The Existence of Chaos for Ordinary Differential Equations with a Center Manifold

Abstract

Ordinary differential equations are considered consisting of two equations with nonlinear coupling where the linear parts of the two equations have equilibria which are, respectively, a saddle and a center. Perturbation terms are added which correspond to damping and forcing. A reduced equation is obtained from the hyperbolic equation by setting to zero the variable from the center equation. Melnikov theory is used to obtain a transverse homoclinic solution, and hence chaos, in the reduced equation. Conditions are then established such that the chaos for the reduced equation is shadowed by chaos for the full equation. The resonant case is also studied when the chaos of the full system is not detected from the reduced equation. The techniques make use of exponential dichotomies.