Bifurcation and Chaos in Coupled BVP oscillators

Bonhoffer–van der Pol(BVP) oscillator is a classic model exhibiting typical nonlinear phenomena in the planar autonomous system. This paper gives an analysis of equilibria, periodic solutions, strange attractors of two BVP oscillators coupled by a resister. When an oscillator is fixed its parameter values in nonoscillatory region and the others in oscillatory region, create the double scroll attractor due to the coupling. Bifurcation diagrams are obtained numerically from the mathematical model and chaotic parameter regions are clarified. We also confirm the existence of period-doubling cascades and chaotic attractors in the experimental laboratory.

[1]  Guanrong Chen,et al.  Bifurcation Analysis of Chen's equation , 2000, Int. J. Bifurc. Chaos.

[2]  H. Kawakami,et al.  Bifurcation of a Modified BVP Circuit Model for Neurons Generating Rectangular Waves (Special Section on Nonlinear Theory and Its Applications) , 1999 .

[3]  Hiroyuki Kitajima,et al.  Bifurcations of Periodic Solutions in a Coupled Oscillator with Voltage Ports , 1998 .

[4]  Hiroshi Kawakami,et al.  Synchronization and Chaos of Coupled Duffing-Rayleigh Oscillators (Special Section on Nonlinear Theory and its Applications) , 1996 .

[5]  Hiroshi Kawakami,et al.  Symmetrical Properties and Bifurcations of the Periodic Solutions for a Hybridly Coupled Oscillator , 1995 .

[6]  H. Kawakami,et al.  Resistively Coupled Oscillators with Hybrid Connection , 1995 .

[7]  Leon O. Chua,et al.  Global unfolding of Chua's circuit , 1993 .

[8]  A. Hodgkin,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve , 1990, Bulletin of mathematical biology.

[9]  L. Chua,et al.  The double scroll family , 1986 .

[10]  L. O. Chua,et al.  The double scroll family. I: Rigorous of chaos. II: Rigorous analysis of bifurcation phenomena , 1986 .

[11]  Hiroshi Kawakami,et al.  Bifurcation of periodic responses in forced dynamic nonlinear circuits: Computation of bifurcation values of the system parameters , 1984 .

[12]  Tetsuro Endo,et al.  Mode analysis of a ring of a large number of mutually coupled van der Pol oscillators , 1978 .

[13]  A. Bautin.  Qualitative investigation of a particular nonlinear system: PMM vol. 39, n≗ 4, 1975, pp. 633–641 , 1975 .

[14]  D. Linkens.  Analytical solution of large numbers of mutually coupled nearly sinusoidal oscillators , 1974 .

[15]  S. Yoshizawa,et al.  An Active Pulse Transmission Line Simulating Nerve Axon , 1962, Proceedings of the IRE.

[16]  R. FitzHugh.  Impulses and Physiological States in Theoretical Models of Nerve Membrane. , 1961, Biophysical journal.

[17]  A. Hodgkin,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve , 1952, The Journal of physiology.