摘要
切换系统动力学是非线性动力学的重要组成部分,它从动力学的角度揭示了切换系统的非线性特征,引起了国内外许多科学领域内众多科学工作者的高度重视,是当前非线性动力学领域的研究热点之一。本论文运用非线性动力学的分岔理论,非光滑动力学分析和数值模拟等方法,深入探讨了不同切换条件下非线性切换系统的动力学行为,并揭示了系统特殊振荡的产生机理。同时,基于Poincare映射以及Floquet乘子理论,讨论了不同切换条件下切换系统分析的方法,并借此研究了时间切换系统以及时间状态混合切换系统随各种参数变化的动力学演化过程,探索了整个系统通往复杂运动的道路。本文的主要研究工作有以下几方面。
基于状态变量切换模式,建立了Rossler振子和Chua's电路之间来回切换的状态切换系统数学模型。通过局部分析,分别给出了两个子系统参数空间诸如fold分岔、Hopf分岔等临界条件,进而考虑两子系统存在不同稳态解时通过状态切换连接下的复合系统的分岔特性,给出了诸如2T-focus/cycle型周期切换振荡、4T-focus/cycle周期切换振荡、混沌切换振荡等复杂振荡行为,并揭示了其相应的产生机理。指出系统的轨迹可以由切换点分割成不同的部分,分别受两个子系统的控制,而随着参数的变化,虽然子系统定性性质保持不变,但切换点数目成倍增加,导致系统由倍周期分岔序列进入混沌。同时,通过子系统平衡点相应特征值的分析,解释了系统存在振荡周期减少序列等现象。
基于时间切换模式,建立了两个Lorenz振子之间参数周期来回切换的时间切换系统数学模型。通过时间切换条件定义的局部截面以及子系统决定的局部映射,构造了整个时间切换系统的Poincare映射,并根据多重打靶法和Runge-Kutta法计算得到Poincare映射在给定参数下的不动点,对应于时间切换系统的各种对称和不对称周期振荡。借助于各子系统平衡态分析,揭示了各种周期振荡相应的产生机理。通过单个参数以及双参数分岔分析,指出切换系统的各种周期运动会经由鞍-结分岔,对称破缺分岔以及倍周期分岔等各种分岔通往混沌。此外,特别指出参数周期切换Lorenz系统的对称闭轨会经由鞍-结分岔后消失直接进入混沌振荡也会经由叉型分岔后失稳新产生一对非对称的同周期闭轨,进而这对非对称的周期闭轨就会各自经由倍周期分岔演化为混沌振荡。
基于时间和状态变量混合切换模式,建立了Duffing振子和van der Pol振子之间来回切换的时间状态混合切换数学模型。由局部截面和局部映射建立了整个时问状态混合切换系统的Poincare映射,指出了由于状态切换条件导致系统周期解周期未知与时间切换周期解周期已知分析时的区别,并得到了其相应雅可比矩阵的形式表达式。通过求解Poincare映射的不动点方程,确定了给定参数下切换系统周期解的位置,并计算得到其相应的Floquet乘子。随着参数的变化,根据Floquet乘子从不同的方向穿过单位元,得到了混合切换系统的双参数分岔曲线,将参数空间分割成具有不同吸引子的各个部分。研究表明,系统的周期解会经由倍周期分岔演化为混沌振荡,而fold分岔连接系统的周期3轨道和混沌运动。
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.
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