Studies in Applied Mathematics February 1981 [antikvár]

Target Patterns and Horseshoes from a Perturbed Central-Force Problem: Some Temporally Periodic Solutions to Reaction-Diffusion Equations* By N. Kopell and L. N. Howard Existence and construction of some time-periodic solutions of a class of reaction-diffusion equations is described. These give examples of center-structures from which emanate trains of waves with outward directed group velocity, in an infinite one-dimensional spatial domain. A discrete set of distinct types of center-structures is found. Similar results are found for sufficiently large finite regions with impermeable boundaries. The existence of many other time periodic solutions corresponding to spatially infinite irregular arrays of center-structures of different types is also demonstrated for these systems. Some numerical examples are presented. 1. Introduction This paper is part of a sequence [1, 2] concerned with pattern formation in the Belousov-Zhabotinskii reaction. The patterns that emerge in the experiments are very complicated (see [1, 3, 4] for pictures), and no pattern is ever exactly repeated. The program continued in this paper is to study idealizations of specific repeatable features of overall patterns, and to find representations of these idealizations which are solutions to the reaction-diffusion equations. These equations are c, = G(c) + DV2c, (1.1) where c is a vector of chemical concentrations, c, = G(c) are the equations of the chemical kinetics, and D is a positive definite matrix of diffusivities. In the process, one hopes to find conditions on the chemical-kinetic equations which Address for correspondence: Professor L. N. Howard, Room 2-377, M.I.T., Cambridge, MA 02139. •Research of N. K. partially supported by National Science Foundation Contract MCS-77-03715 and the Sloan Foundation. That of L. N. H. partially supported by N.S.F. Grant MCS-76-23281. studies in applied mathematics 64:1-56 (1981) Copyright © 1981 by the Massachusetts Institute of Technology Published by Elsevier North Holland, Inc. 1 0022-2526/81/010001 + 56S02.50