Chaos And The Existence Of Singular Cycles Of Piecewise Affine Systems

Chaos as an interesting dynamic phenomenon has been extensively studied in various areas. Shilnikov type theorems provide rigorous tools to study chaos for smooth dynamical systems, the theorems are partially extended to piecewise smooth dynamical systems. How-ever, the existence of homoclinic orbits or heteroclinic cycles is a necessary condition for Shilnikov type theorems. To prove the existence of homoclinic orbits or heteroclinic cycles is generically a remarkably difficult task. Fortunately, one can determine analytically the stable and unstable manifolds and solutions of the subsystems for a piecewise affine system. Piecewise affine systems provide good models for constructing the systems with homoclinic orbits or heteroclinic cycles based on which we can study the existence of chaotic invariant sets. This work committed to study the existence of homoclinic orbits or heteroclinic cycles and chaos and have obtained the following innovation results:(1) The existence of homoclinic orbits of three-dimensional piecewise affine systems. A criterion to ensure the existence of homoclinic orbits in a class three-dimensional piecewise affine systems which cross the switching surface transversally at two points is provided, meanwhile, a mathematical methodology is provided for constructing chaotic systems.(2) The existence of heteroclinic cycles and chaos of three-dimensional piecewise affine systems. A criterion to ensure the existence of heteroclinic cycles in a class three-dimensional piecewise affine systems which cross the switching surface transversally at two points is provided. Based on topological horseshoes theory, the existence of invariant chaotic sets is proved, meanwhile, a mathematical methodology is provided for constructing chaotic systems.(3) The existence of bifocal-homoclinic orbits of four-dimensional piecewise affine sys-tems. A criterion to ensure the existence of bifocal-homoclinic orbits in a class four-dimensional piecewise affine systems which cross the switching surface transversally at two points is provided, meanwhile, a mathematical methodology is provided for constructing chaotic systems.(4) The existence of bifocal-heteroclinic cycles of four-dimensional piecewise affine sys-tems. A criterion to ensure the existence of bifocal-heteroclinic cycles in a class four-dimensional piecewise affine systems which cross the switching surface transversally at two points is provided. Based on the criterion, a four-dimensional piecewise affine system with bifocal heteroclinic cycles is constructed and the existence of chaoticinvariant sets is proved by computer simulations.The paper is organized as follows:Some conceptions and present state of piecewise affine systems are provided in chapter one.The symbolic dynamical systems and topological horseshoe lemma are introduced in chapter two.Chapter three contains a criterion for a class of three-dimensional piecewise affine sys-tems that ensure the existence of homoclinic orbits is given. In addition, a methodology for constructing chaotic dynamical systems is provided. By this method, three chaotic systems with computer simulations are given.Chapter four introduces a criterion to ensure the existence of heteroclinic cycles in a class three-dimensional piecewise affine systems is provided. Based on topological horse-shoes theory, the existence of invariant chaotic sets is proved, meanwhile, a mathematical methodology is provided for constructing chaotic systems. By the method, several chaotic systems with computer simulation are given.Chapter five contains:First, a criterion that ensure the existence of bifocal homoclin-ic orbits for a class of four-dimensional piecewise affine systems is given. In addition, a methodology for constructing chaotic dynamical systems is provided. By this method, a four-dimensional chaotic system with computer simulation is given. Second, a criterion to ensure the existence of bifocal heteroclinic cycles in a class four-dimensional piecewise affine systems is provided. By the method, a four-dimensional system with a heteroclinic cycle and computer simulation results for chaotic invariant sets are provided.Chapter six concludes the conclusions and future plans.