Bifurcation and chaos in discrete FitzHugh–Nagumo system

The discrete FitzHugh–Nagumo system obtained by Euler method is investigated. The conditions of existence for fold bifurcation, flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory, chaotic behavior in the sense of Marotto’s definition of chaos is proved. And numerical simulation results not only show the consistence with the theoretical analysis but also display the new and interesting dynamical behaviors, including attracting invariant circle, period-3, period-6, period-7, period-9, period-15, period-20, period-21, and period-n orbits, an inverse cascade of period-doubling bifurcation in period-3, cascade of period-doubling bifurcation in periods-9, 15, 20 and 21, interior and exterior crisis phenomena, intermittency mechanic, transient chaos in period-window, attracting and non-attracting chaotic attractors. The computations of Lyapunov exponents confirm the chaotic behaviors.