Abstract :
The bifurcations of symmetric cycles in a fourth order reversible system, which is the steady-state system of a reaction-diffusion system introduced by H. Shyldkrot and J. Ross (J. Chem. Phys.82, No. 1, 1985, 113-122), are studied via a normal form analysis. Numerical approximations of spatially periodic steady states of this system, bifurcating from a constant steady state at both strong and subharmonic resonances, were obtained by N. Kazarinoff and J. Yan (Phys. D48, 1991, 147-168). These solutions emerge through bifurcations produced by varying a diffusion coefficient. Here we focus on the bifurcation regime at 1 : 1 resonance. A normal form reduction, through cubic order terms, of the steady-state system is performed. This bifurcation is analyzed by embedding the normal form in a one-parameter family of reversible systems. We conclude that the bifurcation at 1 : 1 resonance is of hyperbolic type. Using the 3-jet of the normal form, the spatial structure of the bifurcating small amplitude symmetric cycles is approximated by a hyperbolic dispersion relation. The results obtained are verified by numerical calculations. Analytic solutions of the linear system are also compared with numerical solutions of the nonlinear system. We observed that certain characteristic features of the spatial pattern are preserved by solutions when nonlinearity takes over. This observation is explained by studying the hyperbolic curves that are the intersections of the manifold of symmetric cycles and the set of fixed points of the involution. An analytic equation approximating the hyperbolic curve is obtained, from which important information about the system is derived.
