一些非光滑系统的分岔及分岔控制问题的研究

项目名称： 一些非光滑系统的分岔及分岔控制问题的研究

项目编号： No.U1204106

项目类型： 联合基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 付士慧

作者单位： 郑州大学

项目金额： 30万元

中文摘要： 本项目以具有形状记忆合金(SMA)约束的干摩擦或碰撞振子，蔡氏电路等非光滑系统为研究对象，将光滑系统与非光滑系统的理论有效地结合，通过理论分析与数值模拟的方法研究周期解或平衡点的分岔及分岔控制。主要内容如下：对具有SMA约束的干摩擦或碰撞振子，探讨系统轨线和分界面的关系，确定周期解的常规分岔和非光滑分岔；与相应的弹性约束的非光滑系统相比，发现SMA具有减振的作用。对蔡氏电路，分析平衡点和平衡流形的稳定性，发现一些新的平衡点的常规分岔和非光滑分岔。对较一般的三维系统给出了平衡点可能出现的分岔类型及其判别方法，还对分段线性化的连续系统得到系统稳定到唯一平衡点的条件。在研究平衡点分岔的基础上，进一步对蔡氏电路等系统，设计出分段线性或非线性的连续函数作为控制器，进行分岔控制的研究，获得所需要的动力学行为。本项目完善非光滑系统理论知识的同时，使非光滑动力学更直接联系工程应用。

中文关键词： 蔡氏电路；平衡点；平衡流形；分岔；控制

英文摘要： By the theoretical analysis and numerical simulation, we studied the bifurcation and bifurcation control of some non-smooth systems such as: dry friction or impact oscillators with shape memory alloy (SMA), Chua’s circuit and so on. For dry friction or impact oscillators with SMA, we dissected the relationship of orbits and switching interfaces, investigated some classical bifurcations and non-smooth bifurcations of the periodic solution, and found vibration reduction can be achieved through the shape memory property of SMA restraints. For Chua’s circuit, we analyzed the stability of equilibrium points and equilibrium manifolds, discovered some new classical bifurcations and non-smooth bifurcations of the equilibrium point. We also obtained bifurcation conditions of the equilibrium point for three-dimensional systems and gave sufficient conditions of non-bifurcation with persistent stability for piecewise linearized continuous systems. Furthermore, we designed piecewise linear or non-linear continuous controllers and investigated the bifurcation control for the dynamical behavior. The project not only made the theory of non-smooth systems more perfect, but also promoted the application of non-smooth systems in engineering.

英文关键词： Chua’s circuit；equilibrium point；equilibrium manifold；bifurcation；control;