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. A fundamental question is then whether the time evolutions of these local computational unit will remain dynamically independent of each other or whether, at some point, they will start to change their states all in the same rhythm. This is the notion of “synchronization”, which we will study throughout this chapter. Starting with the paradigmal Kuramoto model we will learn that synchronization processes may be driven either by averaging dynamical variables or through causal mutual influences.
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Notes
- 1.
In the complex plane \(\psi _{j}(t) = e^{i\theta _{j}(t)} = e^{i(\omega t-\mathit{kj})}\) corresponds to a plane wave on a periodic ring. Equation (9.27) is then equivalent to the phase evolution of the wavefunction ψ j (t). The system is invariant under translations j → 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 \(k = 2\pi n_{k}/N\).
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Gros, C. (2015). Synchronization Phenomena. In: Complex and Adaptive Dynamical Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-16265-2_9
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DOI: https://doi.org/10.1007/978-3-319-16265-2_9
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