The Study Of Limit Cycle Bifurcations In Some Piecewise Smooth Systems And Related Problems

The averaging method and successive function method are often applied to study Hopf, homoclinic and heteroclinic bifurcations in pla-nar smooth dynamical systems. Later, those methods were extended to piecewise smooth systems. For smooth near-Hamiltonian systems, one can use the first order Melnikov function to obtain the number of bi-furcated limit cycles. Liu and Han extended that method to piecewise smooth near-Hamiltonian systems, gave a general formula of the first order Melnikov function.Since dynamical system is relevant with mechanics, physics, biology, economics, and engineering technology and so on. Recently, the scholar and the public always pay attention to it. For enriching results in dynam-ical system, we investigate bifurcation of limit cycles of some piecewise smooth Hamilton dynamical systems, and application in physics about dynamical system. This paper is divided into five chapters and orga-nized as follows:fist chapter states the background of piecewise smooth system, introduce main results, and presents some lemmas to prove the-orems. Second chapter discusses limit cycle bifurcations near homoclinic and heteroclinic loop in piecewise smooth near-Hamiltonian systems, de-rives explicit formulas for the forth coefficients in the expansion of the Melnikov function, and exhibits the concrete conditions of the number of bifurcated limit cycles. Third chapter studies limit cycle bifurcation-s near generalized (double) homoclinic loop in piecewise smooth near-Hamiltonian systems with a hyperbolic saddle on a switch line. Due to a hyperbolic saddle on a switch line, the expansion of the Melnikov function near generalized (double) homoclinic loop is more complicat-ed than in the smooth systems. After computing explicit formulas for the sixth coefficients in the expansion of the Melnikov function near and outside generalized (double) homoclinic loop, the concrete conditions of the number of bifurcated limit cycles are stated. Fourth chapter, a con-crete condition under a piecewise smooth system, consisting of a center, a homoclinic loop and a generalized homoclinic loop, has11limit cycles which is given by the results theotetically obtained in Chapter two and three, by using the qualitative analysis method of differential equation system, fifth chapter gave phase portraits of Modified Kdv, and its ex-plicit traveling solves including soliton, periodic wave, kink and anti-kink wave.