Some Research On Planar Singularly Perturbed Problems With Bifurcations

This thesis aims to investigate planar transcritical type turning point bifur-cation and heteroclinic bifurcation of travelling wave in extended Fisher/KPP equation with cut-off. For the past few years, it has gained great development in studying bifurcations of singularly perturbed systems by means of geometric singular perturbation theory combining the theory of dynamical systems. Such as canards, homoclinic and heteroclinic bifurcation in singularly perturbed systems. Because of the singularity for the singular perturbation problems, the bifurcation theory remains to be further developed and improved. In this thesis, several pla-nar singularly perturbed problems with bifurcations were investigated by means of geometric singular perturbation theory and the theory of dynamical systems, some results of predecessors were extended.The dissertation is divided into three chapters. The main results are outlined as follows:Chapter One introduces the history and actuality for the geometric singular perturbation theory, canard phenomenon and reaction-diffusion equations with cut-off. The work of this thesis is outlined and some remaining problems are given.Chapter Two is devoted to investigate transcritical type turning point bifur-cation on the plane, focus on the birth of canards and the vanish of relaxation oscillations. Canard phenomenon was first found and studied in van der Pol equa-tion in the1980s. Generically, canards can emerge from two possible mechanisms, one is the equilibrium of the reduced problem passing through the fold point of the critical manifold, the other is due to the self-intersection of the critical manifold. An important feature of canard phenomenon is that the transition from small Hopf-type cycles to large relaxation oscillations occurs in an exponentially small parameter interval. The existence of canard and the disappearance of relaxation oscillation in planar transcritical type turning point were proved, asymptotic ap-proximations of the control parameters were obtained, and the configuration of small limit cycles from Hopf bifurcation to canard explosion was given by means of blow-up transformations combined with standard tools of dynamical systems theory. This content is different from the classic canard model, it is a combination of the two mechanisms with complicated dynamic quality.Chapter Three deals with travelling wave in extended Fisher/KPP equation with cut-off. Cut-offs were first introduced by Brunet and Derrida in1997to model fluctuations in propagating fronts that arise in the large-scale (or mean-field) limit of discrete N-particle systems. They found that particle attribute of the discrete systems can be modeled by introducing a small cut-off∈on the reaction term in the deterministic reaction diffusion equations. In this chapter, we prove the existence and uniqueness of travelling wave in extended Fisher/KPP equation with cut-off, derive the asymptotic expansion of the corresponding propagation speed. Further by exponential dichotomies and Evans function method, we prove that the wave with critical speed is locally exponentially stable in some weighted spaces.