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Chaos and bifurcation in the space-clamped FitzHugh–Nagumo system. (English) Zbl 1045.37058
Summary: The discrete space-clamped FitzHugh-Nagumo system obtained by the Euler method is investigated. It is proven that there are SNB and Naimark-Sacker bifurcations and chaotic phenomena in the sense of Marotto. The numerical simulations presented not only show the consistence of our results with the theoretical analysis but also exhibit the complex dynamical behaviors. This includes the period-9, 10, 16, 25, 27 and 42 orbits, quasi-period orbits, cascade of period-doubling bifurcations, pitchfork bifurcation, attractor merging crisis and boundary crisis, intermittency as well as supertransient and chaotic bands. The computation of Lyapunov exponents confirms the dynamical behaviors. We also consider the influence of periodic external signal on the original system via numerical simulations and find that \(\gamma\) and \(\omega\) affect the dynamical behaviors significantly.

MSC:
37N25 Dynamical systems in biology
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37G10 Bifurcations of singular points in dynamical systems
92C20 Neural biology
Software:
DMC
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References:
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