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Chaos behavior in the discrete BVP oscillator. (English) Zbl 1069.65141
The authors apply the forward Euler scheme to the discrete Bonhoeffer-van der Pol (BVP) oscillator where the stimulus is considered as the constant value and investigate this version as discrete dynamical system in $$\mathbb{R}^2$$. They first rigorously prove that this discrete model possesses the chaotic phenomenon in the sense of F. R. Marotto’s definition [J. Math. Anal. Appl. 63, 199–223 (1978; Zbl 0381.58004)] when the stepsize is larger than the critical value. Meanwhile the numerical simulations also perfectly support the theoretical analysis and also exhibit the complex dynamics including the ten-periodic orbit, a cascade of period-doubling bifurcation, quasiperiodic orbits and the chaotic orbits in Marotto’s chaos and intermittent’s chaos.

##### MSC:
 65P20 Numerical chaos 65P30 Numerical bifurcation problems 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior 37M20 Computational methods for bifurcation problems in dynamical systems 37N25 Dynamical systems in biology 92C20 Neural biology 37C27 Periodic orbits of vector fields and flows 37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
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