Relaxation Oscillations And Their Applications In Several Planar Singular Perturbation Systems

Due to the existence of turning points, relaxation oscillation problem has been one of the main research topics in the field of singular perturbation theory. This paper focuses on the existence and the maximum number of relaxation oscillations (including classical relaxation oscillation and canard relaxation oscillation) in sin-gularly perturbed generalized Lienard systems and Hopfield neural network. The thesis is divided into five chapters:In the first chapter, we give some preliminaries of this thesis. Firstly, we give two approximation lemmas and related asymptotic lemmas, and secondly, we introduce the definition of the slow divergence integral and its related propositions.In the second chapter, based on the asymptotic analysis method developed by Eckhaus, we study the existence and the asymptotic properties of relaxation oscillations of regular and canard types in a singularly perturbed generalized Lienard system with non-generic turning point. As applications, we consider two classical and important biological models, namely, a FitzHugh-Nagumo ODE and a two-dimensional predator-prey model with Holling-II response, in which, we prove the existence of regular relaxation oscillations and canard relaxation oscillations in these models and give the parametric conditions.In the third chapter, we study the maximum number of canard limit cycles in a singularly perturbed generalized polynomial Lienard system. By analyzing the multiplicities of the zeroes of the slow divergence integrals and its complete unfolding, we obtain the upper bounds of canard limit cycles bifurcating from the appropriate limit cycle set through the generic Hopf breaking mechanism and the generic jump breaking mechanism.The fourth chapter studies the existence and asymptotic properties of canard limit cycle in a singular perturbation two-neuron Hopfield neural network. Under appropriate assumptions of the critical curve and based on the asymptotic analysis method, we obtain the existence conditions for the duck solution of the system as well as the associated bifurcation curves. At the same time, we also analyze the type the equilibrium point and the global dynamics of the system.The fifth chapter summarize the above research results and states some future work.