Spatiotemporal dynamics near a codimension-two point

Spatiotemporal dynamics resulting from the interaction of two instabilities breaking, respectively, spatial and temporal symmetries are studied in the framework of the amplitude equation formalism. The corresponding bifurcation scenarios feature steady-Hopf bistability with corresponding localized structures but also different types of mixed states. Some of these mixed modes result from self-induced subharmonic instabilities of the pure steady and Hopf modes. The bifurcation schemes are then used to organize the results of numerical simulations of a one-dimensional reaction-diffusion model. These dynamics are relevant to experimental chemical systems featuring a codimension-two Turing-Hopf pc,int but also to any experimental setup where homogeneous temporal oscillations and spatial patterns are obtained for nearby values of parameters.