The behavior of rings of coupled oscillators

Coupled oscillators in a ring are studied using perturbation and numerical methods. Stability of waves with nearest neighbor weak coupling is shown for a class of simple oscillators. Linkens' [23] model for colorectal activity is analyzed and several stable modes are found. Stability of waves with general (non nearest neighbor coupling) is determined and comparisons to the nearest neighbor case are made. Approximate solutions to a ring with inhomogeneities are compared with numerical simulations.

[1]  F. Busse,et al.  On the stability of steady finite amplitude convection , 1965, Journal of Fluid Mechanics.

[2]  T S Nelsen,et al.  Simulation of the electrical and mechanical gradient of the small intestine. , 1968, The American journal of physiology.

[3]  Dynamical Model of the Human Small Intestine , 1973 .

[4]  Nancy Kopell,et al.  Plane Wave Solutions to Reaction‐Diffusion Equations , 1973 .

[5]  Derek A. Linkens,et al.  Mathematical Modeling of the Colorectal Myoelectrical Activity in Humans , 1976, IEEE Transactions on Biomedical Engineering.

[6]  D. A. Linkens,et al.  The stability of entrainment conditions forRLC coupled van der pol oscillators used as a model for intestinal electrical rhythms , 1977 .

[7]  James P. Keener,et al.  On the Validity of the Two-Timing Method for Large Times , 1977 .

[8]  D. Linkens,et al.  Frequency Entrainment of Coupled Hodgkin-Huxley-Type Oscillators for Modeling Gastro-Intestinal Electrical Activity , 1977, IEEE Transactions on Biomedical Engineering.

[9]  F. Hoppensteadt,et al.  Slowly Modulated Oscillations in Nonlinear Diffusion Processes , 1977 .

[10]  Johan Grasman,et al.  Mutually synchronized relaxation oscillators as prototypes of oscillating systems in biology , 1979 .

[11]  J. Neu.  Coupled Chemical Oscillators , 1979 .

[12]  J. Cowan,et al.  Temporal oscillations in neuronal nets , 1979, Journal of mathematical biology.

[13]  P. Holmes,et al.  Bifurcation of periodic motions in two weakly coupled van der Pol oscillators , 1980 .

[14]  M. Kawato,et al.  Two coupled neural oscillators as a model of the circadian pacemaker. , 1980, Journal of theoretical biology.

[15]  A. Winfree.  The geometry of biological time , 1991 .

[16]  P. S. Hagan,et al.  Diffusion-induced morphogenesis in the development of Dictyostelium , 1981 .

[17]  J. Rinzel,et al.  Waves in a simple, excitable or oscillatory, reaction-diffusion model , 1981 .

[18]  G. Ermentrout.  n:m Phase-locking of weakly coupled oscillators , 1981 .

[19]  Patrick S. Hagan,et al.  Spiral Waves in Reaction-Diffusion Equations , 1982 .

[20]  G. Ermentrout.  Asymptotic Behavior of Stationary Homogeneous Neuronal Nets , 1982 .

[21]  R. Traub.  Simulation of intrinsic bursting in CA3 hippocampal neurons , 1982, Neuroscience.

[22]  R. Traub,et al.  Cellular mechanism of neuronal synchronization in epilepsy. , 1982, Science.

[23]  M. Golubitsky,et al.  Bifurcation on the hexagonal lattice and the planar Bénard problem , 1983, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[24]  G. Ermentrout,et al.  Frequency Plateaus in a Chain of Weakly Coupled Oscillators, I. , 1984 .

[25]  J. Alexander,et al.  Patterns at primary Hopf bifurcations at a plexus of identical oscillators , 1986 .