Homoclinic trajectories and chaotic behaviour in a piecewise linear oscillator

Alexei Pokrovskii; Oleg Rasskazov; Daniela Visetti

doi:10.3934/dcdsb.2007.8.943

Article Contents

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

This issue Previous Article On the existence of time optimal controls for linear evolution equations Next Article Attractors for return maps near homoclinic tangencies of three-dimensional dissipative diffeomorphisms

Homoclinic trajectories and chaotic behaviour in a piecewise linear oscillator

1.
Department of Applied Mathematics, University College, Cork

Received: September 2006

Revised: July 2007

Published: November 2007

Abstract Related Papers Cited by

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.

Keywords:

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

Citation:

Article Metrics

HTML views() PDF downloads(63) Cited by(0)

Homoclinic trajectories and chaotic behaviour in a piecewise linear oscillator

Abstract

Access History

Article Metrics

Access History

Other Articles By Authors

Catalog

Homoclinic trajectories and chaotic behaviour in a piecewise linear oscillator

Abstract

Access History

SHARE

Article Metrics

Access History

Other Articles By Authors

Catalog

Export File

Citation

Format

Content