On The Study Of Limit Cycles For Two Non-smooth Systems

Piecewise smooth system as a kind of typical nonlinear system,its related theory has been widely applied in the field of natural science and social science,many scientific issues need to use piecewise smooth dynamical systems theory to analysis.In this paper,we study the existence of limit cycles for a pendulum model with dry friction and impact.First of all,combined with the description language of Filippov system.we explain the reasons why the pendulum will eventually stop on the sliding set when the pendulum swings without additional energy.Secondly,by using the numerical simulation method,we give the conditions of the energy of the limit cycle when the pendulum system has energy supplement,and we prove that there exists a unique stable limit cycle in the pendulum model with the Theory of Poincare-Bendixson.Finally,on the basis of piecewise linear systems with parameter,a class of piecewise-nonlinear systems with parameters is studied.When the parameter changes from more than 0 to less than 0,the equilibrium point of the system is to move the node to the focus,and the existence of limit cycle of the system is proved.