Bulletin of the Belgian Mathematical Society - Simon Stevin

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

Michal Fečkan and Joseph Gruendler

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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.

Article information

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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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34C37: Homoclinic and heteroclinic solutions 37C29: Homoclinic and heteroclinic orbits 37D45: Strange attractors, chaotic dynamics

ordinary differential equations homoclinic solutions bifurcations center manifold


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

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