zbMATH — the first resource for mathematics

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.

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
Full Text: DOI
[1] DOI: 10.1016/0167-2789(93)90084-E · Zbl 0779.34032 · doi:10.1016/0167-2789(93)90084-E
[2] DOI: 10.1016/0303-2647(77)90002-8 · doi:10.1016/0303-2647(77)90002-8
[3] DOI: 10.1016/S0006-3495(61)86902-6 · doi:10.1016/S0006-3495(61)86902-6
[4] DOI: 10.1073/pnas.74.4.1543 · doi:10.1073/pnas.74.4.1543
[5] DOI: 10.1113/jphysiol.1952.sp004764 · doi:10.1113/jphysiol.1952.sp004764
[6] DOI: 10.1007/BFb0103262 · doi:10.1007/BFb0103262
[7] DOI: 10.1016/0022-247X(78)90115-4 · Zbl 0381.58004 · doi:10.1016/0022-247X(78)90115-4
[8] DOI: 10.1142/9789812798732 · doi:10.1142/9789812798732
[9] DOI: 10.1016/0167-2789(93)90211-I · Zbl 0790.93072 · doi:10.1016/0167-2789(93)90211-I
[10] DOI: 10.1016/0960-0779(92)90036-M · Zbl 0768.58032 · doi:10.1016/0960-0779(92)90036-M
[11] Sugie J., Quart. Appl. Maths. 49 pp 543–
[12] Treskov S. A., Quart. Appl. Math. 54 pp 601–
[13] DOI: 10.1080/14786441108564652 · doi:10.1080/14786441108564652
[14] DOI: 10.1007/978-1-4757-4067-7 · doi:10.1007/978-1-4757-4067-7
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.