Self-Synchronization in Globally Coupled Oscillators with Hysteric response

Hisa-Aki Tanaka, Allan J. Lichtenberg, and Shin'ichi Oishi.
Physica D, vol. 100, pp. 279-300, 1997.


We analyze a large system of nonlinear phase oscillators with sinusoidal nonlinearity, uniformly distributed natural frequencies and global all-to-all coupling, which is an extension of Kuramoto's model to second-order systems. For small coupling, the system evolves to an incoherent state with the phases of all the oscillators distributed uniformly. As the coupling is increased, the system exhibits a discontinuous transition to the coherently synchronized state at a pinning threshold of the coupling strength, or to a partially synchronized oscillation coherent state at a certain threshold below the pinning threshold. If the coupling is decreased from a strong coupling with all the oscillators synchronized coherently, this coherence can persist until the depinning threshold which is less than the pinning threshold, resulting in hysteretic synchrony depending on the initial configuration of the oscillators. We obtain analytically both the pinning and depinning threshold and also expalin the discontinuous transition at the thresholds for the underdamped case in the large system size limit. Numerical exploration shows the oscillatory partially coherent state bifurcates at the depinning threshold and also suggests that this state persists independent of the system size. The system studied here provides a simple model for collective behaviour in damped driven high-dimensional Hamiltonian systems which can explain the synchronous firing of certain fireflies or neural oscillators with frequency adaptation and may also be applicable to interconnected power systems.

Download PDF

Figures at a glance


  1. Y. Aizawa, Synergetic approach to the phenomena of mode-locking in nonlinear systems, Prog. Theoret. Phys. 56 (1976) 703-716.
  2. L.L. Bonilla, J.C. Neu and R. Spigler, J. Stat. Phys. 67 (1992) 313.
  3. J.D. Crawford, Scaling and singularities in the entrainment of globally coupled oscillators, Phys. Rev. Lett. 74 (1995) 4341-4344.
  4. F. Crick and C. Koch, Toward a neurobiological theory of consciousness, Seminars Neurosci. 2 (1990) 263-275.
  5. H. Daido, Lower critical dimension for populations of oscillators with randomly distributed frequencies: A renormalization group analysis, Phys. Rev. Lett. 61 (1988) 231-234.
  6. H. Daido, Onset of cooperative entrainment in limit-cycle oscillators with uniform all-to-all interactions: Bifurcation of the order function, Physica D 91 (1996) 24-66.
  7. M. de Sousa Vieira, A.J. Lichtenberg and M.A. Lieberman, Self-synchronization of many coupled oscillators, Int. J. Bifurcation Chaos 4 (1994) 1563-1577.
  8. G.B. Ermentrout, An adaptive model for synchrony in the firefly Pteroptyx malaccae, J. Math. Biol. 29 (1991) 571-585.
  9. C.M. Gray, P. Konig, A.K. Engel and W. Singer, Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflect global stimulus properties, Nature 338 (1989) 334-337.
  10. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, New York. 1983).
  11. P. Hadley and M.R. Beasley, Dynamical states and stability of linear arrays of Josephson junctions, J. Appl. Phys. 50 (1987) 621-623.
  12. FE. Hanson, Pacemaker control of rhythmic flashing of fireflies, in: Cellular Pacemakers, ed. D.O. Carpenter, Vol. 2 (Wiley, New York, 1982) pp. 81-100.
  13. J.J. Hopfield, Pattern recognition computation using action potential timing for stimulus representation, Nature 376 (1995) 33-36.
  14. T. Konishi and K. Kaneko, Clustered motion in symplectic coupled map systems, J. Phys. A. 25 (1992) 6283-6296.
  15. Y. Kuramoto, Self-entrainment of a population of coupled nonlinear oscillators, in: Int. Syrup. on Mathematical Problems in Theoretical Physics, ed. H. Araki, Lecture Notes in Physics, Vol. 39 (Springer, New York, 1975) pp. 420-422.
  16. Y. Kuramoto, Chemical Oscillation, Waves and Turbulence (Springer, Berlin, 1984).
  17. G. Laurent and H. Davidowitz, Encoding of olfactory information with oscillating neural assemblies, Science 265 (1994) 1872-1875.
  18. M. Levi, EC. Hoppensteadt and W.L. Miranker, Dynamics of the Josephson junction, Quart. Appl. Math. (1978) 167-198.
  19. A.J. Lichtenberg and M.A. Lieberman, Regular and Chaotic Dynamics, 2nd Ed. (Springer, New York, 1992).
  20. P.C. Mattews, R.E. Mirollo and S.H. Strogatz, Dynamics of a large system of coupled nonlinear oscillators, Physica D 52 (1991) 293-331.
  21. R.E. Mirollo and S.H. Strogatz, Synchronization of pulse-coupled biological oscillators, SIAM J. Appl. Math. 50 (1990) 1645-1662.
  22. N. Nakagawa and Y. Kuramoto, From collective oscillations to collective chaos in a globally coupled oscillator systems, Physica D 75 (I 994) 74-80.
  23. E. Niebur, H.G. Schuster, D.M. Kammem and C. Koch, Oscillator-phase coupling from different two-dimensional network connectivities, Phys. Rev. A 44 (1991) 6895-6904.
  24. H. Sakaguchi and Y. Kuramoto, A soluble active rotator model showing phase transitions via mutual entrainment, Progr. Theoret. Phys. 76 (1986) 576-581.
  25. F.M.A. Salam, J.E. Marsden and P.P. Varaiya, Arnold diffusion in the swing equations of a power system, 1EEE Trans. CAS 32 (1983) 784-796.
  26. K. Satoh, Computer experiment on the cooperative behaviour of a network of interacting nonlinear oscillators, J. Phys. Soc. Japan 58 (1989) 2010-2021.
  27. S.H. Strogatz, Norbert Wiener's brain waves, in: Lecture Notes in Biomathematics, Vol. 100 (Springer, Berlin, 1994).
  28. S.H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, Reading, MA, 1994) p. 273.
  29. S.H. Strogatz, C.M. Marcus, R.E. Mirollo and R.M. Westervelt, Collective dynamics of coupled oscillators with random pinning, Physica D 36 (1989) 23-50.
  30. S.H. Strogatz and R.E. Mirollo, Phase-locking and critical phenomena in lattices of coupled nonlinear oscillators with random intrinsic frequencies, Physica D 31 (1988) 143-168.
  31. S.H. S trogatz and R.E. Mirollo, Stability of incoherence in a population of coupled oscillators, J. Stat. Phys. 63 (1991) 613-635.
  32. S. Watanabe and S.H. Strogatz, Constants of motion for superconducting Josephson array, Physica D 73 (1994) 197-253.
  33. S. Watanabe and J.W. Swift, Stability of periodic solutions in series arrays of Josephson junctions with internal capacitance, J. Nonlin. Sci., to appear.
  34. C.E. Wayne, Bounds on the trajectories of a system of weakly coupled rotators, Commun. Math. Phys. 104 (1986) 21-36.
  35. K. Wiesenfeid, P. Colet and S.H. Strogatz, Synchronization transitions in a disordered Josephson series array, Phys. Rev. Lett. 76 (1996) 404-407.
  36. A.T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol. 16 (1967) 15-42.
  37. A.T. Winfree, The Geometry of Biological Time (Springer, New York, 1980).
  38. Y. Yamaguchi and H. Shimizu, Theory of self-synchronization in the presence of native frequency distribution and external noises, Physica D 11 (1984) 212-226.
  39. R,A. York and R.C. Compton, Quasi-optical power combining using mutually synchronized oscillator arrays, IEEE Trans. Microwave Theor. and Tech. 39 (1991) 1000-1009.