Analysis of complex and chaotic patterns near a codimension-2 Turing-Hopf point in a reaction-diffusion model

We study a reaction-diffusion system of activator-inhibitor type, and characterize the resulting complex and chaotic spatio-temporal patterns by their Lyapunov spectrum and by a Karhunen-Loeve decomposition into empirical orthogonal eigenmodes. Different periodic patterns corresponding to localized Hopf and Turing modes, and mixed modes including subharmonic spatio-temporal spiking are found near a codimension-2 bifurcation point. The asymptotic patterns are preceded by transient spatio-temporal chaos, before the system abruptly locks into a periodic state in space and time. The Karhunen-Loeve decomposition is shown to be a powerful tool for extracting detailed quantitative information of complex space-time data.