Stability and Pattern of Bifurcating Steady States in Systems with Double Involution

Abstract

A system of ordinary differential equations is said to be a reversible system if there exists an involution such that the vector field is reversed under this involution. The steady-state systems of reaction-diffusion systems (RD-systems) on 1-dimensional space are all reversible ODE systems provided the kinetic terms do not involve the derivatives of component functions. If, in addition, a RD-system exhibits Z(2)-symmetry, then the Z(2)-symmetry introduces an additional involution to its steady-state system. We are interested in how this additional symmetry property affects the temporal stability and pattern of the bifurcating steady states in such RD-systems. We study the class of RD-systems whose steady-state systems are reversible with respect to two linear involutions whose product is a reversal symmetry. The set of fixed points of these involutions are both 2-dimensional planes in the 4-dimensional state space. They intersect at the constant steady states where bifurcating steady states emanate from. The steady states being studied are periodic solutions bifurcated from a constant steady state at 1 : 1 resonance in the steady-state system. Their temporal stability is analyzed through a center manifold reduction on the RD-system. A complete description of their spatial structure is also obtained. Two models, the Lambda-Omega system and a predator-prey model, are studied as examples. We conclude that all small-amplitude bifurcating steady states are temporally stable in the Lambda-Omega system and unstable in the predator-prey model. Numerical approximations of bifurcating steady states in these two models, obtained by solving both the RD-system and the steady-state system, are presented as numerical verification of our analysis. We also conclude that the bifurcation regime near 1:1 resonance in the steady-state system of the predator-prey model is hyperbolic.