Complicated Dynamics And The Mechanism Analysis Of Nonlinear Switched Systems

As one of the significant branch of nonlinear dynamics, switched system dynamics reveal the dynamical laws of nonlinear systems, which have attracted much attention in recent years and have become one of the hot topic in the fields of nonlinear dynamics. In this dissertation, the dynamical behaviors of the switched system and the mechanism of complex oscillation caused by switching condition have been investigated by using bifurcation theory of nonlinear dynamics, non-smooth dynamics analysis and numerical simulations. Meanwhile, based on the theory of Poincare mapping and Floquet multiplier, methods to analyze the switched system with different switching conditions are presented and applied to the investigations to the dynamical behaviors with the change of parameters of the switched system associated with time or state and to explore the road to complex motion. The basic contents of this dissertation are given as following:Switches related to the state variables are introduced, upon which a typical switching dynamical model which alternates between the Rossler oscillator and Chua’s circuit is established. Through the local analysis, the critical conditions such as fold bifurcation and Hopf bifurcation are derived to explore the bifurcations of the compound systems with different stable solutions in the two subsystems. Different types of oscillations of the switched system such as2T-focus/cycle periodic oscillator,4T-focus/cycle periodic oscillator and chaotic oscillator are observed, of which the mechanism is presented to show that the trajectories of the oscillations can be divided into several parts by the switching points, governed by the two subsystems, respectively. With the variation of the parameters, cascading of doubling increase of the switching points can be obtained, leading to chaos via period-doubling bifurcations. Furthermore, period-decreasing sequences have been obtained, which can be explained by the variation of the eigenvalues associated with the equilibrium points of the subsystems.By introducing the periodic parameter-switching scheme to the Lorenz oscillator, a time switched dynamic model is established. The Poincare map of the whole system is defined by suitable local sections related to the time switching condition and local maps determined by the two subsystems. The location of the fixed point corresponding to the periodic solution of the switched system and the parameter values of local bifurcations are calculated by multiple shooting method and Runge-Kutta method. The mechanisms of the periodic oscillators can be understood by analyzing the equilibrium attractors of the two subsystems. Through the one and two parameters analysis, we conclude that the period-doubling bifurcation, symmetry-breaking bifurcation and saddle-node bifurcation play an important role in the generation of various periodic solutions and chaos. Furthermore, Study shows that with the change of the parameter, the stable symmetric periodic trajectory suddenly disappears via saddle-node bifurcation or becomes unstable and a pair of stable asymmetric periodic trajectories are created by pitchfork bifurcation, which may evolve to chaos by the cascade of period-doubling bifurcations and the two chaotic attractors may expand to interact with each other forming an enlarged chaotic solution.The behaviors of system which alternate between Duffing oscillator and van der Pol oscillator are investigated to explore the influence of the switches on dynamical evolutions of system. Switches related to the state and time are introduced, upon which a typical switched model is established. The Poincare map of the whole system is defined by suitable local sections associated to the time and state switching condition and local maps determined by the trajectories governed by the two subsystems, and the formal expression of its Jacobian matrix is obtained. The location of the fixed point and associated Floquet multipliers are calculated. Based on the conditions when the Floquet multiplies of corresponding fixed point associated with the periodic solution pass the unit circle, two-parameter bifurcation sets of the switched system are obtained, dividing the parameter space into several regions corresponding to different types of attractors. It is found that cascading of period-doubling bifurcations may lead the system to chaos, while fold bifurcations determine the transition between period-3solution and chaotic movement.