Summary

Proceedings of the 2012 International Symposium on Nonlinear Theory and its Applications

2012

Session Number:D3L-A

Session:

Number:887

Chaos and Oscillation Death in a Weakly Driven BVP Oscillator

Yoshimasa Shinotsuka,  Naohiko Inaba,  Munehisa Sekikawa,  Tetsuro Endo,  

pp.887-890

Publication Date:

Online ISSN:2188-5079

DOI:10.15248/proc.1.887

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Summary:
We discuss the bifurcation structure of oscillation death and chaos generated in a weakly driven BVP oscillator where the parameter values are chosen such that a stable focus and a stable relaxation oscillation coexist in close proximity when no perturbation is applied. Chaos and oscillation death coexist in the weakly driven BVP oscillator at B1 = 0.002 where B1 is an amplitude of the forcing term. A sudden disappearance of a chaotic oscillation is confirmed as B1 increases, and only oscillation death becomes an attractor. The basin of oscillation death is derived numerically. The boundaries of the basin are extremely complex. When we increase B1 a little, it is found that the basin becomes extremely large, and the solutions for all initial conditions converge to oscillation death at B1 = 0.004. It means that chaos drastically disappears when the amplitude of the forcing term is extremely weak.

References:

[1] A. Rabinovitch and I. Rogachevskii, “Threshold, excitability and isochrones in the Bonhoeffer-van der Pol system,” Chaos, Vol. 9, pp. 880-886, 1999.

[2] K. Shimizu, M. Sekikawa, and N. Inaba, “Mixed-mode oscillations and chaos from a simple second-order oscillator under weak periodic perturbation,” Phys. Lett. A, Vol. 375, pp. 1566-1569, 2011.

[3] M. Sekikawa, K. Shimizu, N. Inaba, H. Kita, T. Endo, K. Fujimoto, T. Yoshinaga, and K. Aihara, “Sudden change from chaos to oscillation death in the Bonhoeffer-van der Pol oscillator,” Phys. Rev. E, Vol. 84, pp. 056209-1-8, 2011.

[4] B. Braaksma and J. Grasman, “Critical dynamics of the Bonhoeffer-van der Pol equation and its chaotic response to periodic stimulation,” Physica D, Vol. 68, pp. 265-280, 1993.

[5] T. Nomura, S. Sato, S. Doi, JP. Segundo, and M. D. Stiber, “A Bonhoeffer-van der Pol oscillator model of locked and non-locked behaviors of living pacemaker neurons,” Biol. Cybern, Vol. 69, pp. 429-437, 1993.

[6] A. Rabinovitch, R. Thieberger, and M. Friedman, “Forced Bonhoeffer-van der Pol oscillator in its excited mode,” Phys. Rev. E, Vol. 50, pp. 1572-1578, 1994.

[7] T. Nomura, S. Sato, S. Doi, J. P. Segundo, and M. D. Stiber, “Global bifurcation structure of a Bonhoeffer-van der Pol oscillator driven by periodic pulse trains,” Biol. Cybern, Vol. 72, pp. 55-67, 1994.

[8] S. Doi and S. Sato, “Mathematical Biosciences, The global bifurcation structure of the BVP neuronal model driven by periodic pulse trains,” Math. Biosci, Vol. 125, pp. 229-250, 1995.

[9] I. Shimada and T. Nagashima, “A numerical approach to Ergordic problem of dissipative dynamical systems,” Prog. Theor. Phys., Vol. 61, pp. 1605-1615, 1979.

[10] N. Inaba and S.Mori, “Chaos via Torus Breakdown in a Piecewise-Linear Forced van der Pol Oscillator with a Diode” IEEE Trans. Circuit Syst., Vol. CAS-38, pp. 398-409, 1991.