Theory And Applications Of Piecewise Smooth Dynamical Systems

In the domains of power electronics,mechanical engineering,control and biology, the models of lots of scientific problems are described by non-smooth functions. Therefore, it is necessary to develop non-smooth (i.e., piecewise smooth) dynamical systems theory to study these problems. Though the piecewise smooth dynamical systems (PSDS) theory has been developing for several decades and has got much progress, there are still many basical problems that have not been solved. That's why the study and application of many practical systems have to be delayed. Under this background, I am committed to studying the PSDS theory and have obtained the following innovation results:(1) The generalized Hopf-bifurcations of planar PSDS for some cases are investigated. Specially speaking, completed results on the generalized Hopf bifurcations are obtained under the following two cases:(a) When there exists a unique discontinuity boundary and the Jocobi matrix of each subsystem has a pair of conjugate complex eigenvalues;(b) When there exist several discontinuity boundaries intersecting at a corner and the Jocabi matrix of each subsystem either has a pair of conjugate complex eigenvalues or two different non-zero real eigenvalues.(2) The number of limit cycles in a class of planar PSDS is studied. The result that there can exist1to3limit cycles provides a negative answer to a guess in the famous international journal 《J. Differential Equations》 that the planar PSDS with one discontinuity boundary can has at most2limit cycles.(3) A class of3-dim piecewise linear (PL) chaotic systems is constructed. The existence of chaotic attractor is theoretically proved, which provides some basic theories and specific plan for design of chaos generators. In addition, as application of the obtained theories, two chaos generators with numerical simulations and circuit compensations are given.The paper is organized as follows:The disadvantage of the smooth systems theory and the motivation, history and present state of the PSDS theory are introduced in Chapter one.In Chapter two, some basic conceptions of smooth dynamical systems theory that can apply to PSDS are first reviewed. Then some basic definitions in PSDS theory are introduced. Finally, the symbolic dynamical systems and a topological horseshoes lemma are introduced as important tools for proving the existence of chaotic attractor.Chapter three includes our innovation results on the generalized Hopf bifurcations for the planar PSDS with several discontinuity boundaries intersecting at a corner and the eigenvalues of each Jocabi matrix either be a pair of conjugate complex numbers or be two different non-zero real numbers. In Chapter four, the number of limit cycles of a calss of planar PL systems with a discontinuity boundary is investigated, and the results that there can exist1to3limit cycles are obtained. Based on the above relults, the generalized Hopf bifurcations under the perturbation of the discontinuity boundary in a calss of planar PL systems are completely discussed.The existence of chaotic attractor in a class of3-dim PL systems was theoretically proved in Chapter five.In Chapter six, two chaos generators with numerical simulations and circuit compensations are given by the results obtained in Chapter five.Chapter seven includes the conclusions of the paper and the future plans.