The Bifurcation Study Of Nonsmooth System Based On Differential Inclusions

The differential equation is widely applied in the dynamical system. The dynamical system is divided into smooth dynamical system and non-smooth dynamical system whether the right-hand side function of the differential equation is continuous or not. Depending on the smooth degree on the right-hand side of the vector field of the DE, the non-smooth dynamical system is divided into non-smooth continuous system, Filippov non-smooth system, pulse non-smooth system.In the applied science and engineering field, there exists collision, impact, friction, and electronic diode of the mechanical system, which makes the study of non-smooth dynamical system more close to the actual application, so researching on the non-smooth system has important practical significance.The set-valued analysis and the theory of differential inclusions are mainly aimed at the study of the DE whose right hand side is not continuous. A new land for the DE whose right hand side is not continuous is developed.In this paper, the following aspects research works are mainly included:The research status at domestic and abroad of the non-smooth system and the differential inclusions theory are introduced in the first chapter. The purpose and significance of choosing this topic, also the problems existed in the field are briefly introduced. A simple description about the arrangement of the main contents of this article is explained.Some basic concepts and theories of the set-valued mappings and differential inclusions are introduced in the second chapter, which included the definition of the set-valued mapping, the existence of the solution about the differential inclusions, and the establishment of the basic solution matrix and the jumping matrix, also the Floquet theory which is used to study the stability of periodic solution. The most important thing is that the right-hand side discontinuous differential equation is converted into the standard form of standardized method and Filippov convex method, and introduced the basic solution matrix, the establishment of the jumping matrix, thus the theory foreshadowing for the following research is offered.In the third chapter, a class containing dry friction of single degree freedom dry friction dynamic system is analyzed based on the second chapter. Mathematical model is established using differential inclusions standardized method and is changed into the standard form of differential inclusion, the fundamental solution matrix and jumping matrix of smooth function which is divided into two subspace by the interface is solved.Then by solving the eigenvalue of the basic solution matrix named Folquet multipliers, the stability of periodic solutions and bifurcation phenomena is researched by Floquet theory. Finally, the correctness of basic theory is verified by numerical simulation.A one-degree-of-freedom vibro impact system with clearance is researched in the fourth chapter by the theoretical method of the third chapter. According to the contact of vibrator and collision surface,the sports behavior can be divided into two situations, and the collision surface is treated as the interface, And then it is transformed into standard differential form through standardized methods, then we discuss its dynamic behavior. Last, we get a series of bifurcation phenomenon about the change of the external excitation frequency numerically.A class of two degrees of freedom dry friction coupling vibration dynamic system dynamics be researched in the fifth chapter. Two degrees of freedom dry friction coupling vibration system has more complex dynamic behavior compared with the single degree of freedom system. In this chapter we take the two mass relative speed as the interface, the mathematical model of the system is transformed into a standard form of differential inclusion. Then the stick-slip vibration dynamic behaviors is studied by using theories of the previous chapter mentioned, we get the condition under different viscosity and sliding conditions. Finally, the numerical simulation verifies the correctness of the basic theory of stick-slip vibration.The sixth chapter of the article is summarized the overall.