# zbMATH — the first resource for mathematics

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
DMC
Full Text:
##### References:
 [1] FitzHugh, R., Impulses and physiological states in theoretical models of nerve membranes, Biophys. J., 1, 445-466, (1961) [2] Nagumo, J.S.; Arimoto; Yoshizawa, S., An active pulse transmission line simulating nerve axon, Proc. IRE, 50, 2061-2071, (1962) [3] Zhang, H.; Holden, A.V., Chaotic meander of spiral waves in the fitzhugh – nagumo system, Chaos, solitons & fractals, 5, 3/4, 661-670, (1995) · Zbl 0925.92028 [4] Biktashev, V.N.; Holden, A.V.; Mironov, S.F.; Pertsov, A.M., Three-dimensional organisation of re-entrant propagation during experimental ventricular fibrillation, Chaos, solitons & fractals, 13, 8, 1713-1733, (2002) [5] Mira, C.; Gardini, L.; Borugola, A.; Cathala, J., Chaos dynamics in two-dimensional noninvertible maps, (1996), World Scientific Singapore [6] Jing, Z.J.; Jia, Z.Y.; Wang, R.Q., Chaos behavior in the discrete BVP oscillator, Int. J. bifurc. chaos, 12, 3, 619-627, (2002) · Zbl 1069.65141 [7] Chen, G.; Hsu, S.B.; Zhou, J., 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, J. math. phys., 39, 2, 6459-6489, (1998) · Zbl 0959.37027 [8] Marotto, F.R., Snap-back repellers imply chaos in Rn, J. math. anal. appl., 63, 199-223, (1978) · Zbl 0381.58004 [9] Lin, W.; Ruan, J.; Zhao, W., On the mathematical clarification of the snap-back repeller in high-dimensional systems and chaos in a discrete neural network model, Int. J. bifurc. chaos, 12, 5, 1129-1139, (2002) · Zbl 1044.37020 [10] Wiggins, S., An introduction to applied nonlinear dynamics and chaos, (1990), Springer-Verlag New York [11] Grebogi, C.; Ott, E.; Yorke, J.A., Chaotic attractors in crisis, Phy. rev. lett, 48, 1507-1510, (1982) [12] Grebogi, C.; Ott, E.; Yorke, J.A., Crises, sudden changes in chaotic attractors and chaotic transients, Physica D, 7, 181-200, (1983) [13] Pomeau, Y.; Manneville, P., Intermittent transition to turbulance in dissipative dynamical systems, Comm. math. phys., 74, 189-197, (1980) [14] Crutchfield, J.P.; Kaneko, K., Are attractors relevant to turbulance?, Phys. rev. lett., 60, 2715-2718, (1988) [15] Kaneko, K., Supertransients, spatiotemporal intermittency and stability of fully developed spatiotemporal chaos, Phys. lett. A, 149, 105-112, (1990) [16] Hastings, A.; Higgins, K., Persistence of transients in spatially structured ecological models, Science, 263, 1133-1136, (1994) [17] Lai, Y.C., Persistence of supertransients of spatiotemporal chaotic dynamical systems in noisy environment, Phys. lett. A, 200, 418-422, (1995) [18] Medio, A.; Lines, M., Nonlinear dynamics: a primer, (2001), Cambridge University Press Cambridge · Zbl 1008.37001 [19] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, (1999), Springer-Verlag New York [20] Ott, E., Chaos in dynamical systems, (1993), Cambridge University Press Cambridge · Zbl 0792.58014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.