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November  2007, 8(4): 943-970. doi: 10.3934/dcdsb.2007.8.943

Homoclinic trajectories and chaotic behaviour in a piecewise linear oscillator

Alexei Pokrovskii 1, , Oleg Rasskazov 1, and  Daniela Visetti 1,

1. 

Department of Applied Mathematics, University College, Cork, Ireland

Received  September 2006 Revised  July 2007 Published  August 2007

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.
Keywords: Chaotic behaviour, topological hyperbolicity, transversal homoclinic orbit..
Mathematics Subject Classification: Primary: 34C28; Secondary: 37D4.
Citation: Alexei Pokrovskii, Oleg Rasskazov, Daniela Visetti. Homoclinic trajectories and chaotic behaviour in a piecewise linear oscillator. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 943-970. doi: 10.3934/dcdsb.2007.8.943
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