Mixed-mode oscillations and chaos from a simple second-order oscillator under weak periodic perturbation

Abstract In this study, we propose a remarkably simple oscillator that exhibits extremely complicated behaviors. The second-order nonautonomous differential equation discussed in this Letter is considered to be one of the simplest dynamics that can produce mixed-mode oscillations (MMOs) and chaos. Our model uses a Bonhoeffer–van der Pol (BVP) oscillator under weak periodic perturbation. The parameter set of the BVP equation is chosen such that a focus and a relaxation oscillation coexist when no perturbation is applied. Under weak periodic perturbation, various types of MMOs and chaos with remarkably complicated waveforms are observed.

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

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

[3]  Bard Ermentrout,et al.  Simulating, analyzing, and animating dynamical systems - a guide to XPPAUT for researchers and students , 2002, Software, environments, tools.

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

[5]  I. Shimada,et al.  A Numerical Approach to Ergodic Problem of Dissipative Dynamical Systems , 1979 .

[6]  Evidence of homoclinic chaos in the plasma of a glow discharge. , 1992, Physical review letters.

[7]  Hayashi.  Mixed-mode oscillations and chaos in a glow discharge , 2000, Physical review letters.

[8]  S. K. Dana,et al.  Shil'nikov chaos and mixed-mode oscillation in Chua circuit. , 2010, Chaos.

[9]  KITDAKORN KLOMKARN,et al.  New Construction of Mixed-Mode Chaotic Circuits , 2010, Int. J. Bifurc. Chaos.

[10]  Georgi S Medvedev,et al.  Chaos at the border of criticality. , 2007, Chaos.

[11]  Georgi Medvedev,et al.  Multimodal oscillations in systems with strong contraction , 2007 .

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

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

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

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

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

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

[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 .