Primary and secondary instabilities of the Mixed mode solution in a reaction diffusion system near the codimension-two Turing-Hopf point

Space-time structures near the codimension-two Turing-Hopf point (CTHP) that combine a spatial pattern with temporal oscillations are studied for a particular reaction diffusion (RD) system, which has been useful to model many different biological processes. We show that in this sytem, these Mixed mode solutions arise in regions of the parameter space where the analysis of the linearized RD system predicts only Turing or Hopf structures. Therefore, a standard weakly non linear analysis is used to find the amplitude equations describing the existence and stability regions of these three primary solutions and to study how they are affected by the proximity to the CTHP, the relative strength of the nonlinear terms of the system and the diffusion coefficient. By means of exhaustive numerical calculations, it is shown that the stability analysis to homogeneous perturbations qualitatively predicts the region where Mixed solutions arise. We also show that if the size of the domain grows, non-homogeneous perturbations become important and secondary instabilities can arise in the system. Two types of secondary structures are numerically identified: (1) spatial patterns that change its effective wave number while they oscillate as an Eckhaus instability, or (2) a chaotic oscillations mixed with a relatively fixed spatial pattern in a Benjamin–Feir–Newell phase turbulence. Our study provides a characterization of the time oscillating patterns observed numerically in the RD system studied here and a panoramic view of the Mixed mode primary and secondary instabilities that arise near the CTHP.