Research On Invariant Sets Of Non-smooth Dynamical Systems

In recent years,non-smooth dynamical systems have attracted more and more attention from scholars at home and abroad.On the one hand,there are many non-smooth dynamic problems in engineering practice,such as impact vi-bration in mechanical structures,stick-slip vibration with dry frictions and the impact vibration in switching circuits systems,etc.On the other hand,there are many dynamical phenomena in non-smooth systems which are different from those in smooth systems,such as the existence of grazing bifurcations and spe-cial routes toward to chaos,etc.The study of these theoretical and practical non-smooth dynamical problems and phenomena greatly enriches the modern dynamical system theory and expands its scope of application.The study of invariant sets of non-smooth dynamical systems is of great significance for un-derstanding the long-term dynamical behavior of the systems.The main research work of this paper is as follows:The existence of Aubry-Mather sets of Fermi-Ulam models are studied.First,in the case of high energy,we give the generating function of the system and prove that the Poincar?e of the system is an exact area-preserving monotone twist map,which is equivalent to the study of its generating function.Next,we use the characteristics of negative quadratic power form of the generation function to prove that for any rational rotation number belonging to the rota-tion interval,there is a corresponding Birkhoff periodic orbit corresponding to the system.Finally,Aubry-Mather set with irrational rotation number can be approximated by Birkhoff periodic orbit with rational rotation number,and the existence of Aubry-Mather set for any rotation number belonging to the rotation interval of the system is obtained.The existence of invariant curves of Fermi-type impact oscillator is stud-ied.First,by using Poincar?e-Cartan integral invariant theory we show that the Poincar?e map of the system has the graph intersection property.Next,by choos-ing an appropriate coordinate transformation,we show that the Poincar?e map of the system is a near integrable twist map in high energy situation.Base on this,by using the Moser's twist theorem,we prove that when the external exci-tation function is a~6class and the norm of the external excitation function is sufficient small,then the system has invariant curves,which implies that all the motions of the system are bounded.Finally,the symmetry of the Poincar?e map is discussed,the condition for the existence of two symmetric invariant curves in the system is given.The existence of Aubry-Mather sets of a piecewise linear system is stud-ied.The generating functions in the two smooth regions of the piecewise smooth system are calculated respectively.By using the properties of connection gen-erating function and Aubry-Mather theory,it is proved that the system has Aubry-Mather set when the external excitation function satisfies~6smoothness condition.The existence of invariant curves of a class of impulsive systems is studied.First,it is proved that,for some set of parameter values,the system has an invariant curve which is discontinuous almost everywhere.Then,we prove that the vertical Lyapunov exponent of the invariant curve is negative,so it is a Strange Nonchaotic Attractors.Finally,it is proved that the invariant curve attracts almost all orbits in the domain of the system.