Chaos and bifurcation in the space-clamped FitzHugh–Nagumo system

Abstract 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 γ and ω affect the dynamical behaviors significantly.

[1]  R. FitzHugh.  Impulses and Physiological States in Theoretical Models of Nerve Membrane. , 1961, Biophysical journal.

[2]  Alfredo Medio,et al.  Nonlinear Dynamics: A Primer , 2001 .

[3]  Sze-Bi Hsu,et al.  Snapback repellers as a cause of chaotic vibration of the wave equation with a van der Pol boundary condition and energy injection at the middle of the span , 1998 .

[4]  Weirui Zhao,et al.  On the Mathematical Clarification of the SNAP-Back-Repeller in High-Dimensional Systems and Chaos in a Discrete Neural Network Model , 2002, Int. J. Bifurc. Chaos.

[5]  E. Ott.  Chaos in Dynamical Systems: Contents , 1993 .

[6]  A. Hastings,et al.  Persistence of Transients in Spatially Structured Ecological Models , 1994, Science.

[7]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[8]  J. Yorke,et al.  Crises, sudden changes in chaotic attractors, and transient chaos , 1983 .

[9]  Kunihiko Kaneko,et al.  Supertransients, spatiotemporal intermittency and stability of fully developed spatiotemporal chaos , 1990 .

[10]  Ruiqi Wang,et al.  Chaos Behavior in the Discrete BVP oscillator , 2002, Int. J. Bifurc. Chaos.

[11]  Crutchfield,et al.  Are attractors relevant to turbulence? , 1988, Physical review letters.

[12]  Ying-Cheng Lai,et al.  Persistence of supertransients of spatiotemporal chaotic dynamical systems in noisy environment , 1995 .

[13]  F. R. Marotto.  Snap-back repellers imply chaos in Rn , 1978 .

[14]  Y. Pomeau,et al.  Intermittent transition to turbulence in dissipative dynamical systems , 1980 .

[15]  J. Yorke,et al.  CHAOTIC ATTRACTORS IN CRISIS , 1982 .

[16]  Arun V. Holden,et al.  Three-dimensional organisation of re-entrant propagation during experimental ventricular fibrillation , 2002 .

[17]  S. Yoshizawa,et al.  An Active Pulse Transmission Line Simulating Nerve Axon , 1962, Proceedings of the IRE.

[18]  A. Holden,et al.  Chaotic meander of spiral waves in the FitzHugh-Nagumo system , 1995 .