LIBRARY " " . MiChigan Stat: University ll!lllllllllfllllllllllllllllllllllllllll THESIS This is to certify that the thesis entitled Bifurcation and Oscillation for Systems of Equations \presented by Beth Angela Barron has been accepted towards fulfillment of the requirements for Ph-D. degree inMathematjcs éL/N/ om Major professor Date _L1un£_22,_1919_ 0-7 639 OVERDUE FINES ARE 25¢ PER DAY PER ITEM '~ Return to book drop to remove this checkout from your record. BIFURCATION AND OSCILLATION FOR SYSTEMS OF EQUATIONS BY Beth Angela Barron A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1979 ABSTRACT BIFURCATION AND OSCILLATION FOR SYSTEMS OF EQUATIONS BY Beth A. Barron This thesis concerns the system of differential equations i + g(x,y) = ulfl(t) + ilk y + h(x,y) = u2f2(t) + Azy when the parameters “l'HZ'Al’AZ are small. It is assumed that all functions are smooth, that f1 and f2 are periodic of period T, and that for (u1.u2.Al,A2) = 0, there exists a periodic solution (p(t),q(t)) of least period T. We investigate the existence and bifurcation of T-periodic solutions near (p(t+a),q(t+a)) (a in [O,T]) when (ul,u2,Kl,A2) is near the origin. In the first part of the thesis, we assume that the corresponding linearized homogeneous equation has a one- dimensional null space and in sectorial regions of a neighborhood of the origin in fig; obtain results on the existence and numbers of solutions which reduce to Beth Angela Barron (p(t+a),q(t+a)) (a E [O,T]) at the origin. In the second part we assume a two dimensional null space and that A1 = A plane in which there exist solutions which reduce to 2 = 0. We then describe regions in the ul-uz (p(t+a),q(t+o)) (a E [O,T]) at the origin. To my parents ii ACKNOWLEDGEMENTS I would like to thank my thesis advisor, Dr. Shui Nee Chow, for his guidance and concern. I am especially grateful for his enthusiasm, which has made the last few years so enjoyable. iii TABLE OF CONTENTS List of Figures. Chapter I: Introduction Chapter II: One Dimensional Null Space. §l. Introduction. §2. Change of Coordinates . . . . §3. Liapunov-Schmidt Method and Scaling §4. Local Results for Some Specific Cases §5. The Undamped Problem. §6. Proofs of Lemmas. §7. The Theorem for Four Parameters Chapter III: Two Dimensional Null Space §l. Preliminaries §2. Scaling §3. The Case for [JL(aO)]*L’(cO)wO # O §4. The Case for [JL(aO)]*L’(orO)wO = O §5. Conditions for Existence of a Solution to the Reduced Prdblem. §6. Statements of Results Bibliography iv Page < s r» 0‘ 13 19 24 30 36 36 38 41 44 48 55 63 Figure Figure Figure Figure Figure Figure LIST OF FIGURES L(a) and the corresponding bifurcation curves. Curve on which solutions are defined. Newton's Polygons for equations (5.4) One region in which solutions exist Solution regions when [JL(GO)]*L’(ao)wO # 0 Solution regions when [JL(a0)]*L’(ao)wO = o Page 29 44 51 57 58 61 CHAPTER I INTRODUCTION Problems in nonlinear oscillation arise frequently in the study of mechanical and electrical systems. Such prdblems have been studied extensively, but due to their complexity, most of the knowledge of these problems is applicable only in specific cases. Perhaps the best known example of nonlinear oscillation is the simple pendulum [ED], for Which the equation is 2« . mt x + mgfi Sin x = 0 where m is the attached mass, I the length, x the angular displacement, and g the acceleration of gravity. Many other examples are discussed by Andronov, Vitt, and Khaikin in Theory of Oscillators [1.]. The general equation (1.1) x + g(x) = 0 describes many such oscillatory motions Which occur with no external forcing or damping. In practice damping, such as friction, and external forcing will be present, at least in small amounts. This may be represented by the equation (1.2) i +g(x) = uf(t) + xx where A and u are near zero. It is usually assumed that (1.1) has a periodic solution x0(t) of the same period as f(t). In recent papers, Loud [‘7] and Hale and Taboas ['6] have considered this equation where A and u vary independently. For (u,l) near the origin and in a sectorial region about a certain line in the u-A plane, Loud observes solutions near x0(t+o) for small values of a. He also observes that these solutions are dis- continuous at the origin [‘7]. Hale and Taboas [ 6] use a different method to obtain bifurcation curves and a characterization of the number of solutions in an entire neighborhood of the origin in the u-A plane. It is important that these solutions do not remain close to x0(t+oo) for a particular value of do but rather to a family {x0(t+o)} where a varies over a compact set. Thus the classical perturbation method cannot be used to investigate these solutions in an entire neighborhood of the origin. Also the solutions will not be continuous at the origin. In this thesis. we examine the system of equations i + g(x.y) ulfl(t) + ilk y-+'h(x.y) u2let) + Azy which describes a coupled system of oscillators. In Chapter II, we assume that the homogeneous linear equation has a null space of dimension one. Our method is similar to that of Hale and Taboas [ 6]. We obtain results for (u1,u2,ll.k2) in a neighborhood of the origin in IR4 , where we allow the parameters to vary independently in the spirit of Chow, Hale, and Mallet-Paret [ 2,3]. Results for a single equation may be obtained as a special case of the main theorem of Chapter II. In Chapter III, we consider the more complex case where the homogeneous linear equation has a two dimensional null space. This case is not meaningful for a single equation, but can be considered for a system of equations. We consider this case without damping, and find that for (u1,u2) near 0, solutions near a known solution of the homogeneous equation occur only on manifolds or in certain narrow regions, which we describe precisely. CHAPTER II ONE DIMENSIONAL NULL SPACE §l. Introduction In all that follows we will be considering the coupled system of second order. scalar equations i + g(x.y) O (1.1) o y + h(x,y) and certain perturbations of this system. We will assume that g, h and all other functions introduced are as smooth as is required. We also assume that (p(t),q(t)) is a periodic solution of (1.1) of least period T > O; in particular, i5(t) + g(p(t).q(t)) o (1.2) c'i(t) + h(a) 5(a) cam) q‘mw: = o 10 and PO -~ c"1 (3.4)(b) Q = O. O CO L 2... where ml and $2 are as defined in (2.3)(b). In this chapter, we are concerned only with the case m = 1, in which S and X are uniquely defined. For any do E [O,T], the Implicit Function Theorem now guarantees the existence of E(t,ul,u2,A1.A2,o), ‘W(t,ul,u2.kl,k2,o) which satisfy (3.4)(a) for (“l’HZ'Al'AZ'm in a neighborhood in IR4 x [O,T] of * ~ [O,O,O,O,a0] . For such “'“1'H2'x1' and A z and 2. ‘W will be the unique solution of (3.5)(a) in a neighbor- hood U : 9% of [0,0]*: moreover E and W’ will depend smoothly on all parameters. In addition, 5 and W are 0(lull+lu2l+lkll+lle) [as (ul.u2.xl,x2)401 uniformly for a E [O,T]. Therefore the prdblem is reduced to one of locating values of o for which the operator Q and the functions O1 and $2, defined by replacing z and w by E and W in (2.3)(b), satisfy (3.4)(b). Using the definitions of $1,m2, and Q and denoting by [r(t+o),r(t+o),s(t+o),s(t+o)] the vector w(t), we may rewrite (3.4)(b) as ll “1 "1 (3.5) L(or) + K + E(u1,u2.>.1.).2,o) = O >. “21 L 2 T T where L(c) = [ j r(t+o)fl(t)dt j s(t+o)f2(t)dt], o o T T K = [ j r(t+o)p(t+a)dt j s(t+o)q(t+o)dt], o o and E(ul.u2.xl.K2.O) = T . . = J‘ {r(t+c1) [x1E+G(t,§,w,o)] + s(t+a) [AZW+H(t.E.W.c)lldt. 0 We note that since G and H are o(]zI-tlw]) and Z,W, and hence : and ‘W are O(]ul]4-2u2]+-]k1]+-]k21), E must be o(]ul]+lu2]+]>.1]+ “.21). We hope to solve equation (3.5) for a in terms of u1,u2,xl.kz in a neighborhood of the origin in 351. However, the Implicit Function Theorem cannot be used with equation (3.5) in its present form. Therefore we shall make a change of scale on the variables ul,u2,ll.x2. In order to guarantee that our results on the scaled prdblem will be applicable to the original prdblem, we prove the following lemma. Lemma 3.2. Let F: IR x Rn 4 mm be smooth and f(ox,n) = B(o)n + E(o,n) where n e JRn,c1 e IR,B(O) is an m x n matrix and E(o,r) = 0(lnl) uniformly for a e [O,T]. Let n = ELE- and F(a,u.~,6) = E-lF(O.€UJ-). Suppose there exists mo and an open set U in IQ] xIR containing (wO.O) and (-wO,O) and a function 5(m,€) 12 defined on U such that 3(wo.0) = 5(-wo.0) = O and 5(w,€) is the unique solution of F(5(w,€),w,€) = O for (w.€) E U. If (wl.€l) and (w2,62) E U, 61 # O and Qlwl = €2w2, then C(wl'el) = 0(w2.€2). Proof. Let E(0,w,6) = E-ZE(G,Ew). Then the first of the following statements is true by hypothesis and the others are equivalent. B(5(wl.el))w1 + 61E(a’ = o 5(5(w1.€1).w2.€2) = 0 By our hypothesis, 5(w2.62) is the unique solution of F(a,w2.€2) = o and thus 5“”1'61) = 3(w2.62). r: The import of this lemma is that we can define 5(n) to be 5(w,€) for any (w,€) such that n = 6w # O, and am) will satisfy mamm) = o. For the remainder of this chapter, we will consider the problem A U (3.5) L(o)[l +K 1 >. x :0 [f2 )‘2 as already reformulated in this section with the change of + E(a.el.u2. 1' 2) scale suggested above. “1 i- All We let = Em and ' ' = 4y where m and v are u 1 ~ ‘2 L 2 J 13 in R2, and define Eia.w.Y.E) = 6-2E(o,6wl.6w2.éyl,€yz). Then (3.5) may be rewritten as (3.6) F(a.w.Y.€) L(cx)w + KY + eE(a.w.y.€) = 0. Because of Lemma 3.2, solving (3.6) for 5(w.Y.E) is equivalent to solving (3.5) for 5(ul,u2,kl,az) in the appropriate neighborhoods. §4. Local Results for Some Specific Cases Before stating a general result, we examine a couple of specific cases. For the remainder of this chapter, we will assume that (wO'YO) is a point in R4 of magnitude one, and that for all a, L(a) ¥ [0,0]. First we suppose there exists a point (a0,wO.YO) such that L(a0)wO + KY0 = O and L’(c10)u)O # O. In terms of F, this means that F(oo,w0,yo,0) = O and DaF(aO’wO'YO'O) # 0. Thus by the Implicit Function Theorem, there exists a unique function 3(w,y,€) defined in a neighborhood V of (WO'YO'O) such that (CO: (.00: YO) N = ~ E . a(0,0,0) GO and F(o(w,y,6),w.Y.€) O in V(co'wo'Yo). We may take to be of the form BX(-E ) V ,6 where B is a ball about (wO'YO) in 351. The value of 60 W111 of course depend on (GO.wO.YO). Next we suppose instead that (GO'WO'YO) is a point which satisfies the conditions L(oo)u¢O + KY0 = O; L'(ao)wO = O; L”(oo)wo # 0. We will assume L”(oo)o;O < O: the result will be similar if L”(oo)wo > O. In terms of 14 F, we have _ . 2 F(0 .0) — O. DaF(a .0) — o, DaF(o O'wO’YO o’wo'Yo O'wO'YO If we assume F is a CCD function, we may apply the Malgrange Preparation Theorem [44]. This theorem implies that in a neighborhood of (o0.wO,YO.O), F may be expressed as F(alinI€) = g(aIWoYoE) [02+Orl(w:Y:€) + r2(wlYI€)] for some COD functions r1,r2 and q where q(oo,wo,vo.0) # O. For almost every (w0,yO,O), there will be either two values of c or no values of o for which F(o.w.v.€) = O. This is essentially the result in which we are interested. but in order to obtain more specific infor— mation, we will prove this by a more direct method. Moreover our proof will not require F to be a CCD function. By the Implicit Function Theorem, there exists a neighborhood V of (wO’YO'O) and a unique function 5(w.Y.€) defined on V such that awa'YO'O) = do and DGF(5(w.Y.€).w.Y.E) = O for (w.Y.E) E V. We define M: V 4 R by M(w.Y.E) = F(3(w.v.€).w.v.6). For fixed (w.Y.E) in V, M(w,y,E) is a local maximum of F(o,w,y,€) with respect to o. .0) < 0. 15 Let m = {(w,y,€) E V: M(w.Y.E) = 01. Since M(wO .0) = F(d .0) = 0. it follows that 1Y0 OowOoYO (wO'YO'O) E m, and it is easily seen that m is a manifold of codimension one. We may assume V is chosen sufficiently small for the following to hold: (4.1) There exists an interval I about dO such that F(d,w,y,€) < M(w,y,€) for all (w,Y,€) 6 V and d E I: moreover if d E I - [do] and (w,y,€) 6 V, then L(a)w + MY # o. (4.2) m. divides V - m into exactly two simply connected regions, V1 and V2, such that in V1. M(w,y,€) < 0 and in V2. M(w.Y.€) > O. (4.3) HIV), the image of V under a} is contained in I. and the distance from div) to the complement of I is at least as large as sup \/2B-1M(w.v,6) where B O (wIYIE) 6V is a positive lower bound for -D§F(d,w.Y.E) in I x V. O (4.4) V is of the form B x (-EO,€O) where B is a ball about (wO'YO) in n51. The value of E0 again depends on (GOIwOIYo) 0 Under these assumptions on V, we can prove the following theorem. Theorem 4.1. Let F: 32 XIRS 4 HR be a C2 function H . g . and (do,iO.YO,O) a p01nt tor which 16 _ . 2 F(aoleIYOIO) “ O7 DaF(aOIwOIYOIO) " 0: DOF(OOIWOIYOIO) < 0° Let the interval I and neighborhoods V,V1,V2 be as defined above and satisfy conditions (4.1) - (4.4). Then there exist exactly two functions 51 and 52 defined on \72 and distinct in v2 such that for i = 1,2, ai(wo.vo.0) = a0 and F(ci(w.Y.€).w.Y.€) = O. (w.Y.E) 6 V2. If (w,y,€) E V then F(d,w,y,€) # O for all d E I. 2' Proof. First let's suppose (w.Y,€) is in V1, and let d E I. In this case we know F(d.w.Y.€) < M(w.Y.€) < 0. Therefore F(G,w,Y,€) < 0 for all points (d,w.y.6) in I x V, and so there is no solution of F(d,w.Y.E) = 0. Next let's suppose (w.Y.€) is an arbitrary point in V2. Recall that so is a positive lower bound for -D§F on I x V. For any d E I the following is true. M(w:Y:€) " F(alwlYI€) F(Elw.y.6).w.y.e) - F(O.w.Y.E) d = j DaF(a,w.Y.E)da 5(w.Y.€) a — = -]’ [DGF(G(w.Y.€).w.Y.€) a(w.Y.E) a r a 1 ~ — + u D2F(b.&.Y.t)db]da — 17 C1 .._ l [ j -o§(b,w.v.6)db1da [a«-Elw.v.t)lst/2. By hypothesis (4.3), I contains values for which [a-aiw.v.€)]BO/2 > M(w.Y.E). Since F(d,w,y,€) < M(w.Y.€) for all d E I, it follows that there exist dl.d2 e I such that d1 < d(w,y,6) < d2: F(di.w.Y.E) = 0,DaF(di.w.Y.E) # 0 for i = 1,2. Since (w.Y.€) was an arbitrary point in V and 2 DaF(di,w,Y,E) # 0, i = 1,2, the Implicit Function Theorem implies the existence of functions 51(w.Y.E) and 82(w,y,6) defined on v2 such that Ftdi(w.y,6).w.y.6) = o for a11 (w.Y.€) in v and i = 1,2. Moreover, lim 5.(w.v.€) = am .y .e ). i = 1.2. (w.Y.€)4(wl.Yl.El)tM 1 l l l O, a similar argument shows that our results hold with V1 and V2 reversed. 19 §5. The Undamped Problem In this section we consider the special case in which there is no damping. This case is of interest both in its own right and also because the proof, while very similar to the one in the general case, is more easily visualized than in the higher dimensional case. This case corresponds to letting y = 0 in (3.6). Thus in this section we will assume that wo is a point in R2 of magnitude one and consider the equation (5.1) F(O,w,€) = L(d)w + eE(a,w,6) = 0 under the following hypotheses: (5.2) L(d) is never 0. (5.3) The set 0 = [(do,w0): L(d0)wO = 0 and I = o . . II L (do)wO 0] is finite and L (don)O # 0 for all (0 )eo. o‘wo (5.4) For a given w there is at most one dO for O' which (comb) E 0.. Each of these hypotheses is generic: moreover (5.4) is actually unnecessary as we shall see in the proof, but it simplifies the counting procedures. Let 0a = [d E [O,T): (d,w) 6 C for some d]. Hypothesis (5.3) implies CO is finite. Let n be the cardinality of Ga. Since (do.wo) E G if and only if ) E 0, it follows that C has 2n elements. 20 For a given w there are two possibilities: 0! (5.5) L(d)wO # 0 for all d. or (5.6) the set Sm0 = [d:L(d)u)O = 0] is finite and each d0 6 SmO satisfies exactly one of the following: (5.6) (a) L’(d0)w0 7! o (5.6) (b) (do.wo) E Q. with at most one d0 6 SmO satisfying (5.6)(b). First we consider an m which satisfies (5.5). 0 Since L(d)wO is never zero, the function [F(d,wo,0)§ achieves a positive minimum for some d E [O,T]. Therefore there exists a neighborhood Vw of (w0,0) such that O F([O,T] x Vw ) is bounded away from zero. We may take Vw O O to be of the form B5(wo)(wo) x (-€(wo),6(wo)) where ) about w in 1R2. O Bb(wo)(w0) is a ball of radius 6(w O For points (do.w0) which satisfy (5.6)(a) or (5.6)(b), we define V(ao'w0) to be the PIOJeCtlon Of the v(d0,w0,0) 0]. defined in the previous section into the space {y We may assume that the neighborhoods V(d about points Oiwo) (d0.wo) in Q are disjoint, so that the function M, which has already been defined for each V(a ‘ ), will be 0"“0 well defined on L} V . We may also assume (d w )6? (Go’wo) o' o ‘ that V(ao'wo) and v(ao'-wo) are symmetric With respect to the origin. Then for wo satisfying (5.6) we may define 21 <2 I I) V . which will then also be of the form Now for each mo 6 1R2 with norm one we have defined a neighborhood Vw . The set {Vw ] forms an Open covering O O of the set [[w) = 1], and thus there is a finite sub- covering, [V ,V ,...V ]. Let ml wz wj W = [wilvw is in the finite subcovering] LMw[(d,w) E 0 for some d]. Furthermore, let A = [(w.€): 1-60 < (w) < 1+6O;l€[ < 60) where and 60 is chosen small enough that A will be contained in L) V . Note that 6 and 6 must be positive but men mi 0 O can be chosen as small as desired. Thus no contradictions will arise if 60 and 60 must be decreased to satisfy an additional condition which will be specified later. We will next state seven lemmas which outline our results for this prdblem. We will then summarize these results in a theorem. which will be proved by proving the seven lemmas. Lemma 5.1. Let (Gi'wi) be in C and let the map Mi and the 2-manifold Vi passing through (TO,0) be as defined in §4. (Actually this is a reduction of what 22 was done in Q4 to the case where y E 0.). Then each mi intersects the boundary of A in a closed curve which is not contractable to a point in the boundary of A, and the Zn 2n manifolds mi divide A - L) mi into 2n connected i=1 components. Lemma 5.2. The number of solutions a (w.€) of (5.1) is constant in each component, and these 3's can be defined smoothly on the entire component. Lemma 5.3. The number of solutions changes by two as (w.€) moves across Mi from one component into another. Lemma 5.4. Let (w1,€l) and (w2.€2) be two points in A such that elwl = Ezwz and El # 0. If a solution 3 is defined for either point, it is defined for both, and 92 (w1,€1) = 5 (w2,62). If M is defined for both points, then 61M (w1,61) = 62M (w2.€2). Let N be a ball of radius 60 about the origin in 1R2, and define a map P ‘Which maps A onto a neighborhood containing N by P(w,€) = 6w. Let C1 = filmi). Then the following will be true. Lemma 5.5. There are exactly n curves Ci given by Ci = 5(mi), each continuous and passing through the origin. Lemma 5.6. For each (Gi'wi) E Q, exactly one of the R) at the (1'\ curves Ci is tangent to the line u = twi (t origin. 23 Lemma 5.7. The n curves Ci divide N into exactly 2n regions. The number of solutions d of the equation “1 (5.7) L(d) “2 + E(a.ul.u2) = o is constant in each region and is the same as the number in the diagonally Opposite region and in the corresponding region of A. The following Theorem and its Corollaries summarize the results and will be proved by proving Lemmas 5.1 - 5.7. Theorem 5.8. Consider the prdblem i + g(x,y) = Hlfl (5.8) u y + h(x.y) = “2f2 where g, h. fl, and f2 are smooth and f1, and f2 are T-periodic functions of t. Suppose that (p(t),q(t)) is a solution of (5.8) and that (p(t),q(t)) is the unique, up to constant multiples, T-periodic solution of 5% + gx(p(t).q(t))x + g (p(t).q(t))y = o y (5.9) ,, y + hx(p(t) .q(t))x + hy(P(t).q(t))y = O. T T Let L(d) = [ j r(t+d)fl(t)dt I s(t+d)f2(t)dt] where o o [r(t),s(t)] is the T-periodic solution of the adjoint equation. Suppose further that L(d) is never zero: the set A'—.( . ‘r = I \ u — .(do,wo). L(do)u.O 0 and L (do)u (of cardinality 2n) and L”(dO)o;O # 0 for all 0 = 0} is finite 24 (d ) E Q; and for a given there is at most one (.00: ) 6 0. Then there exists a neighbor- 0"”0 d for which (d o 0"”0 hood N of the origin in IR2 and exactly n curves Ci which pass through the origin and divide N into 2n regions such that the number of T-periodic solutions of (5.8) is constant in each region and changes by two as u crosses one of the Ci from one region into another. For each me such that (do,w0) 6 0, there is exactly one of the Ci which is tangent to the line u = tw (t 6 R) at O the origin. Proof. This theorem follows immediately from Lemmas 5.1 - 5.7, which will be proven in the next section. B Under the same assumptions. the following corollaries are true and follow from the proofs in the next section. Corollary 5.9. If the winding number of the closed curve L(d) (O g_d g'T) in R2 is nonzero, then for each u E N, there exist at least two T-periodic solutions of (5.8). Corollary 5.10. If 0 is empty, then there exist exactly two T-periodic solutions of (5.8) for each u in N—[O]. §6. Proofs of Lemmas In this section we will prove the seven lemmas stated in §5. 25 Proof of Lemma 5.1. Let (d O) E Q, and consider O.w the function M(w,6) as defined in §4. The set l(w,€): M(w,€) = 0] is the manifold mi passing through (w0.0). DwM(wO.O) = L(GO) and DEM(w0'O) = E(w0.ao.0) which is bounded. Thus the 3-vector [L(do),E(wO.dO,0)] is normal to mi. Now since L(do) is orthogonal to wO' it follows that Wk. intersects the curve [lw0]=l,6=0] transversally. Thus for 60 and 50 chosen sufficiently small, mi is homeomorphic to a disc which intersects the boundary of A in a closed curve which is not contractable to a point in the boundary of A. For each point (di.wi) in Q, the point (di,-wi) is also in 0. Since 0a has cardinality n, 0 has cardinality 2n. For each of the Zn values (ai'wi) in 0 there exists a manifold mi of the form just described. We can choose 60 and 60 as small as necessary to insure that each of the manifolds intersect the boundary of A appropriately and that no two of the mi intersect in A. If follows that the mi will divide A into 2n connected regions. Before continuing, let us prove the following lemma, which will be of use both here and in the next sections. Lemma 6.1. Let 0 be an open connected subset of EU} and F(d,z) a Cl function from [O,T] x 0 into R which is T-periodic in d. If (6.1) )DGF(G,Z)] + [F(c,z)[ s o for (d,z) e [O,T] x s, then the number of solutions d(z) (d(z) + nT considered the same for all integers n) of F(d(z),z) = 0 is 26 constant in O, and these functions may be defined as continuous functions of z in 0. If S = [(0,2) 6 [O,T) x 0: DaF(d,z) = 0], then the number of solutions di(z) for which (di(z)-+nT,z) i S for all z E O is constant. Proof, By the Implicit Function Theorem, a solution at any point at which DaF # 0 is defined uniquely in a neighborhood of that point and may be continued into over- lapping neighborhoods unless DGF(d(z),z) approaches zero at the edge of some neighborhood. Therefore d(z) may be defined continuously in any open subset of O in which DaF(d(z),z) # O, and the number of solutions may change only at a point for which DOF(d,z) = 0. Thus the number of solutions for which DGF(d(z),z) # O in 0 is constant. B Proof of Lemma 5.2. This follows directly from Lemma 6.1 since each region satisfies condition (6.1) on O. [3 Proof of Lemma 5.3. Let (20,60) 6 mi and let d be such that (dO,zO,€O) satisfies O F(d ) = 0; D F(d .E ) = O. a O o'zo,to 0.20 We have seen in §4 that there are two solutions d(z) of F(d(z),z) = 0, d(zo) = do on one side of M1 and none on the other. All other solutions satisfy the conditions of the lemma and so their number is constant. 27 Therefore the total number changes by two as mi is crossed. D Proof of Lemma 5.4. Let (w1,€1) and (w2.€2) be two points in A such that élwl = €2w2 and 61 # 0. First suppose that both (w1,61) and (w2,€2) are in one of the neighborhoods V for which (d ) 6 0. By our results in §4, functions 5' and M o"”o are defined on V, and Lemma 3.2 may be applied to the function 5' on V. Then for i = l or 2, M(wi.€i) L(O(wi.€i))wi + €iE(wi.a(wi.€i).€i) -1 __ ._ 61 [L(d(wi,€i))€iwi + E(Eiwi,d(wi,€i)). Since d(w1,61) = d(w2,€2) by Lemma 3.2 and Elwl = Ezwz. it follows that €1M(w1.61) = €2M(w2.€2). If M is defined for both (w1.61) and (w2,62), and they are not in the same V then it must be that (wl,€l) 6 V and (w2,€2) E V(a0'-w0). In this case we conSider d as defined on V(ao'w0) LJV(OO'_wO) and the remainder of the proof is identical. Because of what we have just proved, all (w1,61) and (w2,€2) for which Elwl = Ezwz Will be on the same or antipodal manifolds mi or will be in the same or anti- 2n ~ podal components of A — Ll mi. Thus if d is defined i=1 for (wl,El), it is defined on the entire component con- taining (wl'él) and on the antipodal component, one of 28 which contains (w2.€2). Therefore, we may apply Lemma 3.2 and this completes the proof. E Proof of Lemma 5.5. We have seen that if Elwl = Ezwz. then and (w1,€1) is on one of the manifolds, (w2,€2) is on the same or antipodal manifold. Thus each manifold is mapped by P to a curve in ZIP.2 . Moreover manifolds which are antipodal are mapped by P to the same curve Ci“ Thus there are exactly n curves Ci' each given by Ci = F(mi). For each mi. there is some mo for which (w0,0) is on mi and so each Ci passes through the origin. [1 Proof of Lemma 5.6. Let Ci = 5(mi) where mi is the manifold passing through (w0.0). Recall that Mi(w.0) = 0 only for w = w and thus lim u): w O 640 O' (w.€)€7f(i This implies that lim (u/lul) = lim P(w.E)/[P(w,€)[ = iwo. uao 6&0 (16Ci (w.€)€mi Therefore Ci is tangent to the line u = tw (t 6 EU 0 at the origin. [3 Proof of Lemma 5.7. From the above description of the Ci' we know that they intersect only at the origin and there transversally, and that each intersects the boundary of N twice. Thus they must split N into exactly 2n regions. Corresponding to each solution 29 3 (w.E) defined in A, we may define E on N—[O] by 5(ul.u2) = 5(m,6) where 6w = u. a is well defined on N—[O] by Lemma 5.4 and is a solution of (5.7) by the way it is defined. Clearly there are no additional solutions of (5.7) in N-lO], since a corre- sponding solution could be defined in A. Therefore a region in N has the same number of solutions as the corresponding region in A and in fact, diagonally opposite regions in N correspond to the same region in A, one for E > 0, the other for 6 < 0. Therefore diagonally opposite regions have the same number of solutions. D Figure l. L(d) and the corresponding bifurcation curves. 3O §7- The Theorem for Four Parameters Next, let us consider a point (d ,0) at which: o'wo'Yo L(do)wO + MYO = 0 I - L (d0)wO - 0 L”(do)wO = 0 L (a0)w0 # 0 ”I ) Without loss of generality we may assume that L (d0 < 0. mo Recall that [w = 1. By arguments similar to those in ( O'YO‘ §4 we can show that there is a unique function 3(w,y,6) 2 = such that DaF(d(w,Y,€),w.Y.€) = O for all (w,y,€) in a neighborhood V, of (wO'YO’O) and d(wo,yo.0) = do. Define M(w,Y.E) = DaF(§(w.Y.E),w,y,€), and 1' and V2 be analogous to V, V1, and V2 in §4. For (w.v.€) 6 V1, fi= Hum/.6) 6 V\fi(w.y.€) = 0). Let \‘7, V there is no solution d to DGF(G,w,Y,E) = 0 and for (w,y,€) 6 V2, there are two solutions, di(w.Y.€) such i = 1,2. Next we define M1 on V2 L’fi by Mi(w.Y.€) = F(Ei(w.y.6).w.v,e) and define Si: [(on06) EVUW‘MiUDIYoE) :0]. Let W: SlUSZ. We claim m is a 3-manifold containing (wO.YO,0). Referring again to the analoqous argument in §4, we see that lim ao—(WIYIE) = 3(a) IY IE )7 (ova-m; y em»? 1 2 2 2 I_I 2' 2’ 2 thus on fi, 5: = 5? = 3. Therefore 81 Fifi = 82 Fifi = Sl p182 Fifi, which is nonempty since 31 (wOIYOIO) E 51 n 52 (17/7. Let Fa(3(u.v.6) .w.Y.€) G(w.Y.€) ,_ F(d.(w.v.6).w.v.€) [L’(do) o] as """'—_' t (0.) I Y 00) = ' Ludo) K.) which has rank two unless K = [0,0]. We will consider this possibility later, so for now we may suppose K # [0,0]. Thus S F‘fis the zero set of G, has 1 codimension two. Furthermore if (w1,v1,€1) is a point such that dl(wl.vl.€1) = d2(wl.vl.El) then at (w1.Y1.61) we also have ad ad2 1 - 6(onoe) (wlIYllel) - 6(w'Y'e) (wl'Yl'El). Since each Si is a manifold of codimension one, m is a manifold of codimension one in IRS , that is, a 4amanifold. As in §5, we may now show that there are two solutions 5(w,y,€) of F(d,w,Y,E) = O, 5(wo,yo.0) = do, on one side of m, and none on the other. In the above, we made the assumption that K # [0,0]. Let us now consider this case under the additional (generic) assumption that L”(do)u)O # 0 whenever L(do)wO = 0 and L’(do)u:O = 0. This case now essentially reduces to the case in §5. For each curve Ci in space {A = 0] defined in @5, we will now have a 3—manifold containing Ci and tangent to the hyperplane u = two (t 6 EU . 32 Finally, we will state a theorem concerning the bifurcation surfaces in the general case. Theorem 7.1. Consider the problem i + g(x,y) ulfl + Alx (7.1) 5; + h(x,y) = uzfz + 12y where g, h, f1, and f2 are smooth and f1 and f2 are T-periodic functions of t. Suppose that (p(t),q(t)) is a solution of (7.1) and that (p(t),q(t)) is the unique, up to constant multiples, T-periodic solution of :2 + gx(p(t).q(t))x + gy(p(t) .q(t))y = o i} + hx(p(t).q(t))x + hy(p(t).q(t))y = o. T T Let L(d) = [I r(t+d)fl(t)dt f s(t+d)f2(t)dt . Suppose O 0 also that K has a null space of dimension one; L(d) ' . = f ' = is never zero. the set 0 ‘(ao'wO'Y0)‘ L(do)wO + KY0 0, I - II _ . . . L (d0)wO - 0, and L (do)wO — O] is finite and Ill . o L (d0)wO ¥ 0 for all (d0.w0.yo) E Q. for a given (wO'YO) there is at most one dO for which (d ) E 0. Then there exists a neighborhood N of GOwOIYO the origin and a 3-surface S C N (i.e. a "surface" of codimension l) which is symmetric with respect to the origin and consists of the origin and the union of 3-manifolds, each of which passes through the origin. If (GO'wO'YO) satisfies L(d )w + Ky = 0, O O O L'(Go)w0 = O: and 3w 1 + 33 then the line t(wo,y0) is tangent to S at the origin. The number of solutions of (7.1) is constant in each connected component of N - S and changes by two as S is crossed transversally. Proof. We begin by defining _ (2 2 2 _ _ C — {(wO.YO.O).lYO( < (me) + (YO) - l. L(G)wO+KYO — O and L’(d)wO = O for some d]. C is a 2-surface on S3 x [0], and each point on C is described by either the results in this section or the second case in §4. Thus through each point (wO'YO'O) 6 C, there is a 4+manifold which divides a neighborhood V of (w 0) QIYOI into two parts, one in Which there are two solutions 3(w.v.€) of F(a.w.Y.€) = o, 3(w0.vo.0) = do (where a is the value for Which L(d0)wO + KY0 = 0 and O L’(do)wO = 0) and the other in which there are none. Now let (wO'YO) be an arbitrary point on 53. Exactly one of the following is true: (7.2) There is no value of d for which L(d)wO + KY0 = 0. . = { I = . (7 3) 8(w0'YO) \G[L(G)wO4-KYO 0] is nonempty and finite. Each di 6 S( satisfies exactly one of the following. (7.3) (a) L’(<:(i)uuO 7‘ 0 ll 0 L" HQ 08 ‘iL O (7.3)(b) I.’U1.)w (7.3)(c) L’(d.)u u o r. 6‘ H. E: u o t“ (7.4) N 34 We have just discussed cases (7.3)(b) and (c) and we dealt with case (7.3)(a) in §4. We may define neighbor- hoods V of the points satisfying case 2 by ((1)0. YO) intersections as in §6. Case (7) is analagous to the corresponding case (5.5) in §5, and in the same manner as there, we may define a neighborhood V of (wo'Yo) (wO'YO'O) in which (3.5) has no solution. We now delete from S3 x I! arbitrarily small balls N1 and N2 about (0,Yo.0) and (0,-YO,O), where (0,y0) satisfies (7.4). What remains, S3 x [0] - (NlLJNZ), is a compact set of points about which we have defined 3 neighborhoods V(w We may cover S x [O] - (NlLJNZ) 0! Y0) with a finite set of V in which we include a (wooYo)' covering of C - (NllJNz). Let 60 be such that if A = [(w,y,€) 6 IRS: d((w,y:€),W) < 60]; the finite covering also covers A. Let 60 also be small enough to guarantee that each component of the set Where 5 intersects itself has an element for which E = 0 (That is C intersects itself only when C does). Because the functions d(w,y,€) which are the basis for defining the manifolds which comprise 5 satisfy the conditions of Lemma 11.2 where the F in the lemma is DGF, the 5' are defined continuously in a neighborhood of C except at points at which D§(O,w,Y,E) = 0. We have already shown that there is a single manifold at these points, and so 5 is itself a manifold through C except at points of intersection where it is locally the intersection of two manifolds. 35 Now let N be a ball about the origin in IR4 of 4 radius 6 and let P: A a it be defined by 0’ P(w,y,E) = (Ew,Ey). P maps A onto a neighborhood of the origin which contains N, and P maps 5 onto a 3-surface which we shall call S, and P(N1LJN2) contains the intersection of the manifolds which make up S. We may again use Lemma 3.2 to define solutions 5(u,v,kl.12) in N. Moreover if (wO'YO’O) 6 C. (u.K) = t(wo,y0) is tangent to S. The proofs are similar to those in §5 and §6. CHAPTER III TWO DIMENSIONAL NULL SPACE §l. Preliminaries In the last chapter we assumed that the equation H “2(t) z + [gx(p(t+c) .q(t+c)).g (p(t+a) oCI(t+G)][ ] = o y w(t) (1.1) = o *z(t) w + [h (p(t+a) .q(t+d)) .h (h(t+0() .q(t+a) ][ ] X Y w(t) had a T-periodic solution (p(t+d),q(t+d)) which was unique up to constant multiples. In this chapter, we will assume that (1.1) has exactly two linearly independent T-periodic solutions, (p(t+d),q(t+d)) and (E(t+a),§(t+d)), and we will consider the equation § + g(x.y) = ulfl (1.2) y + h(x.y) = uzf2 where pl and “2 are near zero. We recall that by Lemma II.3.1, the conditions "21‘ ' 0 1 22 elflh) +G('.z.W.G) (1.3) (a) = %B(I-Q) w ’ O l . Lw2“ L¢2f2(.) +H(°,Z,w,0)_~ 36 37 for some B: (I-Q)9T -0 IR2 such that (woman +VO(O.=p)l[i>(c1) 6(a) 4(a) Ei(a)]* s 0, 5(0) _._ o, [ o 1 (1.3)(b) ulfl(-)-+G(°,z.W.O)E O i ! l I U2f2(.) +H(°ozowoc)-! are necessary and sufficient for (1.2) to have a T-periodic solution. Moreover such a solution must be of the form x(t) p(t+d) + z(t) y(t) q(t+c) + w(t) where (z,z,w,W) satisfy (1.3). As in the previous chapter, the Implicit Function Theorem implies the existence of EB(t,u1,u2,d) and *wa(t,ul,u2,d) which satisfy (1.3)(a) for (u1,u2,d) in a neighbornood of [O,O,d0]* in 1R2 x [O,T]. Note that in this case, B(@) cannot be defined uniquely as in Chapter II, and so E and G will depend on B. This gives rise to a one-parameter family of solutions 3 and G of (1.3)(a). For now, we will assume that we have chosen a particular B which satisfies (1.4) [¢(O)B(cp)+v (O.@)l*[15(a)§(0)q(a)c'i(cx)]* = 0: (8(0) = O. O In §6, we will discuss the results if B is allowed to vary. Let w(t) be the matrix whose rows span the solution space of the adjoint homogeneous equation 38 corresponding to (II.3.2). Let rl(t+d) r1(t+d) s1(t+a) sl(t+d) W(t) = . r2(t+d) r2(t+d) s2(t+d) 32(t+a), Then we may rewrite (1.2) as (11] (1.5) L(d) ( + E(u1.u2.d) = O [42‘ where f r1(t+d)f1(t)dt j s1(t+c)f2(t)dt; O 0 1 L(d) = T T (c f r2(t+d)fl(t)dt j 52(t+d)f2(t)dt§ (.0 O J and [E (H 0“ la) B(Ulvuzpa) 51‘ 1 l 2 1: -E2(H10H215J T Ei(u1,u2,d) = g [ri(t+d)G(t,z,w,d)i—si(t+d)H(t,z,w,d)]dt §2. Scaling As in the previous chapter, we will want to use a scaling procedure, but the type of scaling used here will vary. The following lemma will guarantee that the results of the scaled prdblem may be used to solve the original. Lemma 2.1. Let F: ]R x R2 -0 IR2 be a smooth map 1R2, d E HR, If) given by F(d,n) = B(d)n + E(G,r) where r B(d) is a 2 x 2 matrix, and E(G,T) = O()r]) as 39 n * O uniformly for d E [O,T]. Let U = [Ewl €1+K 1+1 * wzl - '- _ -1 and F(aoLUIE) = F(alwlleOE) "" E F(O.Ewl.€ “32) for a fixed I 2.0. Suppose there exists (do,w0) such that F(Go.w0.0) = O. I. Suppose there exists a neighborhood U in IR2 of .0) and a unique pair of smooth functions, (w20 d(w2,€) and wl(w2,6), defined in U such that w1(w2 .0) = wl . and d(w2 ,0) = 00. O O O (l) (2) . If ((92 .61) and ((112 .62) are both in U and 1+). (1) _ 1+). (2) (l) _ (2) 61 wz — 62 wz . then Elw1(w2 ,El) - 62ml(w2 .62) (2) (1).€1) = C(w2 .62). and for 61.62 # 0, d(w2 II. Suppose there exists a neighborhood U in R2 of and a unique pair of smooth functions, d(wl,E) (wlo') and w2(w1.€) defined in U such that F(G(wl.€).wl.w2(wl.€).€) = 0. w2(w10.0) = wzo and d(w10.0) = 00. If (mil),él) and (wiz),€2) are points in U such that Elwil) = Ezwiz), then Ei+xw2(w{1),€l) = €§+Aw2(w{2).62). and for 61,62 # 0, 0(wil).61) = 0(wiz),E2) Egggf. If 61 = 62 = 0, the result is clear, so let's assume 61,62 # 0, and let the conditions of the lemma hold as in I. Then the first of the following 40 statements is true by our hypotheses and the others are equivalent. ’ (1) w1(w2 .61)] B(G(wél).€1)) i (1) 3 e1""2 J + 61E(w1(w(l) 61) w(l) C(w(1). El).€1) = o (1) (w .6) B(G(w(1),€)) N113) +1 E1 “’2 + EiEXw1(w(l), 61).w(l),d(wé1),€1),€l) = O -l (l) 6 (6 E )w1(w2 .6 ) B(O(w(l) e )) : x2( :11 + 2 62 m2 + B(ez (E;€1)(wl(w(1),€ ).6 61+* uéz’ 0(w(1).€1)) = o -16 (1)6 E (w2 ) B(G(w(l) 61)) 2 E1":1)2 61 1+). 2 E2 u’2 + €2E(€;l€ 1w wl(w(1). E l).E El+xwéz).c(wél).€1).€2) - 0 Therefore (2) _ (1) -1 .(1) s - p U C(wz IE2) - 0((112 '61): and E2 610-} 1(0-(2 (\l) — (391(4. since these are the unique functions defined in U and satisfying the last equation. An analagous proof shows II. :1 41 * I §3. The Case for [JL(OO)] L (a0)w0 7K 0 The L(d) we have defined is a 2 x 2 matrix which is a T-periodic function of G. Let 0 = {d1,d2...dn] = the set of values of d for which L(d) is singular. We will assume that L(d) # [g g] for all d, though it would not affect the local analysis for a particular d 6 0 if L(d) = 02X2 for a different d E Q. In this section, we will consider values dO E 0 for which * l (3.1) [JL(GO)] L (G0)w0 # 0 where wo is a unit eigenvector for L(do) and J = [_3 g] . In this case, we let “1 “’1 - -2 u = g = Em, and let B(w,d,€) = E E(€w .6w .0). 2 Our problem can then be reformulated as (3.2) F(w.d.€) = L(d)w + EE(w,d,E) = 0. For notational convenience, we will let I. (d) .6 (d) L(d) = l 2 } £3(d) £4(d) We have H = v\ = I . F(iO,GO,O) o and DQF(10,GO,O) L (00)u0 # o by (3.1). Therefore by the Implicit Function Theorem, there exists a neighborhood V of (w 0) and a unique function 0' 3(T,E) defined on V such that 42 5(w ,0) = d 0 0 and Fi(w,d(w,6),E) = 0 for all (w.€) E V for either i = l or i = 2. Let's assume i = 1. (For i = 2, the same result is true by a similar argument.) Then for a given w, (3.2) will have a solution if and only if F2(woa(w:€)161 = 0. We know that F2(w0.d(w0.0).0) = O and that ~ I I DwFZ - Dwd(wo.0)[£3(ao) £4(ao)lwo + [13(00) 24(a0)] Since F1(w,a(w.€).6) = O for all (w.€) 6 V, it follows that Dw3(wo.0)[£1(co) 12(a0)1w0 + [21(a0) 22(a0)1 = 0. Thus Dwa(w0'0) = —[£1(GO) £2(QO)]/[Ll(co) 22(Oo)]wo which implies ( . ' )’ .. ‘-. DwFZ - -[£l(ao) (2(c0)]z[£3(a0) £4(ao)luO/[£l(ao) L2(ao]*o~ + [23(d0) 24(d0)]. We have assumed [L1(GO) £2’(GO) 16:0 3‘ O. and 50 Dsz = 0 if and only if -[£ GO).£2(GO)][£3(GO).£4(ao)lwo + [£3(do) £4(do)l ll 43 1 I which is equivalent to £3(d0) -Ll(do) £i(d0) £2(do) I I (”Oslo 24(d0) -£2(oo) 23(d0) 24(d0) or to [J L(do)]*L’(do)u)O # O. This is our assumption and thus Dsz # 0. Therefore, by the Implicit Function Theorem, we may solve uniquely for w1(w2,€) in a neighborhood of (w2 .0) or for O w2(wl.6) in a neighborhood of (ml ,0) or for both to O solve (3.2). Suppose we have defined wl(w2,€). Then 5(ul.w2.E) = 3(wl(w2.6).w2.€) = 6(u2.6) is a function of wz and E. and since 3(wl,w2,é) and wl(w2,€) were uniquely defined, d(w2.€) and wl(w2,E) are the unique pair of functions satisfying F(wl(w2.€).w2.0(w2.€).€) = 0! Wl(w20.O) = wlo.a(w20.0) = 00. Therefore we can apply Lemma 2.1 with A = O to guarantee that the functions f Ewl(w2.€) where cwz = u ul(H2) ’ 2 0 if u = O 2 and 5(u2) = 5(w2,€) where éwz = dz are well defined in a neighborhood of origin. A similar argument may be used to define u2(ul) and 5(u1) in the case that w2(wl,6) is defined such that 44 ~ fi = . F(wl.w2(wl.€).G(wl.w2(wl.s).6),€) 0. In either case, we get a curve C through the origin in the ul-uz plane on which a solution a to (1.3) may be defined. Since u2 sz w20 lim -—-= lim 6w (w 0) = Er-, the vector mo is tangent u-oo “1 e 40 1 2 ' lo ucxiC to C at the origin. ;/0 Figure 2. Curve on which solutions are defined. * ’ = .§4. The Case for [JL(dO)] L (do)u.)O O. In the previous section, we considered solutions of the system under the condition L(do)wO = 0 and [JL(dO)]*L’(dO)u)O # 0. In this section, we suppose [JL(dO)]*L’(dO)wO = O and see what possibilities exist. Ul’o Let wo = 1 be a unit eigenvector for L(GO), and w - 02 let B be a 2 x 2 matrix such that Bwo = [$1 and 5 J det B = 1. We note that B*J*B 45 and so B*J* = J* B-l. Thus the following are equivalent * ’ . = [JL(GO)] L (Go)u0 O L(d )*J*B-1BL’(o )B-le = o o o o (B‘l)*L(d )*B*J*BL’(d )B-le = o o o o (JBL(aO)B'1)*(BL’(dO)B'1)(Bwo) = o. [a (a) a (a) Let A(d) = BL(G)B-l = l l 2 . We have La3(d) a4(d)d A(GO)[O] = O, and so al(d0) = a3(do) = O, and [JA(GO)]*A’(GO)[3] = O which implies 0 0 a£(do) aé(d0) l = 0 . a4(do) -a2(do) a§(doo a;(co) o I I _ . _ 1 Thus a4(do)al(d0) - a2(do)a3(do) - 0. Since Bwo — [ ]. 0 ble + bzwo = O and l 2 b3w0 + b4w0 = 0, but 1 2 blb4 - be3 = 0 and so we get b4 = wl and b3 = -w2 . Moreover O O L(d0)wO = O, and so £l(do)b4 - £2(Go)b3 = 0 £3(do)b4 - 24(d0)b3 = 0 It will be useful to specify as much as possible about the form of A. We know 46 A(OO) lbbL +bb£ -be -be -bb£ 4321 +13% +th 141 243 132 234 121 23 12 124 = 2 2 b3b4£1+b4£3 4:322 -b3b4 44 40219321 -b2b4£3 ”311231)2 +1.11%);4 .0 b2 +bb (I -£)-b2£ 1 2 1 2 4 1 2 3 0 £1+£4 where 2i = £i(d0). Since b1 and b2 need satisfy only the condition blb4 - be3 = l, we may choose b1 and b2 such that (41) b21.(d)+bb(£ (d)-£ (d))—b2£ (d)-b9t’0 ° 120 1240 10 230’ ' We also note that a1’(ao) = b1[b4£1'(d0) +b3£7j(a0)] + b2[b4£3j(ao) —b3£‘;(ao)]. and so as long as I I I I (4.2) b4£1(d0)4#b3£2(do) + b4£3(do)«-b3£4(d0) # 0, we may choose b1 and b2 so that (4.1) holds and ai(do) # 0. Condition (4.2) is the same as L’(dO)u)O # O. (41] H2 + E(ul,u2,d) = 0 if and only if Furthermore L(G)[ _ H _ b 1L(d){ul] + b lB(ul,u2,d) = 0, so we may assume that k 2 0 1 A(d ) = 0 —1 . 0 b (£l(do) + 14(GO)) by replacing E with a multiple of E. Using this and I I _ ' . the fact that a4(do)al(do) — a2(d )a3(d ) — 0, we arrive O O at the following form for A(d): 47 ad + (12ml (d) l + dm2 (d) A(d+do) = 2 2 aCd+dm3(d) c+dd+dm4(d) where mi(d) = 0(1) as d a 0. We may now simplify the original prdblem as follows. The next four equations are equivalent [H1] L(d) ( + B(u1.u2.d) = O (“21 1 ml L(a)B' B + B(ul.u2.a) = O -“2 [u ] b'lsL(d)B’1B l + b-lBE(u1.u2,d) = o “2 L J Full _1 A(d)B + b BE(ul.u2,d) = 0 [“21 Moreover ~ U _ N A (d(u1.u2))B[u1] + b lBEa (ul.u2.d(ul.u2)) = 0 “o 2 o if and only if N “1 A(O(ul.u2)+GO)B u 2 ]+ b-1E(H1IU2IE(U1IH2)+GO) = 0 where (I A O(d) = A(a+do) and an(ul.u2.d) = B(ul.u2.d+do). Therefore 5(ul,u2) solves u il -1 A (G)B[ + b BE (u ,0 ,d) = 0 d0 .02! d0 1 2 if and only if C(ul,u2) + dO solves 48 (11] —1 A(d)B[u J + b 2 BE(U1IH2IG) = O- In the next section we will consider solutions of the problem (1.5) if L(d) has the form ad‘+ d2m1(d) l + dm2(d) 1 2 2 The preceding aco + d m3(d) c + dd + d m4(d)J discussion shows that we will be able to obtain from this results for all cases in which [JL(dO)]*L'(dO)wO = O. §5. Conditions for Existence of a Solution to the Reduced Pr0blem In this section, we consider the prdblem u 1 (5.1) L(a)[t1 ]+ B(Hlluzlc) = OI 2 2 ad + d m1(d) l + dm2(d) M“) = 2 2 aCd + d m3(d) c + dd + d m4(d) a # 0 mi(0) = 0(1), 1 = 1, 4 B(ul.u2.d) = 0(lull+l.u21). Since E = o(]u1[+-]u2[). _ l. 2 .2 . . 1 1 2 + A D2 E(O,O,d)',.:.2 + i D3 E(O,O,O)L.‘.3 2 “2 2 6 dz 1 + i-Dz D. E(O,O,d)u2u 2 t a l 2 1 2 1 M H2 1 2 49 l 3 3 -+ 6 Du2D(O,O,c)u2 l 4 4 + 24 DM.'LE(O,O,cx)ul + As before we wish to use a scaling procedure to convert (5.1) into an equation to which the Implicit Function Theorem may be applied, but a different change of variables is more useful in this case. We let “1 = [ewl x H 65w for K 2_O and A will be 2 ] where 6 = 6 2 chosen so that the scaled equation may possibly have solutions. (5.1) now becomes [w1 - (5.3) L(d) 6m + EE(w,c,6,€) = 0, [2 where EZEXw.o,6,E) = B(Ewl,€6w2,a), Using (5.2), we see that EZEXw,a,6,E) = l-D2 E(O,O,a)€2w2 + D D E(O,O,G)E26w.w 2 ”l l “1 pz 2 + J: D2 E(O,O,O)€252w2 + l- D3 E(O,O,C1)€3U.‘3 2 ”2 2 6 “l l l 2 3 2 + 2 DH Dp E(0,0.d)€ éwlwz l 2 l 2 3 2 2 + 2 DH Du E(0,0,o)€ 5 wlwz l 2 + %D3 E(O,O,a)€363w3 “2 2 l 4 4 4 + 53 Du E(0,0,o)€ wl + ‘l 50 2 2 2 Thus €E(w,o.6.€) = €A20 + 66All + 66 A02 + 6 A30 + E 6A21 2 2 2 3 3 + E 6 A12 + E 6 A03 + E A4O + ... where Am] A.. =A13 I is a function of w .w .0 We may now rewrite 13 A‘(2) 1 2 ij j (5.3) in the following form: (5.4) (a) [(aawl + 6w2) + (02m1(a)w1 +o6m2(c)w2)] (l) (1) 2A (1) + E[A20+6A11 4-6 A02 ] 2 (1) (1) 2 (1) 3A (1) +€[A30+6A21 +6 A12 +6 A03] +€3[A‘£(]5)+...]+ ...=0 (5 .4) (b) [(acowl + c6w2) + (<1sz (c1)wl + do6w2) + 026m4 (c) w2)] (2) (2) A(Z) + €[A20 +6All +62 A02 ] + E 2[A(2)+ 6Aéi)+ 62A{§)+-6 3 A(2)] +€3[Ai(2))+...]+...=0. We will now apply the method of Newton's Polygon as described by W. D. MacMillan [£3]. The slopes of the lines in the polygons represent values of h for which a = 62516) and 6 = 6A may possibly solve (5.4). It is reasonable to hope that solutions of this type may exist. Equation (5.4) is solved by E = O, c = O. and 6w2 = O, and so if we solve for d and 6w2 as functions of 6 in a neighborhood of zero, we expect 0 and 6w2 to have a power of E as a factor. Let X 51 be the highest power of 6 contained in both. Then a = 62516) and 6w2 = €xw2(6). We could use the Implicit Function Theorem to verify this if the Jacobian matrix of equations (5.4) with respect to a and (6w2). awl l] acwl CJ were nonsingular. Since it is not. we do not necessarily expect unique solutions of (5.4) in an entire neighborhood of zero, but solutions of the type described above may be possible under some conditions. 0.6 p h \‘ €t~u~ \\ ‘~\ \- ‘3; - E (a) first equation GI: 0,5) ‘x '\\N\ \ (b) second equation if C ? O (c) second equation if c Figure 3. Newton's polygons for equations (5.4). 52 The solid lines in the diagram indicate possible values of A if DiFl(O,wO) # O for (a) and DiE2(O.wO) # O for (b) and (c). The dotted lines indicate possible values of A if these and perhaps other derivatives of E vanish at (O.wo). If c # O, A will equal one generically, but may be an integer greater than one if apprOpriate derivatives vanish as indicated above. If c = O, A may equal one or one half generically, but may equal an integer multiple of one half if the appro- priate derivatives vanish at (O.w0). We will lock only at the cases where S = l or B =‘5 are possible solutions. These will be the only possibilities with probability one, and the other cases could be dealt with in the same manner as the ones we will discuss. First we consider the case x = 1. We then have 6 = E and o = 65} and so equation (5.4) becomes _. ..2 ._ _. ._ (5.5) (a) [(adwl+ wz) + Ed-ml(6o)wl+ €dm2(6c)w2] (l) (1) (l) 2 (l) (l) (l) + A20 “All +A30 ] + 6 [A02 +A21 +A4o ] + = o (5 5) (b) [(acaw +cw ) + (632m (6636 + eadw )+ 6252111 (ac-{)6 1 ' l 2 3 l 2 4 2 (2) z (2) (2) :2 (2) (2) (2) + A20 + :[All +A3O ] + 6 [A02 +A21 +A4O] + = O which we denote by 61(53wl.w2.é) = 0. We are hoping for a solution for values of E in a neighborhood of zero and 53 so a necessary condition for a solution is the existence of w and 5' (Recall w = 1) such that 20 0 1O 61(00'1'w2 IO) = O; lee. 0 330 + 0.12 + A(zlg(l,w2 .30) - O 0 0 acc—i + cw + A(2)(l w E) - O O 2 20 ' 2 ' O O 0 or a5' + w + 1-D2 E (O 0 O) — O O 2 2 u l ' ' O l aca' + cw + 1-D2 D (o o 0) = o O 2 2 u 2 ' ’ ° 0 1 This will be satisfied by any wz and 2 0 do = -(2a)-1(D F. (0,0,0) +dw ) if and only if “1 1 2o 2 2 (5.6) -cD E (0,0,0) + D E (0,0,0) = 0. “l l “l 2 However in this case, the Jacobian matrix. G(llw2 Iaolo) D (w1.w2.c) O aoO4-E1HH(O,O.O) l a acaO4-E2Hu(0,0,0) c ac has rank 1, and so the existence of solutions to (5.5) will depend on the higher order terms. Next we consider the case A = %— which is possible when c = O. In this case, 6 = 61/2 and d = 61/25} and equation (5.4) becomes —‘ :1/2’2 51/2— M ,1/2— 51/2— \ (5.7)(a) [(aoml4-w2)+-t c ml(\ G)ll+-: om2(t G)u2] .1/2 (1) g (1) ;3/2 (1) (1) _ + : [A20 ] + ‘[All ] + 6 [A02 -+A3O ] + ... _ O 54 2 (5.7) (b) [(3 m (61/25) (61/25m1 + dEwZ) + 61/232111 4 w2] 3 + A(z) + 61/2A(2) 20 11 + “Ad? +A3(c2>)] + €3/2A(2) 21 + ... = 0 Which we shall refer to as Gz(wl.w2,5}E) = 0. Again we recall that wl = l, and so a necessary condition is the O existence of and a- such that 62(1,w ,5',O) = 0, w 20 0 2O 0 or a5. + w = O O 20 -— (2) - (O) + duo 2 + A20 — 0 52m 0 3 O This will be satisfied by = -a5' and an 56 # 0 w 20 0 Which satisfies 52(m (O)-ad) + i-Dz E (0,0,0) = 0 if 0 3 2 “l 2 one exists. Thus a necessary condition for a solution of this type is that either 2 . (5.8) -Du E2(0,0,0)/2(m3(0)-ad) > O 1 (m3(0) -ad 7! O) or m3(O) - ad = O and D2 E (0.0.0) = O. “1 2 The Jacobian matrix D H -— G (l.w' .E'.O) (11.w2.d) 3 2O 0 = ado l a 52 KD+E «>00) Ed 25m W)+G“ om3 2 ’ ' o o 3 “2 L 114 O 55 Note that the determinant of the submatrix consisting of columns two and three is Za'm (O) + dw - ad'd O 3 20 O = 2d0m3(0) - Zacod = 2d0(m3(O)-ad) and Will be nonzero if m3(0) # ad. The determinant of the submatrix con- sisting of columns one and three is 2adom3(0) + aoodw20 - ado 3(O) - aDulD2(O.O.O) = afizm (0) -a§2d-D2 E (o o 0)] O 3 O “l 2 ' ' = a[52[m (O)-—ad] - D2 E (0.0.0). This cannot be zero if 0 3 “l 2 2 O — I .2 Du132(0,0.0) # 0 Since o0 must satisfy oo(m3(O)-ad) + ;-D2 D (0,0,0) = 0. Therefore 2 L11 2 “1 2 (1113(0) -ad) # o and ““3“” _ad) < 0 will be sufficient conditions for a solution. If m3(0) - ad = O, the existence of a solution depends on the higher order terms. §6. Statements of Results In the previous sections, we have considered B to be a fixed map from (I-Q)9T to m2 which satisfied condition (1.4). In this section, we will discuss the result of allowing B to vary. This will allow us to state theorems which have been proved for the most part in §3 or in §5. 56 Theorem 6.1. In the equation i + g(x,y) II t H H1 H (6.1) I ’C H. y + h(XIY) _ 2 2: we assume that g, h. f1. f2 are smooth and f1' f2 are T-periodic functions of t. Suppose that (p(t),q(t)) is a T—periodic solution of i + g(x,y) 0 y + h(x,y) O and that there are exactly two linearly independent, T-periodic solutions of r2(t)' E + [gx(p(t+a).q(t+a)).g (p(t+d).q(t+a))] = y _w(t). (6.2) .. [z(tfl ‘w + [hx(p(t+o),q(t+o)),h (p(t+d),q(t+o))] = Y w(t). Let (rl(t+o),sl(t+o)) and (r2(t+o),s2(t+o)) be the solutions to the adjoint equation, and let "T T. l j r1(t+o)fl(t)dt J sl(t+o)f2(t)dt 0 O L(o) = T 3 j r2(t+o)fl(t)dt J 52(t+o)f2(t)dt -0 O .1 Suppose that (6.3) L(Go)w = 0: [JL(GO)]*L'(GO)wO # O; lmo] = l O O f l] where J = 1 . L-1 o- 57 Then there exist two curves Cl and C2 in a neighborhood of the origin in ]R2 such that for u between Cl and C2, there exists a solution to (6.1). Cl and C2 intersect at the origin where both are tangent to u = two (t 6 EU .fl 1 I (A). —;.a, Figure 4. One region in which solutions exist. Egggf. L(d) is independent of B, and so for each B which satisfies (1.4) we have already shown there exists a curve CB tangent to u = two (t 6 30 at the origin such that (6.1) has a solution for u on CB' It is clear from the definition of X8 (II.3.3) N that z, w, and thus a depend on B smoothly. Thus U CB is a region of the form described Bsatisfies(l.4) 58 and so curves Cl and C2 Which bound this region exist and are as described in the theorem. 2 Such regions will exist for each (do.wo) which satisfies (6.3) and so there may be several regions in which solutions exist. Figure 5. Solution regions when [JL(oO)]*L’(oO)wO ¥ 0. Next, let's look at what was required for a solution under the conditions imposed in §4 and §5. All of the conditions depended on L(d) or on second or higher order derivatives of E. L(d) is independent of B, and so it remains to see how E and its derivatives depend on B. Recall that (6.4) Ei(c,ul,u2) = ri(t+o)G(t,z,w,o) + si(t+o)H(t,z,w,o)dt, OL—Dl'a i = 1,2, where 2, G are functions of t, ul, dz. 59 In fact. 3(u) ulfl4-G(t.z(u).W(u).a) ~ 6(O)B(w) + V0(t.I-Q ». W(u)_ uzfz-tfilt.z(u).W(u).a) (6.5) From (6.5) we see that the derivatives of E and a depend on the derivatives of B at u = 0. Therefore the derivatives of E depend on B also. Therefore in considering the existence of solutions to (6.1) where [JL(dO)]*L’(dO)wO = 0, it is necessary to consider all functions B(m) which satisfy (1.4). Thus we have the following theorems. Theorem 6.2. For the equation i + g(x,y) = ulfl (6.1) .. y + h(x.y) = uzfz. we assume that g, h, f1, f2 are smooth and fl’ f2 are T-periodic functions of t. Suppose that (p(t),q(t)) is a T-periodic solution of 0 x + g(x,y) § + h(x,y) = O and that there are exactly two linearly independent, T-periodic solutions of H (2(t) z + [gX(p(t+0).q(t+G)).g (p(t+d).q(t+0))][ J = O y w(t) (6.2) " [z(t)] w + [hX(p(t+d),q(t+c)).hy(p(t+c).q(t+a))] - = O. 60 Let (rl(t+o),sl(t+c)) and (r2(t+d),sz(t+o) be the solutions to the adjoint equation, and let ('T T 1 j rl(t+o)fl(t)dt j sl(t+o)f2(t)dt o o L(d) = T Tr- g r2(t+d)fl(t)dt g sz(t+o)f2(t)dt and T Ei(B.a.ul.u2) = Ej)ri(t+<1)G(t.zB.w[3.<1t) + si(t+a)H(t.zB.wB.a)dt where EB, GB are functions of t, u which satisfy (2' ’ o ' '2" pf <->+cs(-.z.w.a) = KB(I -Q) 1 l G o 55’) (1121526) +G(-.2.W.G)_] for a function B: (I -Q)9T 4 ]R2 such that (1.4) [@(O)B(cp) +VO(O.w)]*[I3(0)5(a)