Pitchfork and Hopf bifurcations of traveling pulses generated by coexisting front and back waves

Abstract

Abstract. Pitchfork and Hopf bifurcations of traveling pulse solutions bifurcating from coexisting traveling front and back waves of reaction-diffusion systems are studied. It is assumed that the parameter set on which the traveling front exists and the set on which the back wave exists intersect non-transversally. As a result, more complicated bifurcations than the transversal case are proven to occur, including pitchfork and Hopf bifurcations of traveling pulses. Introduction.Recently, bifurcation phenomena of traveling waves and their stability in parabolic systems has been attracting attention. Such systems include reaction-diffusion systems, where the waves are thought to represent qualitative change of propagation of transition layers of chemical substances, and nerve axon equations, where they are thought to represent conduction of electric pulses. These waves are expected to be stable if they are observed experimentally. This problem has already commanded a large body of literature.One way to study this subject is to regard the bifurcations and the linear stability of traveling waves as bifurcations of homoclinic or heteroclinic solutions of ordinary differential equations.A natural strategy in this line of thought is as follows: (1 The subject of this paper is bifurcation from coexisting front and back waves which do not satisfy transversality conditions. This type of degeneracy appears in [13] and belongs to stage (2) above.In [13], the following type of one-dimensional reaction diffusion system is treated:[eTUt=e 2 u xx + f (u,v' i 0,j) \ vt =v xx +g (u,v',6,'y) where x € M. and t > 0. e and r are real positive parameters, with e a small parameter. The nullcline of / intersects with that of g at P - (up,vp), R - (UR,VR) and Q - (UQ^VQ) where vp < VR < VQ. P and Q are stable constant solutions of (1.1), which guarantees the bistability, while R is unstable. *