ŠIL'NIKOV HOMOCLINIC ORBITS IN TWO CLASSES OF 3D AUTONOMOUS NONLINEAR SYSTEMS

Abstract

The Šil'nikov homoclinic theorem provides one analytic criterion for proving the existence of chaos in three-dimensional autonomous nonlinear systems. In applications of the theorem, however, the existence of a homoclinic orbit that usually determines the geometric structure of the chaotic attractor is not easily verified mainly because there are no available efficient methods. In this paper, based on the undetermined coefficient approach we present a framework of how to find homoclinic orbits in two classes of three-dimensional autonomous nonlinear systems of normal forms, including how to set a reasonable form of expanding series of the homoclinic orbit, how to determine all coefficients in the expansion, and how to find a numerical homoclinic orbit. Numerical examples show that the proposed framework in combination with computer simulation is very efficient.