Studies On The Melnikov Method For Global Dynamics And Homoclinic Chaos Control In Planar Non-smooth Systems

In the field of nonlinear dynamics,the subject of global bifurcation and chaotic dynamics of discontinuous dynamical systems has attracted the extensive attention of many scholars.In many practical engineering systems,there are many non smooth factors such as gap,collision,dry friction,etc.Therefore,non-smooth or even discontinuous dynamic models are needed to study the dynamic behavior of the system.In recent years,the classical Melnikov method is an important analytical method for studying the global dynamics of smooth systems,which has a wide range of applications in smooth systems.Because of many non smooth factors in practical engineering,the research on the Melnikov method for generalizing non smooth systems has become a hot issue.The main work is to promote the plane non-smooth system Melnikov method for plane non-smooth and even discontinuous systems to study homoclinic chaos and chaos control.Firstly,the dynamic theory of smooth dynamical system is described,and the derivation process of classical Melnikov method is introduced in detail,the corresponding Melnikov function of Duffing system is calculated.A kind of non-smooth oscillator is introduced,a general form of planar piecewise-smooth oscillators is given to approximatively model many nonlinear restoring force of smooth oscillators subjected to all kinds of damping and periodic excitations.Further the corresponding homoclinic Melnikov function of the non-smooth system is introduced.some new effective methods for suppressing homoclinic chaos in a weak periodically excited nonsmooth oscillator are studied,and the main idea is to modify slightly the Melnikov function such that the zeros are eliminated.In the absence of controls,This analytical tool is useful to detect the threshold of parameters for the existence of homoclinic chaos in the non-smooth oscillator.After some methods of state feedback control,self-adaptive control and parametric excitations control are respectively considered,sufficient criteria for suppressing homoclinic chaos are derived by employing the Melnikov function of non-smooth systems.Finally,a concrete application example is studied.the sufficient condition of chaos control is obtained through Melnikov analysis,and Matlab is used for numerical simulation,theectiveness of strategies for suppressing homoclinic chaos is analytically and numerically demonstrated through a specific example.