We focus on the long time behavior of complex networks of reactiondiffusion (RD) systems. We prove the existence of the global attractor and a L ∞ -bound for a network of n RD systems with d variables each. This allows us to prove the identical synchronization for general class of networks and establish the existence of a coupling strength threshold value that ensures such a synchronization. Then, we apply these results to some particular networks with different structures (i.e. different topologies) and perform numerical simulations. We found out theoretical and numerical heuristic laws for the minimal coupling strength needed for synchronization relatively to the number of nodes and the network topology, and discuss the link between spatial dimension and synchronization.

We focus on the long time behavior of complex networks of reactiondiffusion systems. We prove the existence of the global attractor and the L ∞bound for networks of n reaction-diffusion systems that belong to a class that generalizes the FitzHugh-Nagumo reaction-diffusion equations.

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Spiral solutions or spiral waves can be found in many natural systems. Spiral waves were observed in studies about the potential in brain and heart cells. Their appearance in the human heart is a presentation of arrhythmia. The paper showed how to create spiral solutions of diffusion-reaction system of FitzHugh-Nagumo type and the transition of spiral solutions according to the time step and space step discretization of finite difference method. Decreasing the value of space step discretization makes the spiral wave grow bigger, but if the value of time step discretization is increased at the same given space step, the finite difference method will be explosive, meaning that spiral wave no longer exists.