On Mode Interactions in Reaction Diffusion Equation with Nearly Degenerate Bifurcations

Abstract

Chemical reactions as an open system have been investigated both theoretically and experimentally since nonlinear and non-equilibrium physics became one of current branches of physics. A typical example of these is the Belousov-Zhabotinsky reaction; a cerium ion catalyzes oxidation reaction of malonic acid by acidic bromate. In this reaction it is observed that the ratio of concentrations of Ce'+ and Ce3 '· may temporally oscillate, form a spatially periodic pattern or move as a wave under appropriate conditions. These phenomena are called dissipative structures. The concept of dissipative structures may help us to understand from a unified viewpoint various phenomena in open systems such as the Benard convection, 11 the laser oscillation,21 the temporal rhythm in living systems besides chemical reacting systems.3J Several nonlinear reaction diffusion equations4l have already been studied both theoretically and numerically. It is shown that they have generally the desired solutions to explain those dissipative structures which appear in the B-Z reaction system except the wave mode. The characteristic features of these equations are as follows. First, they exhibit the bifurcation of solution. Secondly their solutions will approach finally to certain states irrespective of initial conditions. Thirdly, bifurcating modes may mutually interact in vanous ·ways. The first and the second have already been studied. Therefore, we study the third in this paper.