Piecewise-linear Bonhoeffer–van der Pol dynamics explaining mixed-mode oscillation-incrementing bifurcations

Piecewise-linear Bonhoeffer–van der Pol dynamics explaining mixed-mode oscillation-incrementing bifurcations Kuniyasu Shimizu1,∗ and Naohiko Inaba2 1Department of Electrical, Electronics and Computer Engineering, Chiba Institute of Technology, Chiba 275-0016, Japan 2Organization for the Strategic Coordination of Research and Intellectual Property, Meiji University, Kawasaki 214-8571, Japan ∗E-mail: kuniyasu.shimizu@it-chiba.ac.jp

[1]  S Sato,et al.  The global bifurcation structure of the BVP neuronal model driven by periodic pulse trains. , 1995, Mathematical biosciences.

[2]  Irving R. Epstein,et al.  Systematic design of chemical oscillators. Part 13. Complex periodic and aperiodic oscillation in the chlorite-thiosulfate reaction , 1982 .

[3]  Takashi Tsubouchi,et al.  Chaos via duck solution breakdown in a piecewise linear van der Pol oscillator driven by an extremely small periodic perturbation , 2004 .

[4]  Munehisa Sekikawa,et al.  Chaos disappearance in a piecewise linear Bonhoeffer–van der Pol dynamics with a bistability of stable focus and stable relaxation oscillation under weak periodic perturbation , 2014 .

[5]  E. Kutafina Mixed mode oscillations in the Bonhoeffer-van der Pol oscillator with weak periodic perturbation , 2013 .

[6]  David J. W. Simpson,et al.  Mixed-mode oscillations in a stochastic, piecewise-linear system , 2010, 1010.1504.

[7]  K Pakdaman,et al.  Dynamics of moments of FitzHugh-Nagumo neuronal models and stochastic bifurcations. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  N. Buric,et al.  Dynamics of FitzHugh-Nagumo excitable systems with delayed coupling. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Kuniyasu Shimizu,et al.  Complex mixed-mode oscillations in a Bonhoeffer–van der Pol oscillator under weak periodic perturbation , 2012 .

[10]  Horacio G. Rotstein,et al.  Introduction to focus issue: mixed mode oscillations: experiment, computation, and analysis. , 2008, Chaos.

[11]  S Sato,et al.  Response characteristics of the BVP neuron model to periodic pulse inputs. , 1992, Mathematical biosciences.

[12]  J. L. Hudson,et al.  An experimental study of multiple peak periodic and nonperiodic oscillations in the Belousov–Zhabotinskii reaction , 1979 .

[13]  Arnaud Tonnelier,et al.  McKean caricature of the FitzHugh-Nagumo model: traveling pulses in a discrete diffusive medium. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  Harry L. Swinney,et al.  Complex periodic oscillations and Farey arithmetic in the Belousov–Zhabotinskii reaction , 1986 .

[15]  I. Rogachevskii,et al.  Threshold, excitability and isochrones in the Bonhoeffer-van der Pol system. , 1999, Chaos.

[16]  Stephen Coombes,et al.  Canard-Like Explosion of Limit Cycles in Two-Dimensional Piecewise-Linear Models of FitzHugh-Nagumo Type , 2012, SIAM J. Appl. Dyn. Syst..

[17]  F. Albahadily,et al.  Mixed‐mode oscillations in an electrochemical system. I. A Farey sequence which does not occur on a torus , 1989 .

[18]  M. Koper Bifurcations of mixed-mode oscillations in a three-variable autonomous Van der Pol-Duffing model with a cross-shaped phase diagram , 1995 .

[19]  Shinji Doi,et al.  A Bonhoeffer-van der Pol oscillator model of locked and non-locked behaviors of living pacemaker neurons , 1993, Biological Cybernetics.

[20]  Kuniyasu Shimizu,et al.  Mixed-mode oscillations and chaos from a simple second-order oscillator under weak periodic perturbation , 2011 .

[21]  Shinji Doi,et al.  Global bifurcation structure of a Bonhoeffer-van der Pol oscillator driven by periodic pulse trains , 2004, Biological Cybernetics.

[22]  Erik S. Van Vleck,et al.  Spatially Discrete FitzHugh--Nagumo Equations , 2005, SIAM J. Appl. Math..

[23]  Takashi Hikihara,et al.  Period-doubling cascades of canards from the extended Bonhoeffer–van der Pol oscillator , 2010 .

[24]  Salvador Balle,et al.  Experimental evidence of van der Pol-Fitzhugh-Nagumo dynamics in semiconductor optical amplifiers. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  Avinoam Rabinovitch,et al.  Resonance effects in the Bonhoeffer-van der Pol system , 1996 .

[26]  Valery Petrov,et al.  Mixed‐mode oscillations in chemical systems , 1992 .

[27]  Shanmuganathan Rajasekar,et al.  Period doubling route to chaos for a BVP oscillator with periodic external force , 1988 .

[28]  H. McKean Nagumo's equation , 1970 .

[29]  Helwig Löffelmann,et al.  GEOMETRY OF MIXED-MODE OSCILLATIONS IN THE 3-D AUTOCATALATOR , 1998 .

[30]  Belinda Barnes,et al.  NUMERICAL STUDIES OF THE PERIODICALLY FORCED BONHOEFFER VAN DER POL SYSTEM , 1997 .

[31]  Raymond Kapral,et al.  Microscopic model for FitzHugh-Nagumo dynamics , 1997 .

[32]  J. G. Freire,et al.  Stern-Brocot trees in cascades of mixed-mode oscillations and canards in the extended Bonhoeffer-van der Pol and the FitzHugh-Nagumo models of excitable systems , 2011 .

[33]  Johan Grasman,et al.  Critical dynamics of the Bonhoeffer–van der Pol equation and its chaotic response to periodic stimulation , 1993 .

[34]  L Schimansky-Geier,et al.  Analytical approach to the stochastic FitzHugh-Nagumo system and coherence resonance. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[35]  K. Aihara,et al.  Array-enhanced coherence resonance and forced dynamics in coupled FitzHugh-Nagumo neurons with noise. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[36]  Kazuyuki Aihara,et al.  Sudden change from chaos to oscillation death in the Bonhoeffer-van der Pol oscillator under weak periodic perturbation. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[37]  G. Flores Stability analysis for the slow traveling pulse of the Fitzhugh-Nagumo system , 1991 .

[38]  Irving R. Epstein,et al.  Systematic design of chemical oscillators. Part 13. Complex periodic and aperiodic oscillation in the chlorite-thiosulfate reaction , 1982 .

[39]  Nikola Popović,et al.  Three Time-Scales In An Extended Bonhoeffer–Van Der Pol Oscillator , 2014, Journal of Dynamics and Differential Equations.

[40]  Nancy Kopell,et al.  Mixed-Mode Oscillations in Three Time-Scale Systems: A Prototypical Example , 2008, SIAM J. Appl. Dyn. Syst..

[41]  Kuniyasu Shimizu,et al.  Experimental study of complex mixed-mode oscillations generated in a Bonhoeffer-van der Pol oscillator under weak periodic perturbation. , 2015, Chaos.