We study the estimation of Poincare maps of three-dimensional vector fields near a hyperbolic critical point, which involves linearization problems. Standard linearization theorems have several defects in applications. They usually require complicated non-resonance conditions on the eigenvalues of the vector field at the critical point. Even when one has these non-resonance conditions, as one gets close to a resonance, the size of the neighborhood where the C1 linearization exists typically gets too small for practical uses. We seek for a linearization theorem that overcomes these shortcomings and may have broad practical applications. We have proved a partial linearization theorem that gives a C1 linearization h near a hyperbolic critical point p on a two-dimensional invariant surface of a three-dimensional vector field X. Let the eigenvalues of DX(p) be a, b and c, where a > 0 > b > c. Essentially our theorem only requires that 2b > c to obtain h in some neighborhood U of p in . In addition, the explicit size of U is found, which depends on the C2 information of X, as well as the C0 and C1 sizes of h. Based on our partial linearization theorem, we obtain desired estimation of Poincare maps from some transversal curve to the stable manifold of p to another transversal surface to the unstable manifold of p. Our estimation of such Poincare maps will have many applications, including an in-depth study of the famous Lorenz equations. For example, it seems likely that we will be able to substantially improve results of Tucker on the existence of the Lorenz strange attractor, and obtain rigorous results on the existence of chaos near the first homoclinic bifurcation as numerically investigated in the well-known book of Colin Sparrow.
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