Homoclinic trajectories and chaotic behaviour in a piecewise linear oscillator

Alexei Pokrovskii; Oleg Rasskazov; Daniela Visetti

doi:10.3934/dcdsb.2007.8.943

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2007, Volume 8, Issue 4: 943-970. Doi: 10.3934/dcdsb.2007.8.943

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Homoclinic trajectories and chaotic behaviour in a piecewise linear oscillator

1.
Department of Applied Mathematics, University College, Cork

Received: August 31, 2006

Revised: June 30, 2007

Published: October 2007

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Abstract

In this paper we consider the equation $\ddot x+x=\sin(\sqrt{2}t)+s(x)\,$ where $s(x)$ is a piece-wise linear map given by min$\{5x,1\}$ if $x\ge0$ and by max$\{-1, 5x\}$ if $x<0$. The existence of chaotic behaviour in the Smale sense inside the instability area is proven. In particular transversal homoclinic fixed point is found. The results follow from the application of topological degree theory the computer-assisted verification of a set of inequalities. Usually such proofs can not be verified by hands due to vast amount of computations, but the simplicity of our system leads to a small set of inequalities that can be verified by hand.

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Mathematics Subject Classification: Primary: 34C28; Secondary: 37D45.

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