Study On The Stability Of Periodic Solution And Grazing Bifurcation Of A Class Of Piecewise Smooth System

For some decades, non-smooth systems are very popular in applied mathe-matics and engineerings. For example, dry friction vibration systems and non-linear control systems can be accurate modeled by non-smooth systems. In fact, non-smooth systems even have special dynamical behavior, such as grazing b-ifurcation, corner-collision bifurcation and sliding bifurcation. Moreover, these bifurcations provide new routes toward to chaotic motions. In this paper we discuss the dynamics of solutions which are intersection about a collision system of a collision surface, and the grazing periodic solutions of a collision surface. This paper is organized as follows:In chapter 1, we give a simple account of the research status about nonlinear systems. Moreover, nonlinear systems are classified.In chapter 2, we study a class of two-degree-of-freedom piecewise linear system. First, we develop a normal form of local discontinuity mapping and give a jump matrix by using a zero-time discontinuity mapping. Then, the stability of periodic solutions is discussed by using Floquet theory and numerical simulation.In chapter 3, we study a class of one-degree-of-freedom vibro-impact system and the grazing bifurcation.In chapter 4, we study a class of one-degree-of-freedom vibro-impact collision system. The singularity in this impact system caused by grazing bifurcation is discussed. We show that the square root singularity will only exist in the trace of the Jacobian matrix in some special conditions and should not exist in the determinant in any condition while grazing condition is satisfied.