Abstract
The onset of collective synchronous behavior in globally coupled ensembles of oscillators is discussed. We present a formalism that is applicable to general ensembles of heterogeneous, continuous time dynamical units that, when uncoupled, are chaotic, periodic, or a mixture of both. A discussion of convergence issues, important for the proper implementation of our method, is included. Our work leads to a quantitative prediction for the critical coupling value at the onset of collective synchrony and for the growth rate of the resulting coherent state.
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E. Ott, Chaos in Dynamical Systems (Cambridge Univ. Press, 1993), Chapter 3. For a given Ω, the natural measure μΩ for an attractor A of the uncoupled system dx/dt = G(x, Ω) gives the fraction of time μΩ(S) that a typical infinitely long orbit originating in B(A) (the basin of attraction of A) spends in a subset S of state space. By the word typical we refer to the supposition that there is a set of initial conditions x(0) in B(A) where this set has Lebesgue measure (roughly volume) equal to the Lebesgue measure of B(A) and such that each initial condition in this set gives the same value (i.e., the natural measure) for the fraction of time spent in S by the resulting orbit.
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Ott, E., So, P., Barreto, E., Antonsen, T. (2003). Synchrony in Globally Coupled Chaotic, Periodic, and Mixed Ensembles of Dynamical Units. In: Pikovsky, A., Maistrenko, Y. (eds) Synchronization: Theory and Application. NATO Science Series, vol 109. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0217-2_8
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DOI: https://doi.org/10.1007/978-94-010-0217-2_8
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-1417-8
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