Relaxation autowaves (Q731544)

This paper is a first step in the use of underlying ideas of the theory of relaxation oscillations in ordinary differential equations as a tool that can help to understand the intrinsic essence of self-organization modes in reaction-diffusion systems. The paper deals with a bilocal model of the classical Hutchinson equation with diffusion and with large Malthusian parameter of linear growth. First, the author develops a method for constructing the asymptotics of the relaxation Hutchinson cycle with arbitrary accuracy. Then, for appropiate values of the migration constant and the Malthusian parameter, and choosing the initial values of the population in a suitable space, a Poincaré map can be defined so that the corresponding periodic solution can be referred to as the self-organization mode of the considered Hutchinson system.