Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

1∶1 and 2∶1 phase entrainment in a system of two coupled limit cycle oscillators

  • 92 Accesses

  • 29 Citations


A model of a pair of coupled limit cycle oscillators is presented in order to investigate the extent of, and the transition between, 1∶1 and 2∶1 phase entrainment, a phenomenon exhibited by numerous diverse biological systems. The mathematical form of the model involves a flow on a phase torus given by two coupled first order nonlinear ordinary differential equations which govern the oscillators' phase angles (i.e. their respective positions around their limit cycles). The regions corresponding to 1∶1 and 2∶1 phase entrainment in an appropriate parameter space are determined analytically and numerically. The bifurcations occurring during the transition from 1∶1 to 2∶1 phase entrainment are discussed.

This is a preview of subscription content, log in to check access.


  1. 1.

    Cohen, A. H., Holmes, P. J., Rand, R. H.: The nature of the coupling between segmental oscillators of the lamprey spinal generator for locomotion: A mathematical model. J. Math. Biol. 13, 345–369 (1982)

  2. 2.

    Ermentrout, G. B., Kopell, N.: Frequency plateaus in a chain of weakly coupled oscillators, I. SIAM J. Math. Anal. 15, 215–237 (1984)

  3. 3.

    Cole, J. D.: Perturbation Methods in Applied Mathematics. Waltham: Blaisdell 1968

  4. 4.

    Denjoy, A.: Sur les courbes définies par les équations differentielles a la surface du tore. J. Math. 17 (IV) 333–375 (1932)

  5. 5.

    Glass, L., Mackey, M. C.: Pathological conditions resulting from instabilities in physiological control systems. Ann. N.Y. Acad. Sci. 316, 214–235 (1979)

  6. 6.

    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York: Springer 1983

  7. 7.

    Guevara, M. R., Glass, L.: Phase locking, period doubling bifurcations, and chaos in a mathematical model of a periodically driven oscillator: A theory for the entrainment of biological oscillators and the generation of cardiac dysrhythmias. J. Math. Biol. 14, 1–23 (1982)

  8. 8.

    Hoppensteadt, F. C., Keener, J. P.: Phase locking of biological clocks. J. Math. Biol. 15, 339–349 (1982)

  9. 9.

    Keith, W. L.: Phase Locking and Entrainment in Two Coupled Limit Cycle Oscillators. Ph.D. Thesis, Cornell University, pp. 17–32, 1983

  10. 10.

    Rand, R. H.: Computer Algebra in Applied Mathematics: An Introduction to MACSYMA. Boston: Pitman 1984

  11. 11.

    Rand, R. H., Holmes, P. J.: Bifurcation of periodic motions in two weakly coupled van der Pol oscillators. Int. J. Nonlinear Mech. 3, 387–399 (1980)

  12. 12.

    Rand, R. H., Storti, D. W., Upadhyaya, S. K., Cooke, J. R.: Dynamics of coupled stomatal oscillators. J. Math. Biol. 15, 131–149 (1982)

  13. 13.

    van der Pol, B., van der Mark, J.: The heart beat considered as a relaxation oscillation and electrical model of the heart. Phil. Mag. 6, 763–775 (1928)

  14. 14.

    Minorsky, N.: Nonlinear oscillations. New York: Kreiger 1974

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Keith, W.L., Rand, R.H. 1∶1 and 2∶1 phase entrainment in a system of two coupled limit cycle oscillators. J. Math. Biology 20, 133–152 (1984). https://doi.org/10.1007/BF00285342

Download citation

Key words

  • Oscillators
  • limit cycles
  • phase entrainment
  • bifurcations