Impulsive homoclinic chaos in Van der Pol Jerk system

Abstract

A 3D continuous autonomous chaotic system is reported, which contains a cubic term and six system parameters. Basic dynamic properties of the new Van der Pol Jerk system are studied by means of theoretical analysis and numerical simulation. Based on the Silnikov theorem, the chaotic characterisitics of the dynamic system are discussed. Using Cardano formula and series solution of differential equation, eigenvalue problem and the existence of homoclinic orbit are studied. Furthermore, a rigorous proof for the existence of Silnikov-sense Smale horseshoes chaos is presented and some conditions which lead to the chaos are obtained. The formation mechanism indicates that this chaotic system has impulsive homoclinic chaos, and numerical simulation demonstrates that there is a route to chaos.

Keywords

chaotic system / homoclinic orbit / Silnikov theorem

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