Exact Periodic Solutions of a Kind of Quintic Nonlinear Liénard Equation and Their Evolution with Hamilton Energy

In this paper, we study the exact periodic solutions of a kind of Liénard equation with cubic and quintic nonlinearities and the evolution of periodic solutions with Hamilton energy. Firstly, the qualitative analysis of the equation is carried out by using the dynamic system theorem, then the existence conditions, the number and the approximate positions of the periodic solutions and other bounded solutions for this equation are given. By applying the analysis approach on the basis of the first integral and several appropriate transformations, we obtain all seven families of elliptic function periodic solutions of the equation under different parameters conditions. By studying the limit of these periodic solutions on the Hamilton energy evolution and directly using the analysis approach based on the first integral, all ten pairs of bell-shaped solutions and kinkshaped solutions (corresponding to homoclinic and heteroclinic orbits) for the studied equation under different parameter conditions are also given. In this paper, we establish the relationship between the Hamilton energy of the system and the periodic solution, bell-shaped solution and kink-shaped solution of the system. From the analysis in this paper, it can be seen that the reason why the Liénard equation we studied has periodic solutions, bell-shaped solutions and kink-shaped solutions is essentially determined by the energy value-taken of the Hamilton system corresponding to the equation in different ranges. As the preliminary application of the results in this paper, the periodic wave solutions and the solitary wave solutions of the typical Ablowitz equation are also given.