Homoclinic Orbits in Several Classes of Three-Dimensional Piecewise Affine Systems with Two Switching Planes

Yanli Chen,Lei Wang,Xiaosong Yang

doi:10.3390/math9243285

Yanli Chen, Lei Wang + Show 1 more

Open Access

https://doi.org/10.3390/math9243285

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Abstract

The existence of homoclinic orbits or heteroclinic cycle plays a crucial role in chaos research. This paper investigates the existence of the homoclinic orbits to a saddle-focus equilibrium point in several classes of three-dimensional piecewise affine systems with two switching planes regardless of the symmetry. An analytic proof is provided using the concrete expression forms of the analytic solution, stable manifold, and unstable manifold. Meanwhile, a sufficient condition for the existence of two homoclinic orbits is also obtained. Furthermore, two concrete piecewise affine asymmetric systems with two homoclinic orbits have been constructed successfully, demonstrating the method’s effectiveness.

Highlights

Because many chaotic phenomena in physics and engineering systems can be associated with homoclinic orbits or heteroclinic cycle, the existence of homoclinic orbits or heteroclinic cycle is critical in chaos research
We investigate the existence of homoclinic orbits to saddle-focus equilibrium point in three-dimensional three-zone piecewise affine systems with two switching planes regardless of symmetry
We present an analytic method for determining the existence of homoclinic orbits in two classes of three-dimensional piecewise affine systems with two switching planes

Summary

Introduction

Because many chaotic phenomena in physics and engineering systems can be associated with homoclinic orbits or heteroclinic cycle, the existence of homoclinic orbits or heteroclinic cycle is critical in chaos research. We investigate the existence of homoclinic orbits to saddle-focus equilibrium point in three-dimensional three-zone piecewise affine systems with two switching planes regardless of symmetry. 2. The Existence of Single Homoclinic Orbit We consider the following class of three-dimensional piecewise affine systems: Ax + a, x ≤ d1,. It can be seen that system (14) meets all the conditions in Theorem 1, so it has a homoclinic orbit to the equilibrium point p. There are some conclusions similar to Shil’nikov Theorem when Shil’nikov-like conditions are satisfied. System (15) satisfies all the conditions in Theorem 1, it has a homoclinic orbit to the equilibrium point p.

The Existence of Two Homoclinic Orbits

Conclusions