Bulletin of the Belgian Mathematical Society - Simon Stevin
- Bull. Belg. Math. Soc. Simon Stevin
- Volume 11, Number 1 (2004), 77-94.
The Existence of Chaos for Ordinary Differential Equations with a Center Manifold
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.
Bull. Belg. Math. Soc. Simon Stevin, Volume 11, Number 1 (2004), 77-94.
First available in Project Euclid: 23 March 2004
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Fečkan, Michal; Gruendler, Joseph. The Existence of Chaos for Ordinary Differential Equations with a Center Manifold. Bull. Belg. Math. Soc. Simon Stevin 11 (2004), no. 1, 77--94. doi:10.36045/bbms/1080056162. https://projecteuclid.org/euclid.bbms/1080056162