Periodic Solutions And Homoclinic And Heteroclinic Solutions In Several Singularly Perturbed Systems

The researches on the existence of periodic orbits and homoclinic and hete-roclinic orbits in chemical,biological and physical models have been one of the important topics in the field of singular perturbation.In this thesis,based on some recent works,we study the existence of periodic solutions and homoclinic and hetero-clinic solutions in several singular perturbed systems,including a Gierer-Meinhardt(G-M)equation with two slow nonlinearities and some coupled Fitzhugh-Nagumo equations(FHNs)by combining geometric singular perturbation theory,phase plane analysis and Melnikov method.The thesis is divided into four chapters:In chapter 1,we study the existence of pulses and monotonous and non-monotone fronts in a Gierer-Meinhardt equation with two slow nonlinearities.In fact,this problem is equivalent to the study on the existence of homoclinic and heteroclinic solutions of a four-dimensional singularly perturbed system with two fast and slow variables.Based on geometric singular perturbation theory,we analysis the global dynamics of the layer system and the degenerate system on the slow manifolds by respectively using the Melnikov function and the method of phase plane analysis.Then we suitably connect the fast and slow orbits and set up the conditions for the transversally intersections between them.Finally,we get the existence of pulses and monotonous and non-monotone fronts in this G-M equation.In chapter 2.motivated by the determination of multiple equilibriums and their bifurcations for a coupled FHN equation,we study a three-dimensional singularly perturbed system with two fast variables and a slow variable.By defining the sin-gular saddles and singular nodes,we proved that the singular saddles and singular nodes persist after(singular)perturbations and become the saddles and nodes of the above-mentioned three-dimensional singularly perturbed system by using lin-earization and the implicit function theorem.At the same time,using the above conclusions applied to analyse and judge the existence of the equilibrium and its type of FHN equation.In chapter 3,we study the existence of periodic orbits and homoclinic orbits of a coupled FHN equation.Firstly,we give the explicit representations of the periodic orbits and homoclinic orbits of the associated layer system.Then we prove the existence of periodic orbits and homoclinic orbits of a coupled FHN equation and get the parameter conditions by using the Melnikov functions.Finally in chapter 4,we summarize the results obtained above and give some future works.