Reaction-Diffusion Systems and Propagation of Limit Cycles with Chaotic Dynamics

Travelling wave solutions to reaction-diffusion systems are considered from the standpoint based on chaos functions. Firstly, the Fisher-KPP equation, which describes a model for the propagation of gene as nonlinear dynamics, is introduced and is transformed into a two-dimensional (2-D) system of nonlinear differential equations. Then, 2-D solvable chaos maps for the 2-D system are derived from chaos functions, and the bifurcation diagrams are numerically calculated to find a system parameter for limit cycles with discrete and chaotic properties. Finally, the chaotic dynamics are discussed by presenting the so-called entrainment and synchronization, and by illustrating the propagation of limit cycles as travelling waves on a phase plane corresponding to the original plane.