Stability Theory of Synchronized Motion in Coupled-Oscillator Systems. III: Mapping Model for Continuous System

By starting with a reaction-diffusion equation a mapping model for the continuous system is proposed. The transition from the uniform state to the non-uniform one occurs at the same vaalue of the diffusion constant for the mapping model as for the original reaction-diffusion equation if the transition exists. The mapping model is further studied by adopting the logistic model in one-dimensional space with a periodic boundary condition. Equal time spectra in wave number space and power spectra for several values of wave numbers are numerically obtained. A comparison of the numerical results of the equal time spectra with a simple theory is made to give a satisfactory agreement for large wave numbers.