The Number Of Limit Cycles For Several Nonlinear Systems

Nonlinear systems in many fields such as physics, biology have their wide application-s. Many phenomena in these disciplines, such as vibration, predator-prey, species growth often requires molded by nonlinear systems. Thus, the study of related property of the solution of the nonlinear system for understanding the dynamic behavior of these systems has important theoretical significance and practical significance. In this paper, we mainly investigate phase portraits, Hopf bifurcation, Poincare bifurcation, Homoclinic bifurcation and Heteroclinic bifurcation of several nonlinear systems, obtaining some new and inter-esting results.We first deduce the explicit expression of first order Melnikov function for smooth and non-smooth near-Hamiltonian systems with double parameters. By this expression, one can study the limit cycle bifurcation of smooth and non-smooth near-Hamiltonian systems. Firstly, we study the maximal number of limit cycles of a piecewise quadratic system and a cubic polynomial system, which have been investigated by [Llibre and Mereu, JMAA(2014)] and [Li and Zhao, IJBC(2014)] respectively. Compared with the result in the above two references, one more limit cycle is found by our method. Secondly, we consider the problem of limit cycle bifurcation for quadratic polynomial systems near a triangle. It is proved that the cyclicity of the triangle in quadratic systems is at least two. In other words, we provide a proof of Theorem 5.2 of [Wang and Han, JMAA(2015)]. Further, together with some new technical skills (take property Z(n,m,l) for example), we study the maximal number of limit cycles of some kinds of polynomial Lienard systems with arbitrary degree and obtain some new lower bounds for the Hilbert number of these systems, which improve truly the certain existing results.Recently, [Han et al. JDE(2009)] has provided the expansion of the first order Mel-nikov function for general Hamiltonian systems with a cuspidal loop having order m. They also gave some formulas for coefficients with m= 1. Later on, [Atabaigi et al. NAT-MA(2012)] obtained ones for m=2. In this paper, some criteria and formulas are derived for any given m, which can bo used to obtain first-order coefficients in the; expansion. In this paper, we deduce the first-order coefficients for the case m= 3 and give the corresponding conditions of existing several limit cycles. Clearly, the cusp is the higher order singular point. In general, the higher order singular point presents more complicated structures around it. This paper will investigate the phase portrait near a higher order singular point of a integrable system. Under the assumption that the homoclinic loop passes through a higher order singular point at the origin, we study the asymptotic expansion of the Mel-nikov function along the level curves of the first integral inside the homoclinic loop near the loop. Meanwhile, the formulas of the first coefficients in the expansion are given, which can be vised to exhibit the sufficient condition of the existence of limit cycles.Finally, we investigate the number of limit cycles in a piecewise polynomial system. First, we give 42 different phase portraits of the unperturbed system with at least one closed orbit. Then, we perturb one phase portrait of them by piecewise polynomials, and consider lower bounds for the maximal numbers of limit cycles emerging from the origin and generalized homoclinic loop, respectively. Second, by establishing the Poincare map, some stability criteria are derived for a homoclinic loop in the non-smooth system under study. Further, based on the theory of stability-changing of a homoclinic loop, a new approach is proposed to find limit cycles for the non-smooth system. Finally, several examples are provided to illustrate the obtained results, obtaining an alien limit cycle and giving its general definition.