
{\bf The existence of chaos for ordinary differential equations with a center manifold}

{\it Michal Fe{\v c}kan; Joseph Gruendler}


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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.


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{\it AMS classification (1991): }
34C37, 37C29, 37D45

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{\it Keywords: }
ordinary differential equations, homoclinic solutions, bifurcations, center manifold

