Abstract
Chimera states arising in the classic Kuramoto system of two-dimensional phase coupled oscillators are transient but they are "long" transients in the sense that the average transient lifetime grows exponentially with the system size. For reasonably large systems, e.g., those consisting of a few hundreds oscillators, it is infeasible to numerically calculate or experimentally measure the average lifetime, so the chimera states are practically permanent. We find that small perturbations in the third dimension, which make system "slightly" three-dimensional, will reduce dramatically the transient lifetime. In particular, under such a perturbation, the practically infinite average transient lifetime will become extremely short, because it scales with the magnitude of the perturbation only logarithmically. Physically, this means that a reduction in the perturbation strength over many orders of magnitude, insofar as it is not zero, would result in only an incremental increase in the lifetime. The uncovered type of fragility of chimera states raises concerns about their observability in physical systems.
Highlights
We find that any infinitesimal deviation from the equator in the oscillator dynamics makes the long-transient chimera state extremely short
The transient time increases exponentially with the system size N, making numerical simulations infeasible to observe the collapse of the chimera state for, e.g., N > 60
A finite time interval in which g0 is approximately constant while the distribution of Di has two peaks signifies the existence of a transient chimera state
Summary
A research frontier in complex and nonlinear dynamical systems is chimera states [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58], a phenomenon of spontaneous symmetry breaking in spatially extended systems in which coherent and incoherent groups of oscillators coexist simultaneously. Chimera states have been studied in diverse systems such as regular networks of phase-coupled oscillators with a ring topology [2,3,5], networks hosting a few populations [6,10], two-dimensional (2D) [4,11] and three-dimensional (3D) lattices [37], torus [16,28], and systems with a spherical topology [38]. Because the focus of our study is on the transient nature of such high-dimensional chimera states, we set the initial condition to be a chimera state in two dimensions as in the classical Kuramoto model and examine how long the state can survive under such perturbations This is done for two cases: the perturbations are such that the local phase space of each oscillator becomes three or four dimensional, respectively. Our finding of a similar scaling law but with respect to deterministic, dimension-augmenting perturbations is further indication of the fragility of chimera states
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