Existence Of Traveling Wave Solutions And Chaos For The Several Class Of Perturbed Systems

Based on geometric singular perturbation theory,Melnikov function and bifurcation theory of dynamic systems,the existence of traveling wave solutions of shallow water wave equations with singular perturbation is studied:One is(1+1)dimensional dispersion longwave equations with KS perturbation,and the other is a regular long-wave equation with distributed delays under two different perturbations.Secondly,based on the bifurcation theory of dynamical systems and Melnikov function,the chaotic dynamical behaviors of two kinds of systems subject to regular damping and periodic perturbations are studied respectively:one is the piecewise-smooth hybrid system with impulse effect,the other is the double Sine-Gordon equation.This paper is consist of seven chapters,the main contents are as follows:The first chapter is the introduction,we introduce some common perturbations in mathematical physics equations,the background,issue,results and innovations.In chapter 2,we introduce some preliminary knowledge including geometric singular perturbation theory,bifurcation theory of dynamical systems-phase plane analysis and Meinikov function,etc.In chapter 3,we study the persistence of the traveling wave solutions for a perturbed(1+1)-dimensional dispersive long wave equation.The methods are based on the invariant manifolds of geometric singular perturbation(GSP,for short)approach,Melnikov integral,and bifurcation theory of dynamical system method.Not only the analytical expression of Melnikov integral is directly obtained for a perturbed PDE,but also the analytical expression of the limit wave speed c is obtained.The results show that:the solitary wave solution exists at a suitable wave speed c for the bifurcation parameterκ ∈(0,1(?))∪(1+(?),2),while the kink and anti-kink wave solutions exist at a unique wave speed c*=(?)/3 for κ=0 or κ=2.Finally,the numerical simulations are utilized to verify our mathematical results.In chapter 4,based on the method of previous section,we investigate the existence of solitary wave solutions of the regularized long wave equation with small perturbations.Two different kinds of the perturbations are considered in this paper:one is the Kuramoto-Sivashinsky perturbation,the other is the Marangoni effects.Indeed,the solitary wave persists under small perturbations.Furthermore,the different perturbations do affect the proper wave speed c ensuring the persistence of the solitary waves.Finally,numerical simulations are utilized to confirm the theoretical results.In chapter 5,we study the chaotic behavior of a class of hybrid piecewise-smooth system incorporated into an impulsive effect(HPSS-IE)under a periodic perturbation.More precisely,we assume that the unperturbed system with a homoclinic orbit,it transversally jumps across the first switching manifold by an impulsive stimulation and continuously crosses the second switching manifold.Then the corresponding Melnikov-type function is derived.Based on the new Melnikov-type function,the bifurcation and chaotic threshold of the perturbed HPSS-IE are analyzed.Furthermore,numerical simulations are precisely demonstrated through a concrete example.The results indicate that it is an extension work of previous references.In chapter 6,we analyze the chaotic motion of the driven and damped double SineGordon equation.We detect the homoclinic and heteroclinic chaos by Melnikov method.The corresponding Melnikov functions are derived.A numerical method to estimate the Melnikov integral is given and its effectiveness is illustrated through an example.Numerical simulations of homoclinic and heteroclinic chaos are precisely demonstrated through several examples.Further,we employ a state feedback control method to suppress both chaos simultaneously.Finally,numerical simulations are utilized to prove the validity of control methods.In chapter 7,we summarize this paper and look forward to the future work.