Abstract
Two weakly coupled oscillators are studied and the existence of n:m phase-locked solutions is shown. With the use of a slow time scale, the problem is reduced to a two-dimensional system on an invariant attracting torus. This system is further reduced to a one-dimensional dynamical system. Fixed points of this system correspond to n:m phase-locked solutions. The method is applied to a forced oscillator, linearly coupled λ-ω systems, and a pair of integrate and fire neuron models.
Similar content being viewed by others
References
Boiteux, A., Goldbeter, A., Hess, B.: Control of oscillating glycolysis of yeast by stochastic, periodic, and steady source of substrate: A model and experimental study. Proc. Natl. Acad. Sci. USA 72, 3829–3833 (1975)
Ermentrout, G. B., Rinzel, J.: Waves in a simple, excitable or oscillatory, reaction-diffusion model. J. Math. Biol. 11, 269–294 (1981)
Fenichel, N.: Persistence and smoothness of invariant manifolds of flows. Ind. Math. J. 21, 193–226 (1971)
Flaherty, J. E., Hoppensteadt, F. C.: Frequency entrainment of a forced van der Pol oscillator. Stud. Appl. Math. 58, 5–15 (1978)
Glass, L., Mackey, M.: A simple model for phase locking of biological oscillators. J. Math. Biol. 7, 339–352 (1979)
Grasman, J., Veling, E. J. M., Willems, G. M.: Relaxation oscillations governed by a van der Pol equation. SIAM J. Appl. Math. 31, 667–676 (1976)
Grasman, J., Jansen, M. J. W.: Mutually synchronized relaxation oscillators as prototypes of oscillating systems in biology. J. Math. Biol. 7, 171–197 (1979)
Hale, J. K.: Ordinary differential equations. New York: Wiley-Interscience 1969
Holmes, P. J.: A nonlinear oscillator with a strange attractor. Phil. Trans. Roy. Soc. London A292, 419–448 (1979)
Holmes, P. J.: Averaging and chaotic motions in forced oscillations. SIAM J. Appl. Math. 38, 65–80 (1980)
Holmes, P. J.: Phaselocking and chaos in coupled limit cycle oscillators. Preprint (1981)
Keener, J. P., Hoppensteadt, F. C., Rinzel, J.: Integrate-and-fire models of nerve membrane response to oscillatory input. SIAM J. Appl. Math. Preprint (1981)
Levinson, N.: Small periodic perturbations of an autonomous system with a stable orbit. Ann. Math. 52, 727–738 (1950)
Littlewood, J. E.: On nonlinear differential equations of the second order. III. Acta. Math. 97, 267–308 (1957)
Neu, J. C.: Nonlinear oscillations in discrete and continuous systems. Ph.D. Thesis, California Institute of Technology, June 1978
Perkel, D. H., Schulman, J. H., Bullock, T. H., Moore, G. P., Segundo, J. P.: Pacemaker neurons: Effect of regularly spaced synaptic input. Science 145, 61–63 (1964)
Rand, R. H., Holmes, P. J.: Bifurcation of periodic motions in two weakly coupled van der Pol oscillators. Preprint (1981)
Reid, J. V. O.: The cardiac pacemaker: Effects of regularly spaced nervous input. Amer. Heart J. 78, 58–64 (1969)
Stein, P. S. G.: Application of the mathematics of coupled oscillator systems to the analysis of the neural control of locomotion. Fed. Proc. 36, 2056–2059 (1977)
Wever, R. A.: The arcadian system of man. New York: Springer 1979
Winfree, A. T.: The geometry of biological time. Biomathematics Vol. 8. New York: Springer 1980
Yamanishi, J., Kawato, M., Suzuki, R.: Two coupled oscillators as a model for coordinated fingertapping by both hands. Biol. Cybern. 37, 219–227 (1980)
Guttman, R., Feldman, L., Jakobsson, E.: Frequency entrainment of squid axon membrane. J. Memb. Biol. 56, 9–18 (1980)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ermentrout, G.B. n:m Phase-locking of weakly coupled oscillators. J. Math. Biology 12, 327–342 (1981). https://doi.org/10.1007/BF00276920
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00276920