Synchronization Phenomena

Gros, Claudius

doi:10.1007/978-3-031-55076-8_9

Claudius Gros²

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Abstract

Complex systems are based on interacting local computational units may show non-trivial emerging behaviors. Examples are the time evolution of an infectious disease in a certain city that is mutually influenced by an ongoing outbreak of the same disease in another city, or the case of a neuron firing spontaneously while processing the effects of afferent axon potentials.

A fundamental question is whether the time evolutions of interacting local units remain dynamically independent of each other, or whether they will change their states simultaneously, following identical rhythms. This is the notion of synchronization, which we will study throughout this chapter. Starting with the paradigmatic Kuramoto model we will learn that synchronization processes may be driven either by averaging dynamical variables, or through causal mutual influences. On the way, we will visit piecewise linear dynamical systems and the reference model for infectious diseases, the SIRS model.

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Notes

1.
“Secular perturbation theory” deals with time-dependent amplitudes, \(a=a(t)\). See Sect. 3.2 of Chap. 3.
2.
Note, that \(\int dx \cos ^2(x) = [\cos {}(x)\sin {}(x)+x]/2\), modulo a constant.
3.
One has \(\int dx \cos ^2(x)\sin ^2(x) = x/8 - \sin {}(4x)/32\), plus an integration constant.
4.
More about critical scaling in Sect. 6.1 of Chap. 6.
5.
An introduction into the intricacies of time-delayed dynamical systems is given in Sect. 2.5 of Chap. 2.
6.
In the complex plane, \(\psi _j(t)=\mathrm {e}^{i\theta _j(t)}= \mathrm {e}^{i(\omega t-kj)}\) corresponds to a plane wave on a periodic ring. Eq. (9.28) is then equivalent to the phase evolution of the wavefunction \(\psi _j(t)\). The system is invariant under translations \(j\to j+1\), which implies that the discrete momentum k is a good quantum number, in the jargon of quantum mechanics. The periodic boundary condition \(\psi _{j+N}=\psi _j\) is satisfied for \(k = 2\pi n_k/N\).
7.
Initial functions for time delay systems are discussed in Sect. 2.5 of Chap. 2.
8.
The connection of the adjacency matrix to the graph spectrum is discussed in Sect. 1.2 of Chap. 1.
9.
Chimera states are shortly discussed on page 336.
10.
For an illustration of the logistic map see Fig. 2.18 in Chap. 2.
11.
Maximal Lyapunov exponents are discussed together with the theory of discrete maps in Chap. 2.
12.
The divergence of the flow is equivalent to the relative contraction/expansion of phase space, \(\varDelta V/V\), as discussed in Sect. 3.1.1 of Chap. 3.
13.
The van der Pol oscillator, treated in Sect. 3.2 of Chap. 3, has to regimes, corresponding respectively to small/large adaptive term.
14.
The methods of time delay systems are laid down in Sect. 2.5 of Chap. 2.

References

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 ADS MathSciNet Google Scholar
He, D., & Stone, L. (2003). Spatio-temporal synchronization of recurrent epidemics. Proceedings of the Royal Society London B, 270, 1519–1526.
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
Panaggio, M. J., & Abrams, D. M. (2015). Chimera states: coexistence of coherence and incoherence in networks of coupled oscillators. Nonlinearity, 28, R67.
Article ADS MathSciNet Google Scholar
Pikovsky, A., & Rosenblum, B. (2015). Dynamics of globally coupled oscillators: Progress and perspectives. Chaos, 25, 9.
Article MathSciNet Google Scholar
Pikovsky, A., Rosenblum, M., & Kurths, J. (2003). Synchronization: A universal concept in nonlinear sciences. Cambridge University Press.
Google Scholar
Somers, D., & Kopell, N. (1993). Rapid synchronization through fast threshold modulation. Biological Cybernetics, 68, 398–407.
Article Google Scholar
Sterratt, D., Graham, B., Gillies, A., Einevoll, G., & Willshaw, D. (2023). Principles of computational modelling in neuroscience. Cambridge University Press.
Book 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 ADS MathSciNet Google Scholar

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Institute for Theoretical Physics, Goethe University Frankfurt, Frankfurt/Main, Germany
Claudius Gros

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Gros, C. (2024). Synchronization Phenomena. In: Complex and Adaptive Dynamical Systems. Springer, Cham. https://doi.org/10.1007/978-3-031-55076-8_9

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DOI: https://doi.org/10.1007/978-3-031-55076-8_9
Published: 14 May 2024
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-55075-1
Online ISBN: 978-3-031-55076-8
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