Bifurcation Theory Of Piecewise Smooth Dynamical Systems

Many dynamical systems that occur naturally in the description of physical processes are piecewise-smooth, such as switching circuits in power electronics, impact and dry frictions in mechanical systems. Piecewise smooth systems are also used extensively to describe the problems in the domains of control biology, economy, medicine, Aerospace and Aviation Management, railway scheduling, population and so on. Therefore, it is necessary to develop the theory of piecewise-smooth dynamical systems to study these problems. The dynamical theory for piecewise smooth systems and maps and its applications has been developed in the last two decades and has got much progress. Lots of basical problems of piecewise-smooth dynamical systems have not yet been solved. I devoted myself to studying the bifurcation theory for piecewise-smooth dynamical systems and have obtained innovation results as follows.(1) We investigate the dynamics of the simplest kind of switched Hamiltonian system-switched simple pendulum, the only force acting on which is gravity. Taking advantage of the Hamiltonian function of each subsystem, we find that the dynamics of the switched pendulum is much more complex than that of the simple pendulum. The switched pendulum is able to rotate more and more rapidly or settle down except oscillating periodically.(2) We consider a class of piecewise-smooth planar parameter dependent dynamical systems where the switching boundaries or discontinuity boundaries are defined by curves passing through the origin. Trajectories near the origin do not involve sliding motion. With the help of Poincare return map technique, we prove that under certain conditions the system undergoes a generalized Hopf bifurcation.(3) We study the bifurcation of limit cycles from periodic orbits of a four dimensional system when the perturbation is piecewise linear with two switching boundaries. Take advantage of the averaging method, our main result shows that when the parameter is sufficiently small that at most six limit cycles can bifurcate from periodic orbits in a class of asymmetric piecewise linear perturbed systems and at most three limit cycles can bifurcate from periodic orbits in another class of asymmetric piecewise linear perturbed systems., Moreover, there are perturbed systems having six limit cycles or three limit cycles.(4) A two-dimensional piecewise linear continuous model with four parameters is analyzed. It reflects the dynamics occurring in a quite simple circuit proposed as a chaos generator by Laura Gardini, et al. There is at most only one fixed point of this system. When the fixed point is unstable, there is another kind of attractor. The total parameter space is investigated in order to classify completely regions of the existence and stability of the fixed point and boundary period-k points.