2 +,
2

+e−2ν(ω1 +ω2 ) (|a− | + |a+ |)). (4.2.96)
2
2

where

·

denotes the sup norm.

Remark 4.9. Similarly during the proof of Lemma 4.7 it is easy to see that the solutions
±
− +
v2 (ω1 , ω2 , a− , a+ , g2 , g2 , θ)(z) depends on (a− , a+ , h− , h+ ) linearly.
2 2
2 2
2
2

The next step is to couple the two solutions

±
±
− +
xi (ω1 , ω2 )(z) := qi (z; θ) + vi (ω1 , ω2 , a− , a+ , gi , gi )(z).
i
i

113

(4.2.97)

near the equilibrium point q1 . For the preparation for the next step we need the following
technical lemma [61].
Lemma 4.8. Suppose the fixed point qi has the leading stable eigenvalue λs (θ) and the leading
i
ss
s
u
unstable eigenvalue λu (θ). Then we introduce constants αi , αi , αi such that
i

u
s
ss
{λss (θ)} < αi < λs (θ) < αi < 0 < αi < λu (θ) < {λuu (θ)},
i
i
i
i

(4.2.98)

where {λss (θ)} and {λuu (θ)} denotes the strong stable and strong unstable eigenvalues of the
i
fixed point respectively.There exists constants ω0 > 0, c0 > 0 and M > 0 such that for any
ωi ≥ ω0 and |θ| < c0
−
R4 = R Q+ (ω2 ; θ) ⊕ R P2 (−ω2 ; θ),
1

(4.2.99)

−
R4 = R Q+ (ω1 ; θ) ⊕ R P1 (−ω1 ; θ).
2

(4.2.100)

Furthermore

¯
Qi,ωi (θ)

≤ M,

(4.2.101)

¯
Pi,ωi (θ)

≤ M,

(4.2.102)

¯
¯
¯
where Pi,ωi (θ) denotes the projection on R Pi− (−ωi ; θ) and Qi,ωi (θ) := I − Pi,ωi (θ).
s u

¯
Qi,ω − Qi

≤ Ce− min{−αi ,αi }ωi .

¯
Pi,ω − Pi

≤ Ce− min{−αi ,αi }ωi .

s u

Now we are ready to couple the two solutions xi :
114

(4.2.103)
(4.2.104)

Lemma 4.9. (Lin’s orbit for linear equation)There exists constants ω0 > 0 and c0 > 0.
Then for any ωi > ω0 , |θ| < c0 , g ∈ Xw and d = (d1 , d2 ) ∈ R2 , then there exist a unique
− + − +
solution v(z) = v(ω1 , ω2 , d, g, θ)(z) with v = (v1 , v1 , v2 , v2 ) ∈ Xω satisfying the equation

(4.2.66),

±
±
vi ± = A± (z; θ)vi + gi (z),
˙
i

with the jump conditions

±
vi (0) ∈ Zi ⊕ Yi ,

(4.2.105)

+
−
vi (0) − vi (0) ∈ Zi ,

(4.2.106)

+
−
v1 (ω2 ) − v2 (−ω2 ) = d1 ,

(4.2.107)

+
−
v2 (ω1 ) − v1 (−ω1 ) = d2 .

(4.2.108)

and the boundary condition

And
±
vi ≤ C(|d| + g ).

(4.2.109)

±
±
Proof. From Lemma 4.6 and Lemma 4.7 we have the solution vi (z) = v(ω1 , ω2 , a± , gi , θ)(z)
i

of (4.2.66) denoted by vi . In particular Lemma 4.6 and Lemma 4.7 also indicate that vi
¯±
¯±

115

satisfy the two jump conditions (4.2.105) and (4.2.106). From (4.2.107) we can obtain

+
a+ − a− = d1 + Q− (−ω2 ; θ)¯2 (−ω2 ) − P1 (ω2 ; θ)¯1 (ω2 ),
v−
v+
1
2
2

a− − a+ =
1
2

+
:=b− ∈R Q− (−ω2 ;θ)
:=b+ ∈R P1 (ω2 ;θ)
2
1
2
+
d2 + P2 (ω1 ; θ)¯2 (ω1 ) − Q− (−ω1 ; θ)¯1 (−ω1 ) .
v
v−
1
+
:=b+ ∈R P2 (ω1 ;θ)
2

(4.2.110)

(4.2.111)

:=b− ∈R Q− (−ω1 ;θ)
1
1

Applying projections Pi,ωi (θ) and Qi,ωi (θ) defined in Lemma 4.8 we obtain
a− = P1,ω1 (d2 + b+ − b− ),
1
2
1

(4.2.112)

a+ = Q2,ω2 (d1 + b− − b+ ),
1
2
1

(4.2.113)

a− = P2,ω2 (−d1 − b− + b+ ),
2
2
1

(4.2.114)

a+ = Q1,ω1 (−d2 − b+ + b− ).
2
2
1

(4.2.115)

− +
From the remarks 4.8, 4.9 we know that b± are linear in (a− , a+ , gi , gi ) . Then we may
i
i
i

write
a = L1 (θ)a + L2 (θ)g + L3 (θ)d,

(4.2.116)

− + − +
where a = (a− , a+ , a− , a+ ), g = (g1 , g1 , g2 , g2 ) and d = (d2 , d1 , d1 , d2 ). Here Li (θ) are
1 1 2 2

linear operators smoothly depending on θ. From (4.2.76) we know that if ωi be sufficiently
large then L1 (θ) < 1. So the operator I − L1 (θ) is invertible and we could uniquely solve
±
(4.2.116) and a = a(g, θ, d). Hence vi (z) = vi (ω1 , ω2 , a± (ω1 , ω2 , g, d, θ), g, θ)(z) solve the
¯±
i

equation (4.2.66) and it is clearly in Xω . We denote these solutions by vi . The estimate
ˆ±
(4.2.109) can be obtained from (4.2.116) and (4.2.95).

116

Now we are ready to construct Lin’s orbits for the nonlinear equation (4.2.56)

±
±
vi = A± (z; θ)vi + h± (z, vi , θ),
˙±
i
i

±
where A± (z; θ) := Dx f (qi (z; θ), θ) and
i

±
±
±
h± (z, vi , θ) = f (x, θ) − f (qi , θ) − A± (z; θ)vi .
i
i

±
We also require the solutions vi satisfy the conditions (L1)-(L3).

Proof of Theorem 4.1. From Lemma 4.9 it follows that there exist solutions

v

= v (ω, h(v, θ), d(ω, θ)),
ˆ

(4.2.117)

:= S(v, ω, θ).

(4.2.118)

− + − +
satisfying the equations (4.2.56) and the conditions (L1)-(L2). Here v = (v1 , v1 , v2 , v2 ),

h = (h− , h+ , h− , h+ ), ω = (ω1 , ω2 ), and d = (d1 , d2 ). This is a fixed point problem and it
1
1
2
2
can be solved by contraction mapping theorem. Denote the RHS of (4.2.117) by S and we
will show that S(v, ω, θ) is a contraction map on v from Br → Br in the space Xω . Here
Br denotes the ball centered at 0 with radius r.

117

From the definition (4.2.57) of h(v, θ) it follows that |h(v, θ)| ≤ C(v 2 + |θ| + v|θ|),
|Dv h(z, v, θ)| ≤ C(v + |θ|) for |v| ≤ r and |θ| ≤ c0 , then combining with (4.2.109) it yields
S(v, h, θ) ω ≤ C ( h(v, θ) ω + |d|) ,
≤ C

v 2 + |d| + |θ| ,
ω

≤ C r2 + c0 + |d| .

(4.2.119)
(4.2.120)
(4.2.121)

±
From the definition (4.2.59) (4.2.60) of d and the conditions (Q2) and (Q3) of qi we can

choose ωi be sufficiently large so that |d| ≤ c0 . So
S(v, h, θ) ω ≤ C(r2 + c0 ).

(4.2.122)

We can also choose r be small enough and c0 < r so that

S(v, h, θ) ω ≤ r.

(4.2.123)

Moreover, from (4.3.39) and the definition (4.2.57) of h we obtain

Dv S(v, h, θ) ω ≤ C( Dv h(v, θ) ω ),

(4.2.124)

≤ C( v ω + |d|),

(4.2.125)

≤ C r.

(4.2.126)

We choose r small enough such that

Dv S(v, h, θ) ω < 1.
118

(4.2.127)

Applying the Banach fixed point theorem we can deduce the existence and uniqueness of the
solution v(ω, θ).

4.3

Estimates for the Jump

By Corollary 2 we already construct ‘Lin’s orbit’. However, ‘Lin’s orbit’ is actually not the
homoclinic orbit that we are looking for since ‘Lin’s orbit’ has the jumps ξi at Σi . If we
could merge those jumps, i.e., ξi = 0, then we can construct the expected homoclinic orbit.
Moreover, from remark 4.7 it is sufficient to merge only one of those jumps to zero.
Theorem 4.2. Under the assumptions (H) and (S’)-(1.1.4) there exists constants ω0 > 0
and c0 > 0. Then for any ωi > ω0 , |θ| < c0 , there exists a unique solution v = v(ω, θ) ∈ Xω
of (4.2.56) satisfying (L1)-(L3). The jump ξi can be written as
s

u

ξ1 (ω, θ) = M1 θ + cs (θ)e2ω1 λ1 + cu (θ)e−2ω2 λ2 +
s

u

+o(e−2ω2 λ2 ) + o(e−2ω1 λ1 ).

(4.3.1)

+
−
where ξ1 =< ψ1 , v1 (0) − v1 (0) > and M1 = R < ψ1 (s), Dθ f (q1 (s), 0) > ds = 0. Here λs
i

and λu are the leading stable and unstable eigenvalues at equilibrium point pi . The functions
i
cu (θ) and cs (θ) are smooth in θ.

119

Proof. From the definition of ξ1 then we have

−
ξ1 = < ψ1 , Q+ (0; θ)x+ (0) − P1 (0; θ)x− (0) >,
1
1
1
+
−
−
= < ψ1 , Q+ (0; θ)q1 (θ)(0) − P1 (0; θ)q2 (θ)(0) > +
1
∞
=ξ1

+
−
−
+ < ψ1 , Q+ (0; θ)v1 (θ)(0) − P1 (0; θ)v2 (θ)(0) > .
1

(4.3.2)

ω
:=ξ1

From (4.2.53) we have
∞
ξ1 = M1 θ.

(4.3.3)

ω
Addressing the first term of ξ1 , from (4.2.80) we obtain

+
v1 (0) = T1 (0, ω2 ; θ)a+ −
1

ω2
0

+
T1 (0, s; θ)Q+ (s; θ)h+ (s, v1 (s; ω, θ), θ)ds.
1
1

(4.3.4)

From (4.2.110) and (4.2.60) it yields

−
+
a+ − a− = q2 (θ)(−ω2 ) − q1 (θ)(ω2 ) +
1
2
−
+
+
+Q− (−ω2 ; θ)v2 (−ω2 ) − P1 (ω2 ; θ)v1 (ω2 ).
2

(4.3.5)

So

−
+
¯
a+ = Q2,ω2 (θ) q2 (θ)(−ω2 ) − q1 (θ)(ω2 )+
1
+
−
+
+Q− (−ω2 ; θ)v2 (−ω2 ) − P1 (ω2 ; θ)v1 (ω2 ) .
2

120

(4.3.6)

Plugging into (4.3.4) it yields

+
−
+
¯
Q+ (0; θ)v1 (0) = T1 (0, ω2 ; θ)Q+ (ω2 ; θ)Q2,ω2 (θ) q2 (θ)(−ω2 ) − q1 (θ)(ω2 )+
1
1
−
+
+
+Q− (−ω2 ; θ)v2 (−ω2 ) − P1 (ω2 ; θ)v1 (ω2 ) +
2

−

ω2

0

+
T1 (0, s; θ)Q+ (s; θ)h+ (s, v1 (s; ω, θ), θ)ds
1
1

(4.3.7)

−
−
Similarly we can also compute the expression for P1 (0; θ)v1 (0)

−
−
−
−
+
¯
P1 (0; θ)v1 (0) = T1 (0, −ω1 ; θ)P1 (−ω1 ; θ)P1,ω1 (θ) q1 (θ)(−ω1 ) − q2 (θ)(ω1 )+
+
+
−
+P2 (ω1 ; θ)v2 (ω1 ) − Q− (−ω1 ; θ)v1 (−ω1 ) +
1
0

+
−ω1

−
−
T1 (0, s; θ)P1 (s; θ)h− (s, v1 (s; ω, θ), θ)ds.
1

(4.3.8)

ω
From (4.3.7) and (4.3.8) we can obtain the expression for ξ1

+,∗
−
+
ω
∗
¯
ξ1 = < T1 (0, ω2 ; θ)Q1 (0; θ)ψ1 , Q2,ω2 (θ) q2 (θ)(−ω2 ) − q1 (θ)(ω2 ) > +

+<

H1
+,∗
∗
¯
T1 (0, ω2 ; θ)Q1 (0; θ)ψ1 , Q2,ω2 (θ)

− < ψ1 ,

ω2
0

−
+
+
Q− (−ω2 ; θ)v2 (−ω2 ) − P1 (ω2 ; θ)v1 (ω2 ) > +
2
H2

+
T1 (0, s; θ)Q+ (s; θ)h+ (s, v1 (s; ω, θ), θ)ds > +
1
1
H3
−,∗

∗
+ < T1 (0, −ω1 ; θ)P1

+<

−
+
¯
(0; θ)ψ1 , P1,ω1 (θ) q1 (θ)(−ω1 ) − q2 (θ)(ω1 ) > +

H4
−,∗
∗
¯
T1 (0, −ω1 ; θ)P1 (0; θ)ψ1 , P1,ω1 (θ)

+
+
−
P2 (ω1 ; θ)v2 (ω1 ) − Q− (−ω1 ; θ)v1 (−ω1 ) > +
1
H5

0

+ < ψ1 ,

−ω1

−
−
T1 (0, s; θ)P1 (s; θ)h− (s, v1 (s; ω, θ), θ)ds >,
1
H6

121

(4.3.9)

where ∗ denotes the adjoint operator with respect to the inner product < ·, · >. In the
following we need to analyze the estimate for each term of (4.3.9). We will show later that
ω
ω
H1 and H4 account for the leading order of ξ1 and the rest terms of ξ1 are higher order

terms. We will show the proof of these estimates later. From (4.3.31), (4.3.30) we have

+,∗

∗
T1 (0, ω2 ; θ)Q1 (0; θ)ψ1 = Ψp1 (ω2 , 0; θ)η + (ψ1 , θ) + h.o.t.
−,∗

∗
T1 (0, −ω1 )P1

(0; θ)ψ1 = Ψp2 (−ω1 , 0; θ)η − (ψ1 , θ) + h.o.t.

(4.3.10)
(4.3.11)

−
−
¯
Q2,ω2 (θ)q2 (θ)(−ω2 ) = Φp1 (−ω2 , 0; θ)η u (q2 (0), θ) + h.o.t.

(4.3.12)

+
+
¯
P1,ω1 (θ)q2 (θ)(ω1 ) = Φp2 (ω1 , 0; θ)η s (q2 (0), θ) + h.o.t.

(4.3.13)

u
where the h.o.t denote the terms of higher order than e−λ ω . Ψpi (t, s; θ) is the transition

˙
matrix of the equation ψ = −(Dx f (pi , θ))∗ ψ and Φpi (t, s; θ) is the transition matrix of the
˙
equation φ = Dx f (pi , θ)φ. η + (ψ1 , θ) (resp. η − (ψ1 , θ)) is the eigenvector of −(Dx f (p1 , θ))∗
(resp. −Dx f (p2 , θ)∗ ) with respect to the eigenvalue −λu (θ) (resp. −λs (θ)). Similarly
−
−
η u (q2 (0), θ) (resp. η s (q2 (0), θ)) is an eigenvector of Dx f (p1 , θ) (resp. Dx f (p2 , θ)) with

respect to the eigenvalue λu (θ) (resp. λs (θ)). Then we have

∗
Ψp1 (ω2 , 0; θ)η + (ψ1 , θ) = Gθ e−Λ ω2 G−1 η + (ψ1 , θ),
θ

(4.3.14)

where Gθ is a nonsingular matrix and −Λ∗ is the block diagonalization of the −(Dx f (p1 , θ))∗ .
We also have

−
−
Φp1 (−ω2 , 0; θ)η u (q2 (0), θ) = (G∗ )−1 e−Λω2 G∗ η u (q2 (0), θ),
θ
θ

122

(4.3.15)

where Λ is the block diagonalization of Dx f (p1 , θ). By construction we know that G−1 η + (ψ1 , θ)
θ
∗
−
and G∗ η u (q2 (0), θ) are in the stable space of e−Λ z and unstable space of eΛz respectively.
θ

Then combining the results of (4.3.10), (4.3.12), (4.3.14) and (4.3.15) we may obtain the
expression for the first term in H1

+,∗
−
∗
¯
< T1 (0, ω2 ; θ)Q1 (0; θ)ψ1 , Q2,ω2 (θ)q2 (θ)(−ω2 ) >
∗
−
= < Gθ e−Λ ω2 G−1 η + (ψ1 , θ), (G∗ )−1 e−Λω2 G∗ η u (q2 (0), θ) > +h.o.t.,
θ
θ
θ
∗
−
= < e−Λ ω2 G−1 η + (ψ1 , θ), e−Λω2 G∗ η u (q2 (0), θ) > +h.o.t.,
θ
θ
u

−
= e−2λ2 (θ)ω2 < G−1 η + (ψ1 , θ), G∗ η u (q2 (0), θ) > +h.o.t.,
θ
θ
u

+
u −
= e−2λ2 (θ)ω2 < η2 (ψ, θ), η2 (q2 (0), θ) > +h.o.t.

(4.3.16)

:=cu (θ)

From (4.3.16) and (4.3.34) we obtain that

u

H1 = e−2λ2 (θ)ω2 cu (θ) + h.o.t.

(4.3.17)

Similarly we can also compute the second term in H4 and obtain
s

H4 = e2λ1 (θ)ω1 cs (θ) + h.o.t.

(4.3.18)

Combining (4.3.17) and (4.3.18) we can recover the estimate (4.3.1).
Before we calculate the estimates for H1-H6. We will introduce two important lemma.
They are concerned with the leading order terms of an orbit of the system x = f (x, θ) that
˙
approaches the equlibrium in the weak stable manifold.

123

Lemma 4.10. Let x = 0 be an asymptotically stable fixed point of a C 2 vector field

x = f (x, θ),
˙

(4.3.19)

where f (·, θ) : Rk → Rk , θ ∈ Rp . Suppose the spectrum of the fixed point be σ(Dx f (0, θ) =
λs (θ) ∪ λss (θ)). Then we introduce constants αss , αs such that
{λss (θ)} < αss < λs (θ) < αs < 0,

(4.3.20)

where {λss (θ)} denotes the strong stable eigenvalues of the fixed point. Let E s (θ), E ss (θ)
be the stable, strong stable subspace of A(θ) respectively. Also Ps (θ) denotes the projection
onto E s (θ) along E ss (θ). Then there exists a constant δ > 0 such that for all solutions x(·)
of (4.3.19) with |x(0)| < δ the limit
η(x(0), θ) := lim Φ(0, z; θ)Ps (θ)x(z)
z→∞

(4.3.21)

exists, where Φ(0, z; θ) is the transition matrix of

x = Dx f (0, θ)x,
˙

(4.3.22)

from z to 0. Furthermore, there exists a constant C such that

ss
s
|x(z) − Φ(z, 0; θ)η(x(0), θ)| ≤ C e− min{|α |,2|α |}z .

(4.3.23)

Remark 4.10. In Lemma 4.10 we assume that 0 is an asymptotically stable fixed point. If 0

124

is a hyperbolic fixed point then Lemma 4.10 describes the behavior of solutions in the stable
manifold. By reversing ‘time’ we can obtain a similar lemma for solutions in the unstable
manifold.
Remark 4.11. Let the assumptions of Lemma 4.10 hold. Then η(x, θ) = 0 if and only if
x ∈ Wθ (0)
/ ss
We will also need the following lemma that make the same assertion as previous lemma
4.10 for non-automomous perturbed linear system.
Lemma 4.11. Let x = 0 be a hyperbolic fixed point of a C 2 vector field

x = A(θ)x + B(z, θ)x,
˙

(4.3.24)

where A(θ), B(·, θ) ∈ GL(k, R), θ ∈ Rp . Suppose the fixed point has the leading stable
eigenvalue λs (θ) and the leading unstable eigenvalue λu (θ). Then we introduce constants
αss , αs , αu such that

{λss (θ)} < αss < λs (θ) < αs < 0 < αu < λu (θ),

(4.3.25)

where {λss (θ)} denotes the strong stable eigenvalues of the fixed point. Let E s (θ), E ss (θ),
E u (θ) be the leading stable, strong stable, leading unstable subspace of A(θ) respectively.
Suppose that there is a constant β < 0 such that |B(z, θ)| < eβz and |αs + β| > |λs (θ)| for
small θ. Also Ps (θ) denotes the projection onto E s (θ) along E ss (θ) ⊕ E u (θ). Then there
exists a constant δ > 0 such that for all solutions x(·) of (4.3.24) with |x(0)| < δ the limit
η(x(0), θ) := lim Φ(0, z; θ)Ps (θ)x(z)
z→∞

125

(4.3.26)

exists, where Φ(0, z; θ) is the transition matrix of

x = A(θ)x,
˙

(4.3.27)

from z to 0. Furthermore, there exists a constant C such that

ss
s
|x(z) − Φ(z, 0; θ)η(x(0), θ)| ≤ C e− min{|α |,|α +β|}z .

(4.3.28)

+,∗

∗
Estimate of T1 (0, ω2 ; θ)Q1 (0; θ)ψ1

+,∗

∗
∗
From the definition of ψ1 we have that T1 (0, ω2 ; θ)Q1 (0; θ)ψ1 = T1 (0, ω2 ; θ)ψ1 solv-

˙
ing ψ = −(A± (z, θ))∗ ψ. This equation has the exponential dichotomies on R− , R+ with
1
±,∗

projections P1

±,∗

(z; θ), Q1 (z; θ). Also

s
{w : sup |Ψ(z, 0; θ)w| < ∞} = (Tq+ (θ)(0) Wθ (p1 ))⊥ ,

(4.3.29)

1

z∈R+

+,∗

s
and R Q1 (z; θ) = (Tq+ (θ)(0) Wθ (p1 ))⊥ . Note that σ(−(Dx f (p1 , θ))∗ ) = −σ(Dx f (p1 , θ)).
1

Then by Lemma 4.11 we obtain

+,∗

∗
∗
T1 (0, ω2 ; θ)Q1 (0; θ)ψ1 = T1 (0, ω2 ; θ)ψ1 ,
uu
u
= Ψp1 (ω2 , 0; θ)η + (ψ1 , θ) + O(e− min{|α1 |,2α1 }ω2 ),

(4.3.30)

˙
where Ψp1 (z, 0; θ) is the transition matrix of the equation ψ = −(Dx f (p1 , θ))∗ ψ and η + (ψ1 , θ)

126

is defined in (4.3.26). Similarly we also have

−,∗

∗
T1 (0, −ω1 ; θ)P1

∗
(0; θ)ψ1 = T1 (0, −ω1 ; θ)ψ1 ,
ss
s
= Ψp2 (−ω1 , 0; θ)η − (ψ1 , θ) + O(e− min{|α2 |,2α2 }ω1 ),

(4.3.31)

˙
where Ψp2 (z, 0; θ) is the transition matrix of the equation ψ = Dx f (p2 , θ)ψ and η + (ψ1 , θ)
is defined in (4.3.26).
−
+
¯
Estimate of Q2,ω2 (θ) q2 (θ)(−ω2 ) − q1 (θ)(ω2 )

From Lemma 4.10 we obtain

+
s +
q1 (θ)(z) = Φp1 (z, 0; θ)η1 (q1 (θ)(0), θ) +
ss
s
+O(e− min{|α1 |,2|α1 |}z ),

(4.3.32)

−
s −
q2 (θ)(−z) = Φp1 (−z, 0; θ)η1 (q2 (θ)(0), θ) +
uu
u
+O(e− min{|α1 |,2|α1 |}z ),

(4.3.33)

s +
where Φpi (θ, ·, ·) is the transition matrix of the equation x = Dx f (pi , θ)x and η1 (q1 (θ)(0), θ)
˙
u −
(resp. η1 (q2 (θ)(0), θ))is in the leading unstable subspace of Dx f (p1 , θ). Combining the

results of (4.3.32), (4.3.33) and (4.2.103) it yields

s u
ss
s
+
¯
Q2,ω2 (q1 (θ)(ω2 )) = O(e−(min{−α1 ,α1 }+min{|α1 |,2|α1 |})ω2 ),

(4.3.34)

−
u −
¯
Q2,ω2 (q2 (θ)(−ω2 )) = Φp1 (−ω2 , 0; θ)η1 (q2 (θ)(0), θ) +
s u
ss
s
+O(e−(min{−α1 ,α1 }+min{|α1 |,2|α1 |})ω2 ).

127

(4.3.35)

Similarly we also have

+
s +
¯
P1,ω1 (q2 (θ)(ω1 )) = Φp2 (−ω1 , 0; θ)η2 (q2 (θ)(0), θ) +
s u
ss
s
+O(e−(min{−α2 ,α2 }+min{|α2 |,2|α2 |})ω1 ),
s u
ss
s
−
¯
P1,ω1 (q1 (θ)(−ω1 )) = O(e−(min{−α2 ,α2 }+min{|α2 |,2|α2 |})ω1 ).

(4.3.36)
(4.3.37)

−
+
+
¯
Estimate of Q2,ω2 (θ) Q− (−ω2 ; θ)v2 (−ω2 ) − P1 (ω2 ; θ)v1 (ω2 )
2

From (4.2.76), (4.2.96) and (4.2.116) we obtain

u

s

−
+
+
|Q− (−ω2 ; θ)v2 (−ω2 )| + |P1 (ω2 ; θ)v1 (ω2 )| ≤ C e−2 min{α1 ,−α1 }ω2 |a|+
2

+ h+ + h−
1
2
u

,
s

≤ C e−2 min{α1 ,−α1 }ω2 d +
+ h+ + h−
1
2

.
(4.3.38)

±
From the definition we know that h± = O( vi 2 ) and hence for v small enough, from
i

(4.2.109) we obtain
±
v i ≤ C di .

(4.3.39)

Using (4.3.32) and (4.3.33) we have

s u

di = O(e− min{−α1 ,α1 }ωi ).

128

(4.3.40)

So
s u

±
vi = O(e− min{−α1 ,α1 }ωi ),

(4.3.41)

and
s u

h± = O(e−2 min{−α1 ,α1 }ωi ).
i

(4.3.42)

Combining with (4.3.38) we obtain

s u

−
+
+
|Q− (−ω2 ; θ)v2 (−ω2 )| + |P1 (ω2 ; θ)v1 (ω2 )| = O(e−2 min{−α1 ,α1 }ωi ).
2

(4.3.43)

Applying (4.2.101) it yield

−
+
+
¯
Q2,ω2 (θ) Q− (−ω2 ; θ)v2 (−ω2 ) − P1 (ω2 ; θ)v1 (ω2 )
2

s u

= O(e−2 min{−α1 ,α1 }ωi ).

(4.3.44)

Similarly we could also have

+
+
−
¯
P1,ω1 (θ) P2 (ω1 ; θ)v2 (ω1 ) − Q− (−ω1 ; θ)v1 (−ω1 )
1

s u

= O(e−2 min{−α2 ,α2 }ωi ).

(4.3.45)

ω

+
Estimate of 0 2 T1 (0, s; θ)Q+ (s; θ)h+ (s, v1 (s; ω, θ), θ)ds
1
1

+,u
+,s
+
+
Decompose v1 into v1 = v1 + v1 , where

+,s

+
+
:= P1 (z; θ)v1 (z),

(4.3.46)

+,u

+
:= Q+ (z; θ)v1 (z).
1

(4.3.47)

v1
v1

129

Following the similar proof in, we have

+,u

+
|Q+ (z; θ)h+ (z, v1 (z; ω, θ))| ≤ C|v1
1
1

+,s

+,u

(z)| |v1 (z)| + |v1

(z)| .

(4.3.48)

Exploiting the properties of exponential dichotomies and (4.3.48) it yields
:=H7
ω1
0

+
T1 (0, s; θ)Q+ (s; θ)h+ (s, v1 (s), θ)ds
1
1
u

≤ Cω2 e−α1 ω2
≤

u
Cω2 e−α1 ω2

u

+
sup {eα1 (ω2 −s) Q+ (z; θ)h+ (z, v1 (z; ω, θ)) },
1
1

s∈[0,ω2 ]
u
+,u
+
v1
sup {eα1 (ω2 −s) |v1 (s)|}.
s∈[0,ω2 ]

(4.3.49)

u
Choose δ < 0 such that δ + α1 > 0 and ω2 so large that eδα2 ω2 < 1. Then

u

+
H7 ≤ Ce−(δ+α1 )ω2 v1

u

+,u

sup {eα1 (ω2 −s) |v1

(s)|}.

(4.3.50)

s∈[0,ω2 ]

From (4.3.41) it follows

u

s u

H7 ≤ Ce−(δ+α1 )ω2 e− min{−α1 ,α1 }ω2

u

+,u

sup {eα1 (ω2 −s) |v1

(s)|} .

(4.3.51)

s∈[0,ω2 ]

:=H8

Addressing H8 term we want to obtain the estimate for it. From (4.2.78) it yields

+,u

v1

(s) = −

ω2
s

+
T1 (z, τ )Q+ (τ ; θ)h+ (τ, v1 (τ ), θ)dτ + T1 (z, ω2 )Q+ (ω2 ; θ)a+ . (4.3.52)
1
1
1
1

130

From the property of exponential dichotomies and (4.3.48) we obtain

+,u

|v1

u

(s)| ≤ Ce−α1 (ω2 −s) |a+ | +
1

ω2
u
+,u
e−α1 (τ −s) |v1 (τ )|

+C

s
u
−α1 (ω2 −s)

≤ Ce

+Cω2

+,s

+,u

|v1 (τ ) + v1

(τ )| dτ,

|a+ | +
1
u

+,u

e−α1 (τ −s) |v1

sup
τ ∈[s,ω2 ]

+,s

+,u

(τ )| |v1 (τ ) + v1

(τ )|

.(4.3.53)

Then

H8 ≤ C|a+ | + Kω2
1

sup
τ ∈[0,ω2 ]
+,u

|v1

sup
τ ∈[0,ω2 ]

u

+,u

e−α1 (τ −ω2 ) |v1

(τ )| ·

+,s

(τ )| + |v1 (τ )| .

(4.3.54)

From (4.3.41) we choose ω2 be sufficiently large such that

sup
τ ∈[0,ω2 ]

+,u

|v1

+,s

(τ )| + |v1 (τ )| <

1
.
2C

(4.3.55)

Applying (4.3.55) to (4.3.54) we obtain

H8 ≤ 2C|a+ |.
1

(4.3.56)

Combining the results of (4.2.116), (4.3.40) and (4.3.42) we conclude

|a+ | ≤ C( d + h ),
1
s u

≤ Ce− min{−α1 ,α1 }ω2 .

131

(4.3.57)

Then
s u

H8 ≤ Ce− min{−α1 ,α1 }ω2 .

(4.3.58)

Plugging (4.3.58) into (4.3.51) we obtain

s u

H7 ≤ Ce−2 min{−α1 ,α1 }ω2 .

(4.3.59)

Similarly we could also get the estimate
0
−ω1

s u

−
−
T1 (0, s; θ)P1 (s; θ)h− (s, v1 (s), θ)ds ≤ Ce−2 min{−α1 ,α1 }ω2 .
1

(4.3.60)

Let’s summarize what we have found in the last few sections.
•
uu
u
+,∗
∗
T1 (0, ω2 ; θ)Q1 (0; θ)ψ1 = Ψp1 (ω2 , 0; θ)η + (ψ1 , θ) + O(e− min{|α1 |,2α1 }ω2 ), (4.3.61)

•
−
+
¯
Q2,ω2 (θ) q2 (θ)(−ω2 ) − q1 (θ)(ω2 )

u −
= Φp1 (−ω2 , 0; θ)η1 (q2 (θ)(0), θ) +
s u
ss
s
+O(e−(min{−α1 ,α1 }+min{|α1 |,2|α1 |})ω2 ),

(4.3.62)

132

•
s u

−
+
+
¯
Q2,ω2 (θ) Q− (−ω2 ; θ)v2 (−ω2 ) − P1 (ω2 ; θ)v1 (ω2 ) = O(e−2 min{−α1 ,α1 }ωi ),
2

(4.3.63)
•
ω2
0

s u

+
T1 (0, s; θ)Q+ (s; θ)h+ (s, v1 (s; ω, θ), θ)ds = O(e−2 min{−α1 ,α1 }ω2 ),
1
1

(4.3.64)

•
−,∗

∗
T1 (0, −ω1 ; θ)P1

ss
s
(0; θ)ψ1 = Ψp2 (−ω1 , 0; θ)η − (ψ1 , θ) + O(e− min{|α2 |,2α2 }ω1 ),

(4.3.65)
•
+
−
s +
¯
P1,ω1 (q2 (θ)(ω1 ) − q1 (θ)(−ω1 )) = Φp2 (−ω1 , 0; θ)η2 (q2 (θ)(0), θ) +
s u
ss
s
+O(e−(min{−α2 ,α2 }+min{|α2 |,2|α2 |})ω1 ),

(4.3.66)

•

s u

+
+
−
¯
P1,ω1 (θ) P2 (ω1 ; θ)v2 (ω1 ) − Q− (−ω1 ; θ)v1 (−ω1 ) = O(e−2 min{−α2 ,α2 }ωi ),
1

(4.3.67)
•

0
−ω1

s u

−
−
T1 (0, s; θ)P1 (s; θ)h− (s, v1 (s), θ)ds = O(e−2 min{−α1 ,α1 }ω2 ).
1

133

(4.3.68)

Remark 4.12. The above estimates are sufficient to obtain the estimates of H1 -H6 which
have been used in Theorem 4.2. It is easy to see that only H1 and H4 account for the leading
ω
order term of ξ1 and the rest of terms are all high-order term.

Remark 4.13. Since we are looking for the homoclinic solution for (4.1.9), then ω1 = ∞
and ξ1 in Theorem simplifies to
u

u

ξ1 (ω2 , θ) = M1 θ + cu (θ)e−2ω2 λ2 (θ) + o(e−2ω2 λ2 (θ) ).

4.4

(4.3.69)

Solving the bifurcation equation

In this section we will discuss how to solve the bifurcation equation (4.3.69),

u

u

M1 θ + cu (θ)e−2ω2 λ2 (θ) + o(e−2ω2 λ2 (θ) ) = 0.

(4.4.1)

Before we solve (4.4.1) directly we need some preliminary lemmas to compute M1 and cu (0)
first. From the definition (4.2.1) of q1 and the fact that φh is the heteroclinic solution of the
second order differential equation (1.1.8)


 φh


 φ
 h
q1 (z) = 

 φh


φh


√
  − µ+ 

 

 
 √
 
µ+
 − µ+ z
 
,
˙
e
=
3 
 
  −(µ+ ) 2 

 

 
2
µ+




134

(4.4.2)

as z → ∞ and



 φh


 φ
 h
q1 (z) = 

 φh


φh

as z → −∞. Similarly we also have






√

µ−
 
 
 
  µ−
 
˙
=
3
 
  (µ− ) 2
 
 
µ2
−

 −φh (−z)


 φ (−z)
 h
q2 (z) = 

 −φh (−z)


φh (−z)
as z → ∞ and








 √
 µ− z
,
e






√
  − µ− 

 

 
 √
 
µ−
 − µ− z
 
,
˙
e
=
3 
 
  −(µ− ) 2 

 

 
µ2
−


 −φh (−z)


 φ (−z)
 h
q2 (z) = 

 −φh (−z)


φh (−z)

(4.4.3)





(4.4.4)




√
µ+ 
 
 


 
  µ+  √
 
 µ+ z
˙
,
=
e
3 
 
  (µ+ ) 2 
 

 

µ2
+

(4.4.5)

as z → −∞. Non-degeneracy condition (4.2.24) implies that the adjoint variational equation
w = − (Dx f (q1 (s), 0))∗ w,
˙

0
0
0
H1,4



 −1 0 0
− 2φ W (φh )

= 

 0 − 1 0 − 2W (φh ) + η
˜


0
0
−1
0

135








 w,





(4.4.6)

(4.4.7)

has a unique, up to constant multiples, bounded solution ψ1 (z) and

H1,4 = η W (φh ) − W (φh )
˜

2

(φh )(φh )2 + 2W (φh )φh .

−W

(4.4.8)

From (4.4.7) the bounded solution ψ1 (z) of (4.4.6) is given by

ψ1 =

−Ψ + 2W (φh )φh Ψ − −2W (φh ) + η Ψ − 2φh W (φh )Ψ,
˜
Ψ + (−2W (φh ) + η )Ψ, −Ψ , Ψ
˜

T

.

(4.4.9)

In order to obtain ψ1 (z) it is sufficient to look for Ψ. Recall that the linearization of (4.1.1)
about φh for θ = 0 is
(Lh + η ) Lh u = 0,
˜

(4.4.10)

Lh (Lh + η ) v = 0,
˜

(4.4.11)

with the adjoint equation given by

which Ψ should satisfy. It is easy to see that Ψ = φh is the solution of (4.4.11). Plugging it
into (4.4.9) and using (4.4.2) we obtain


˜
 (µ+ − η )µ+

 √
 − µ+ (˜ − µ+ )
η

ψ1 = 
˙


−µ+


√
− µ+

136






 √
 − µ+ z
.
e





(4.4.12)

as z → ∞. From the definition (4.3.16) of cu (θ), (4.4.5) and (4.4.12) we deduce


 
√
µ+   (µ+ − η )µ+
˜


 

 
 µ+   −√µ+ (˜ − µ+ )
η

 
cu (0) = < 
,
3  

 (µ+ ) 2  
−µ+

 

 
√
− µ+
µ2
+
3

2˜
= −2µ+ η .








 >,




(4.4.13)

Combining the results of the definition (4.2.54) of M1 , (4.4.9) and (4.1.7) we obtain

< ψ1 (s), Dθ f (q1 (s), 0) > ds,

M1 =
R

=
R

φh ds,

= b+ − b− .

(4.4.14)

We summarize what we have proved above:
Lemma 4.12. Under the assumptions (H) and (S’)-(1.1.4) we have

M1 = b+ − b− ,

(4.4.15)

3

2˜
cu (0) = −2µ+ η .

(4.4.16)

Now we are ready to solve the bifurcation equation (4.4.1).
Lemma 4.13. Under the assumptions (H) and (S’)-(1.1.4) there exists constants c0 > 0

137

such that for any −c0 < θ < 0 there exists a ω2 = ω2 (θ) satisfying
ξ1 (ω2 (θ), θ) = 0.

(4.4.17)

Proof. Solving the leading order term in (4.4.1) we have
ln
ω2 = −

−M1 θ
cu (0)
2λu (0)
2

.

(4.4.18)

Combining the results of (4.1.13), (4.4.15) and (4.4.16) we can further simplify (4.4.18)


ω2 = −



(b −b )θ
ln  + 3 − 
2˜
2µ+ η

√
2 µ+

.

(4.4.19)

In particular we need to choose θ < 0 to make sure
(b+ − b− )θ
3
2˜
2µ+ η

> 0.

(4.4.20)

Remark 4.14. From (4.1.11) and the fact that θ < 0 we conclude that β has to be negative,
which is stated in assumption (S’)-(1.1.4).
Theorem 4.3. Let η , δ and double well W be given and satisfy (H1) and (S’)-(1.1.4). Then
˜
there exists δ0 > 0 such that for all δ ∈ (0, δ0 ) there exists a homoclinic solution of (1.0.8)
denoted by Φm which is homoclinic to b where b satisfies (1.0.9).
Proof. By Theorem 4.1 and Lemma 4.5 we prove that there exists a unique ‘Lin’s orbit’
with the jumps satisfying ξ1 = S ξ2 . Lemma 4.13 indicates that we can solve the bifurcation
equation ξ1 = 0 so ξ2 = 0, which finish the proof of the theorem.
138

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