The amplitude of periodic plane waves depends on initial conditions in a variety of lambda - omega systems

Abstract

lambda - omega systems are an important prototype for the study of reaction-diffusion equations whose kinetics possess a stable limit cycle. Here I use such systems to investigate the evolution of periodic plane waves from exponentially decaying initial data. I begin by considering the case in which lambda (.) and omega (.) are both monotonic functions. I show that in this case the reaction diffusion solution consists of a wave front moving across the domain, with periodic plane waves behind. I show that the amplitude of these periodic plane waves is the unique solution of a simple algebraic equation. For some parameter values, the waves of this amplitude are unstable as partial differential equation solutions and, in this case, the periodic plane waves degenerate into more irregular oscillations. I go on to consider a case in which lambda (.) is a cubic polynomial, of a form so that the algebraic equation governing periodic plane wave amplitude has more than one solution. I show that it is the waves of smallest possible amplitude that appear in the reaction-diffusion solutions, and that there is a novel bifurcation in the partial differential equations as these waves appear.