Analyses On Analytical Solutions Of Nonlinear Traveling Wave Equations And Bifurcations Of Limit Cycles On Equivariant Planar Vector Field

Research and application for bifurcation of nonlinear dynamical system theory get rapid development in nearly30years. The theory has been widely used in chemistry, physics, fluid mechanics, vibration mechanics, celestial mechanics, ecology, biology, fi-nance and other social sciences, and a large number of mathematical models in these areas are described by nonlinear dynamic system. It is a challenging and significant task for re-searchers to use qualitative method and the branch of nonlinear dynamical systems theory to study the mathematical models, and to obtain important results in social production, engineering application and scientific research.This doctoral dissertation is devoted to study two problems of branch of nonlinear dynamical system theory. One is the analytical solutions of nonlinear traveling wave equations, and the other is the number and distribution of limit cycles for a Zq equivariant planar vector field. The historical background, the recent advance, and the classic methods for those two problems are stated in the first chapter.Three nonlinear traveling wave equations are considered for the former. Firstly, the dypamic system method is used to study the (2+1)-dimentional Davey-Stewartson-type equation. Under different parameter conditions, the phase diagram of equation is ana-lyzed. When the power exponent n equal1,2or general integer, the existence of periodic wave solution, solitary solution and so on are discussed and some analytical parameter representations are obtained respectively. Three representations are chosen for numerical simulation in order to discuss the influence of parameter about the solutions.Secondly, a Non-Local Hydrodynamic-Type model is considered. The model is a singular traveling wave system and three-step method is employed. In this part, the exis-tences of periodic wave solutions, peakons, compactons, and periodic cusp wave solutions are discussed. Especially, when the isochoric Gruneisen coefficient equal1, some exact analytical solutions such as smooth bright solitary wave solution, smooth and non-smooth dark solitary wave solution and periodic wave solutions, as well as uncountably infinitely many breaking wave solutions, are obtained. Lastly, the ac-driven complex Ginzburg-Landau equation is studied under the given parameter conditions. The method of dynamical systems is applied to analyze the dynami-cal behavior of the stationary solutions and their bifurcations depending on the parameters of systems. All bounded exact solutions are obtained. To guarantee the existence of those solutions, the constraint parameter conditions are given.Another problem of this doctoral dissertation belongs to weakly Hilbert’s16th prob-lem. Bifurcations of limit cycles in a Z7and Z6equivariant planar vector field of degree7are considered. With the help of numerical analysis, bifurcation theory of planar dy-namical systems and the method of detection function, this part intent to find the maximal number of closed orbits and the maximal number of limit cycles after perturbing the sys-tem. Following the special consideration of Z6and Z7equivariant vector fields of degree7,35and37limit cycles are obtained and the configuration of compound eyes are shown, respectively. How the variable q and perturbed term influence the problem is discussed in the end.At the end of this dissertation, the summary of this thesis and the prospect of future research are given.