Abstract
Macroscopic rhythms are often signatures of healthy functioning in living organisms, but they are still poorly understood on their microscopic bases. Globally interacting oscillators with heterogeneous couplings are here considered. Thorough theoretical and numerical analyses indicate the presence of multiple phase transitions between different collective states, with regions of bi-stability. Novel coherent phases are unveiled, and evidence is given of the spontaneous emergence of macroscopic rhythms where oscillators’ phases are always found to be self-organized as in Bellerophon states, i.e. in multiple clusters with quantized values of their average frequencies. Due to their rather unconditional appearance, the circumstance is paved that the Bellerophon states grasp the microscopic essentials behind collective rhythms in more general systems of interacting oscillators.
Highlights
Rhythmic behaviours are ubiquitous in nature, where we witness events of a rare beauty as, for instance, the acoustic synchrony in cricket’s choruses in the summer or the choreographic dancing of starling flocks in the fall1,2
Synchronization represents a natural scaffold for capturing the microscopic features of these emergent behaviours, and large attention was paid in analyzing the routes towards synchrony in ensembles of dynamical systems6–8
There, the presence of correlations between the natural frequencies of the oscillators and the coupling strength may lead to first-order like (a.k.a. explosive15) phase transitions (PT’s)17,18, where the backward threshold stays fixed, while the forward one can be adequately tuned by choosing the system’s parameters (Typically the median of the natural frequency distribution)
Summary
Rhythmic behaviours are ubiquitous in nature, where we witness events of a rare beauty as, for instance, the acoustic synchrony in cricket’s choruses in the summer or the choreographic dancing of starling flocks in the fall1,2. The predicted behaviors are compared with the numerical solutions obtained integrating directly Eq [1] (Unless otherwise specified, the following stipulations are chosen: (i) the strength of couplings for conformist oscillators is kept fixed to κ2 = 5; (ii) a Lorentzian frequency distribution g(ω) = γ/π/ [(ω − ω0)2 + γ2] with γ = 0.05 is adopted; (iii) numerical integrations are performed with a fourth-order Runge-Kutta method with integration time step Δt = 0.01; (iv) the initial conditions for the phase variables are randomly taken; (v) the typical number of oscillators in the ensemble is N = 5 × 104; (vi) a sufficiently long time interval (much larger than the oscillation period Tf = 2π/Ωf) is used for the average of the order parameter).
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