Stability Analysis Of Traveling Waves In Two Types Of Excitable Slow-Fast Systems

In non-equilibrium systems,due to the interaction of local kinetic processes and diffu-sion transport,a variety of pattern phenomena are generated.One of these is the phenomenon of traveling wave in excitable systems.In this dissertation,we study stabilities of traveling waves in two types of excitable slow-fast systems:a PDE approximation of coupled arrays of Chua’s circuit and a reaction-diffusion-mechanics model.Both of them share some common properties of Fitz Hugh-Nagumo system,which is a paradigm of excitable slow-fast sys-tem.In this dissertation,by using two different methods:Evans function and Lin-Sandstede method,respectively,we prove that the traveling waves（traveling back or traveling pulse）of these two systems are stable.The main work of this dissertation is divided into the following parts:（1）As a classic chaotic circuit,Chua’s circuit became a popular topic of research.We study the partial differential equation approximation of coupled arrays of Chua’s circuit.This system admits traveling front,back,pulse as well as periodic and chaotic waves.However,the stability problem of these waves have rarely been investigated.The question of stability can be reduced to an eigenvalue problem in R⁴.The Evans function method is applied,along with the geometric singular perturbation theory and the smooth linearization.Dividing the half complex plane into three regions,we show that there exists no nontrivial eigenvalue near0 and 0 is a simple eigenvalue by using a decomposition of Evans function.Furthermore,we prove that there are no other unstable spectra.This means that the traveling back is linearly stable.Combining this with some foregone results,we prove that the traveling back is exponentially stable.（2）We analyze the stability of traveling pulse in a reaction-diffusion-mechanics system,which is derived by Matt Holzer,Arjen Doelman and Tasso J.Kaper^[1].This system consist-s of a modified Fitz Hugh-Nagumo system bidirectionally coupled with an elasticity equa-tion.For parameters different from[1],we analyze the spectrum of traveling pulse in this reaction-diffusion-mechanics system by using geometric singular perturbation theory and Lin-Sandstede method,and we prove that the traveling pulse is linearly stable.Specifically,we prove that there is at most a nontrivial eigenvalue near the origin,which determines the stability.In this progress,we construct a candidate eigenfunction,which is piecewise con-tinuous.Furthermore,we provide an approximation of this eigenvalue and confirm that it is negative.In short,the stabilities of traveling front of a partial differential equation approximation of coupled arrays of Chua’s circuit and traveling pulse of a reaction-diffusion-mechanics model are studied.By using Evans function method and Lin-Sandstede method respective-ly,we prove that the traveling front is exponentially stable and the traveling pulse is linearly stable.