Traveling waves, impulses and diffusion chaos in excitable media

• FitzHugh–Nagumo type systems can have an infinite number of different stable wave solutions.
• Such systems can have also an infinite number of different states of spatiotemporal chaos.
• Wave solutions can travel along the space axis with arbitrary speeds.
• Chaotic solutions describe chemical and biological turbulence in such systems.
• Such solutions are singular attractors from the Feigenbaum–Sharkovskii–Magnitskii theory.

In the present work it is shown, that the FitzHugh–Nagumo type system of partial differential equations with fixed parameters can have an infinite number of different stable wave solutions, traveling along the space axis with arbitrary speeds, and also traveling impulses and an infinite number of different states of spatiotemporal (diffusion) chaos. Those solutions are generated by cascades of bifurcations of cycles and singular attractors according to the FSM theory (Feigenbaum–Sharkovskii–Magnitskii) in the three-dimensional system of ordinary differential equations (ODEs), to which the FitzHugh–Nagumo type system of equations with self-similar change of variables can be reduced.