Pattern formation in the turing-Hopf codimension-2 phase space in a reaction-diffusion system

We systematically investigate the behaviour of pattern formation in a reaction-diffusion system when the system is located near the Turing-Hopf codimension-2 point in phase space. The chloride-iodide-malonic acid (CIMA) reaction is used in this study. A phase diagram is obtained using the concentration of polyvinyl alcohol (PVA) and malonic acid (MA) as control parameters. It is found that the Turing-Hopf mixed state appears only in a small vicinity near the codimension-2 point, and has the form of hexagonal pattern overlapped with anti-target wave; the boundary line separating the Turing state and the wave state is independent of the concentration of MA, only relies on the concentration of PVA. The corresponding numerical simulation using the Lengyel-Epstein (LE) model gives a similar phase diagram as the experiment; it reproduces most patterns observed in the experiment. However, the mixed state we obtain in simulation only appears in the anti-wave tip area, implying that the 3-D effect in the experiments may change the pattern forming behaviour in the codimension-2 regime.