Skip to main content
SpringerLink
Log in

Amplitude Death, Synchrony, and Chimera States in Delay Coupled Limit Cycle Oscillators

  • Chapter
  • First Online:
Complex Time-Delay Systems

Part of the book series: Understanding Complex Systems ((UCS))

Abstract

In this chapter we will discuss the effects of time delay on the collective states of a model mathematical system composed of a collection of coupled limit cycle oscillators. Such an assembly of coupled nonlinear oscillators serves as a useful paradigm for the study of collective phenomena in many physical, chemical, and biological systems and has therefore led to a great deal of theoretical and experimental work in the past [1–6]. Examples of practical applications of such models include simulating the interactions of arrays of Josephson junctions [7, 8], semiconductor lasers [9, 10], charge density waves [11], phase-locking of relativistic magnetrons [12], Belousov–Zhabotinskii reactions in coupled Brusselator models [2, 13–15], and neural oscillator networks for circadian pacemakers [16].

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. T. Winfree. The Geometry of Biological Time. Springer-Verlag, New York, 1980.

    MATH  Google Scholar 

  2. Y. Kuramoto. Chemical Oscillations, Waves and Turbulence. Springer, Berlin, 1984.

    MATH  Google Scholar 

  3. S. H. Strogatz. From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D, 143:1–20, 2000. and references therein.

    Article  MathSciNet  MATH  Google Scholar 

  4. K. Satoh. Computer experiment on the cooperative behavior of network of interacting nonlinear oscillators. J. Phys. Soc. Japan, 58:2010–2021, 1989.

    Google Scholar 

  5. G. B. Ermentrout. Oscillator death in populations of “all to all” coupled nonlinear oscillators. Physica D, 41:219–231, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  6. A. A. Brailove and P. S. Linsay. An experimental study of a population of relaxation oscillators with a phase-repelling mean-field coupling. Int. J. Bifurcation Chaos, 6:1211–1253, 1996.

    Article  Google Scholar 

  7. P. Hadley, M. R. Beasley, and K. Wiesenfeld. Phase locking of Josephson-junction series arrays. Phys. Rev. B, 38:8712–8719, 1988.

    Article  Google Scholar 

  8. K. Wiesenfeld, P. Colet, and S. H. Strogatz. Synchronization transitions in a disordered Josephson series array. Phys. Rev. Lett., 76:404–407, 1996.

    Article  Google Scholar 

  9. P. M. Varangis, A. Gavrielides, T. Erneux, V. Kovanis, and L. F. Lester. Frequency entrainment in optically injected semiconductor lasers. Phys. Rev. Lett., 78:2353–2356, 1997.

    Article  Google Scholar 

  10. A. Hohl, A. Gavrielides, T. Erneux, and V. Kovanis. Localized synchronization in two coupled nonidentical semiconductor lasers. Phys. Rev. Lett., 78:4745–4748, 1997.

    Article  Google Scholar 

  11. G. Grüner and A. Zettl. Charge-density wave conduction – a novel collective transport phenomenon in solids. Phys. Rep., 119:117–232, 1985.

    Article  Google Scholar 

  12. J. Benford, H. Sze, W. Woo, R. R. Smith, and B. Harteneck. Phase locking of relativistic magnetrons. Phys. Rev. Lett., 62:969–971, 1989.

    Article  Google Scholar 

  13. I. Schreiber and M. Marek. Strange attractors in coupled reaction diffusion cells. Physica, 5D:258–272, 1982.

    MathSciNet  Google Scholar 

  14. M. F. Crowley and I. R. Epstein. Experimental and theoretical studies of a coupled chemical oscillator: Phase death, multistability, and in-phase and out-of-phase entrainment. J. Phys. Chem., 93:2496–2502, 1989.

    Article  Google Scholar 

  15. M. Dolnik and I. R. Epstein. Coupled chaotic chemical oscillators. Phys. Rev. E, 54:3361–3368, 1996.

    Article  Google Scholar 

  16. M. Kawato and R. Suzuki. Two coupled neural oscillators as a model of the circadian pacemaker. J. Theor. Biol., 86:547–575, 1980.

    Article  MathSciNet  Google Scholar 

  17. Y. Kuramoto. Cooperative dynamics of oscillator community. Prog. Theor. Phys. Suppl., 79:223–240, 1984.

    Article  Google Scholar 

  18. H. Daido. Intrinsic fluctuations and a phase transition in a class of large populations of interacting oscillators. J. Stat. Phys., 60:753–800, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  19. S. Strogatz. Sync: The Emerging Science of Spontaneous Order. Hyperion, New York, 2003.

    Google Scholar 

  20. Y. Aizawa. Synergetic approach to the phenomena of mode-locking in nonlinear systems. Prog. Theor. Phys., 56:703–716, 1976.

    Article  Google Scholar 

  21. M. Shiino and M. Frankowicz. Synchronization of infinitely many coupled limit-cycle oscillators. Phys. Lett. A, 136:103–108, 1989.

    Article  MathSciNet  Google Scholar 

  22. M. Poliashenko and S. R. McKay. Chaos due to homoclinic orbits in two coupled oscillators with nonisochronism. Phys. Rev. A, 46:5271–5274, 1992.

    Article  Google Scholar 

  23. J. L. Rogers and L. T. Wille. Phase transitions in nonlinear oscillator chains. Phys. Rev. E, 54:R2193–R2196, 1996.

    Article  Google Scholar 

  24. D. G. Aronson, G. B. Ermentrout, and N. Kopell. Amplitude response of coupled oscillators. Physica D, 41:403–449, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  25. R. E. Mirollo and S. H. Strogatz. Amplitude death in an array of limit-cycle oscillators. J. Stat. Phys., 60:245–262, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  26. K. Bar-Eli. On the stability of coupled chemical oscillators. Physica D, 14:242–252, 1985.

    Article  MathSciNet  Google Scholar 

  27. P. C. Matthews, R. E. Mirollo, and S. H. Strogatz. Dynamics of a large system of coupled nonlinear oscillators. Physica D, 52:293–331, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  28. H. G. Schuster and P. Wagner. Mutual entrainment of two limit cycle oscillators with time delayed coupling. Prog. Theor. Phys., 81:939–945, 1989.

    Article  MathSciNet  Google Scholar 

  29. E. Niebur, H. G. Schuster, and D. Kammen. Collective frequencies and metastability in networks of limit-cycle oscillators with time delay. Phys. Rev. Lett., 67:2753–2756, 1991.

    Article  Google Scholar 

  30. Y. Nakamura, F. Tominaga, and T. Munakata. Clustering behavior of time-delayed nearest-neighbor coupled oscillators. Phys. Rev. E, 49:4849–4856, 1994.

    Article  Google Scholar 

  31. S. Kim, S. H. Park, and C. S. Ryu. Multistability in coupled oscillator systems with time delay. Phys. Rev. Lett., 79:2911–2914, 1997.

    Article  Google Scholar 

  32. D. V. Ramana Reddy, A. Sen, and G. L. Johnston. Time delay induced death in coupled limit cycle oscillators. Phys. Rev. Lett., 80:5109–5112, 1998.

    Article  Google Scholar 

  33. D. V. Ramana Reddy, A. Sen, and G. L. Johnston. Time delay effects on coupled limit cycle oscillators at Hopf bifurcation. Physica D, 129:15–34, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  34. D. V. Ramana Reddy, A. Sen, and G. L. Johnston. Experimental evidence of time-delay induced death in coupled limit-cycle-oscillators. Phys. Rev. Lett., 85:3381–3384, 2000.

    Article  Google Scholar 

  35. R. Dodla, A. Sen, and G. L. Johnston. Phase-locked patterns and amplitude death in a ring of delay-coupled limit cycle oscillators. Phys. Rev. E, 69:056217, 2004.

    Article  MathSciNet  Google Scholar 

  36. M. P. Mehta, A. Sen, and G. L. Johnston. Amplitude death states of a ring of nonlocally-coupled limit cycle oscillators with time delay. Proceedings of the National Conference on Nonlinear Systems and Dynamics, Chennai, Feb.5-8, 2006, 2006.

    Google Scholar 

  37. G. C. Sethia, A. Sen, and F. M. Atay. Clustered chimera states in delay-coupled oscillator systems. Phys. Rev. Lett., 100:144102, 2008.

    Article  Google Scholar 

  38. P. C. Matthews and S. H. Strogatz. Phase diagram for the collective behavior of limit cycle oscillators. Phys. Rev. Lett., 65:1701–1704, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  39. S. Peles and K. Wiesenfeld. Synchronization law for a van der Pol array. Phys. Rev. E, 68:026220, 2003.

    Article  Google Scholar 

  40. L. L. Bonilla, C. J. Perez Vicente, F. Ritort, and J. Soler. Exactly solvable phase oscillator models with synchronization dynamics. Phys. Rev. Lett., 81:3643–3646, 1998.

    Article  Google Scholar 

  41. H. Daido. Multibranch entrainment and scaling in large populations of coupled oscillators. Phys. Rev. Lett., 77:1406–1409, 1996.

    Article  Google Scholar 

  42. G.B. Ermentrout. The behavior of rings of coupled oscillators. J. Math. Biol., 23:55–74, 1985.

    MathSciNet  MATH  Google Scholar 

  43. P. C. Bressloff, S. Coombes, and B. de Souza. Dynamics of a ring of pulse-coupled oscillators: Group theoretic approach. Phys. Rev. Lett., 79:2791–2794, 1997.

    Article  Google Scholar 

  44. M. Barahona and L.M. Pecora. Synchronization in small-world systems. Phys. Rev. Lett., 89:054101, 2002.

    Article  Google Scholar 

  45. Y. Kuramoto. Scaling behavior of oscillators with non-local interaction. Prog. Theor. Phys., 94:321–330, 1995.

    Article  Google Scholar 

  46. Y. Kuramoto and H. Nakao. Origin of power-law spatial correlations in distributed oscillators and maps with nonlocal coupling. Phys. Rev. Lett., 76:4352–4355, 1996.

    Article  Google Scholar 

  47. Y. Kuramoto and D. Battogtokh. Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. Nonlinear Phenomena in Complex Systems, 5:380–385, 2002.

    Google Scholar 

  48. Y. Kuramoto. Nonlinear Dynamics and Chaos:Where Do We Go from Here?, p 209. Institute of Physics, Bristol, UK, 2003.

    Google Scholar 

  49. J. R. Phillips, H. S. J. van der Zant, J. White, and T. P. Orlando. Influence of induced magnetic-fields on the static properties of josephson-junction arrays. Phys. Rev. B, 47:5219–5229, 1993.

    Article  Google Scholar 

  50. S. I. Shima and Y. Kuramoto. Rotating spiral waves with phase-randomized core in nonlocally coupled oscillators. Phys. Rev. E, 69:036213, 2004.

    Article  Google Scholar 

  51. J.D. Murray. Mathematical Biology. Springer, New York, 1989.

    MATH  Google Scholar 

  52. B. Ermentrout, J. Campbell, and G. Oster. A model for shell patterns based on neural activity. Veliger, 28:369–388, 1986.

    Google Scholar 

  53. N.V. Swindale. A model for the formation of ocular dominance stripes. Proc. R. Soc. London B, 208:243–264, 1980.

    Article  Google Scholar 

  54. D. Tanaka and Y. Kuramoto. Complex Ginzburg-Landau equation with nonlocal coupling. Phys. Rev. E, 68:026219, 2003.

    Article  Google Scholar 

  55. D. M. Abrams and S. H. Strogatz. Chimera states for coupled oscillators. Phys. Rev. Lett., 93:174102, 2004.

    Article  Google Scholar 

  56. D. M. Abrams and S. H. Strogatz. Chimera states in rings of nonlocally coupled oscillators. Int. J. Bifurcation Chaos, 16:21–37, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  57. F. M. Atay. Distributed delays facilitate amplitude death of coupled oscillators. Phys. Rev. Lett., 91:094101, 2003.

    Article  Google Scholar 

  58. M. Dhamala, V. K. Jirsa, and M. Ding. Enhancement of neural synchrony by time delay. Phys. Rev. Lett., 92:74104, 2004.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Plasma Research, Bhat, Gandhinagar, 382428, India

    Abhijit Sen & Gautam C. Sethia

  2. Department of Biology, University of Texas at San Antonio, San Antonio, TX, 78249, USA

    Ramana Dodla

  3. EduTron Corp., 5 Cox Road, Winchester, MA, 01890, USA

    George L. Johnston

Authors
  1. Abhijit Sen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ramana Dodla
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. George L. Johnston
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Gautam C. Sethia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Abhijit Sen .

Editor information

Editors and Affiliations

  1. Wissenschaften, MPI für Mathematik in den, Inselstr. 22-26, Leipzig, 04103, Germany

    Fatihcan M. Atay

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Sen, A., Dodla, R., Johnston, G.L., Sethia, G.C. (2009). Amplitude Death, Synchrony, and Chimera States in Delay Coupled Limit Cycle Oscillators. In: Atay, F. (eds) Complex Time-Delay Systems. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02329-3_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-02329-3_1

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02328-6

  • Online ISBN: 978-3-642-02329-3

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics

Search

Navigation