Skip to main content
SpringerLink
Log in

Synchronization Phenomena

  • Chapter
  • First Online:
Complex and Adaptive Dynamical Systems

  • 918 Accesses

Abstract

Here we consider the dynamics of complex systems constituted of interacting local computational units that have their own non-trivial dynamics. An example for a local dynamical system is the time evolution of an infectious disease in a certain city that is weakly influenced by an ongoing outbreak of the same disease in another city; or the case of a neuron in a state where it fires spontaneously under the influence of the afferent axon potentials.

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 59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    In the complex plane \(\psi_j(t)=e^{i\theta_j(t)}=e^{i(\omega t-kj)}\) corresponds to a plane wave on a periodic ring. Eq. (7.26) is then equivalent to the phase evolution of the wavefunction \(\psi_j(t)\). The system is invariant under translations \(j\to j+1\) and the discrete momentum k is therefore a good quantum number, in the jargon of quantum mechanics. The periodic boundary condition \(\psi_{j+N}=\psi_j\) is satisfied for the momenta \(k = 2\pi n_k/N\).

Further Reading

  • A nice review of the Kuramoto model, together with historical annotations, has been published by Strogatz (2000), for a textbook containing many examples of synchronization see Pikovsky et al. (2003). Some of the material discussed in this chapter requires a certain background in theoretical neuroscience, see e.g. Dayan and Abbott (2001).

    Google Scholar 

  • We recommend that the interested reader takes a look at some of the original research literature, such as the exact solution of the Kuramoto (1984) model, the Terman and Wang (1995) relaxation oscillators, the concept of fast threshold synchronization (Somers and Kopell, 1993), the temporal correlation hypothesis for cortical networks (von der Malsburg and Schneider, 1886), and its experimental studies (Gray et al., 1989), the LEGION network (Terman and Wang, 1995), the physics of synchronized clapping (Néda et al., 2000a, b) and synchronization phenomena within the SIRS model of epidemics (He and Stone, 2003). For an introductory-type article on synchronization with delays see (D’Huys et al., 2008).

    Google Scholar 

  • Dayan, P., Abbott, L.F. 2001 Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems. MIT Press, Cambridge.

    MATH  Google Scholar 

  • D’Huys, O., Vicente, R., Erneux, T., Danckaert, J., Fischer, I. 2008 Synchronization properties of network motifs: Influence of coupling delay and symmetry. Chaos 18, 037116.

    Article  MathSciNet  ADS  Google Scholar 

  • Gray, C.M., König, P., Engel, A.K., Singer, W. 1989 Oscillatory responses in cat visual cortex exhibit incolumnar synchronization which reflects global stimulus properties. Nature 338, 334–337.

    Article  ADS  Google Scholar 

  • He, D., Stone, L. 2003 Spatio-temporal synchronization of recurrent epidemics. Proceedings of the Royal Society London B 270, 1519–1526.

    Article  Google Scholar 

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

    Book  MATH  Google Scholar 

  • Néda, Z., Ravasz, E., Vicsek, T., Brechet, Y., Barabási, A.L. 2000a Physics of the rhythmic applause. Physical Review E 61, 6987–6992.

    Article  ADS  Google Scholar 

  • Néda, Z., Ravasz, E., Vicsek, T., Brechet, Y., Barabási, A.L. 2000b The sound of many hands clapping. Nature 403, 849–850.

    Article  ADS  Google Scholar 

  • Pikovsky, A., Rosenblum, M., Kurths, J. 2003 Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  • Somers, D., Kopell, N. 1993 Rapid synchronization through fast threshold modulation. Biological Cybernetics 68, 398–407.

    Article  Google Scholar 

  • Strogatz, S.H. 2000 From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators. Physica D 143, 1–20.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Strogatz, S.H. 2001 Exploring complex networks. Nature 410, 268–276.

    Article  ADS  Google Scholar 

  • Terman, D., Wang, D.L. 1995 Global competition and local cooperation in a network of neural oscillators. Physica D 81, 148–176.

    Article  MathSciNet  MATH  Google Scholar 

  • von der Malsburg, C., Schneider, W. 1886 A neural cocktail-party processor. Biological Cybernetics 54, 29–40.

    Article  Google Scholar 

  • Wang, D.L. 1999 Relaxation oscillators and networks. In Webster, J.G. (ed.), Encyclopedia of Electrical and Electronic Engineers, pp. 396–405. Wiley, New York.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universität Frankfurt, Institut für Theoretische Physik, Max-von-Laue-Str. 1, 60438, Frankfurt, Germany

    Prof. Dr. Claudius Gros

Authors
  1. Prof. Dr. Claudius Gros
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Claudius Gros .

Exercises

Exercises

7.1.1 The Driven Harmonic Oscillator

Solve the driven harmonic oscillator, Eq. (7.1), for all times t and compare it with the long time solution \(t\to\infty\), Eqs. (7.3) and (7.4).

7.1.2 Self-Synchronization

Consider an oscillator with feedback,

$$\dot\theta(t)\ =\ \omega_0\,+\,K\sin[\theta(t-T)-\theta(t)] .$$

Discuss the self-synchronization in analogy to Sect. 7.3, the stability of the steady-state solutions and the auto-locking frequencies in the limit of strong self-coupling \(K\to\infty\).

7.1.3 Synchronization of Chaotic Maps

The Bernoulli shift map \(f(x)=ax\ \textrm{mod}\ 1\) with \(x\in[0,1]\) is chaotic for \(a>1\). Consider with

$$\begin{array}{rcl} x_1(t+1)& =& f\Big((1-\kappa)x_1(t)+\kappa x_2(t-T)\Big) \\ x_2(t+1)& =& f\Big((1-\kappa)x_2(t)+\kappa x_1(t-T)\Big) \end{array}$$
((7.46))

two coupled chaotic maps, with \(\kappa\in[0,1]\) being the coupling strength and T the time delay, compare Eq. (7.30). Discuss the stability of the synchronized states \(x_1(t)=x_2(t)\equiv \bar x(t)\) for general time delays T. What drives the synchronization process?

7.1.4 The Terman–Wang Oscillator

Discuss the stability of the fixpoints of the Terman–Wang oscillator, Eq. (7.38). Linearize the differential equations around the fixpoint solution and consider the limit \(\beta \to 0\).

7.1.5 The SIRS Model – Analytical

Find the fixpoints \(x_t\equiv x^*\) of the SIRS model, Eq. (7.44), for all τ R , as a function of a and study their stability for \(\tau_R=0,1\).

7.1.6 The SIRS Model – Numerical

Study the SIRS model, Eq. (7.44), numerically for various parameters a and \(\tau_R=0,1,2,3\). Try to reproduce Figs. 7.13 and 7.14.

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Gros, C. (2011). Synchronization Phenomena. In: Complex and Adaptive Dynamical Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04706-0_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-04706-0_7

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-04705-3

  • Online ISBN: 978-3-642-04706-0

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

Publish with us

Policies and ethics

Search

Navigation