Homoclinic And Heteroclinic Branches Of Switched Differential Systems With Invariant Line

Switched differential systems have a wide range of applications in physics,biomathematics,control engineering and other fields.Studying the dynamic behavior of switched differential systems can provide theoretical support and technical means for interesting phenomena in these disciplines,such as vibration,population ecology,and electronic circuits.In this paper,we mainly explore and discuss the homoclinic and heteroclinic bifurcations of switched differential systems with invariant straight lines.Firstly,according to the theory of Melnikov function method for continuous system and 2-piecewise smooth system,this study gives the proof and derivation of the general first-order Melnikov function expression for n+1-piecewise smooth system.Secondly,based on the Melnikov function method,this study discusses the homoclinic bifurcation of a class of 3-piecewise linear differential systems with two switching lines m-hen the perturbation order is n.In this part,we give the phase portrait on the plane according to the Hamilton function of the undisturbed system,and then get the explicit expression of the first-order Melnikov function of the perturbed system by means of analytical techniques.It is obtained that the system can be found 2n+3[n+1/2]limit cycles near the generalized homoclinic loop.Finally,the Melnikov function method is applied again to study the heteroclinic bifurcation of a class of 3-piecewise linear differential systems,and it can be found that the system have 2 limit cycles near the heteroclinic loop.Compared with the 2-piecewise linear differential system,1 limit cycle is added.The above research results enrich the existing understanding of the limit cycle bifurcation of switched differential systems,and have certain guiding significance for the application fields of circuits and control systems.