Critical Switching in Globally Attractive Chimeras

Abstract

We report on a new type of chimera state that attracts almost all initial conditions and exhibits power-law switching behavior in networks of coupled oscillators. Such switching chimeras consist of two symmetric configurations, which we refer to as subchimeras, in which one cluster is synchronized and the other is incoherent. Despite each subchimera being linearly stable, switching chimeras are extremely sensitive to noise: arbitrarily small noise triggers and sustains persistent switching between the two symmetric subchimeras. The average switching frequency increases as a power law with the noise intensity, which is in contrast with the exponential scaling observed in typical stochastic transitions. Rigorous numerical analysis reveals that the power-law switching behavior originates from intermingled basins of attraction associated with the two subchimeras, which in turn are induced by chaos and symmetry in the system. The theoretical results are supported by experiments on coupled optoelectronic oscillators, which demonstrate the generality and robustness of switching chimeras. arXiv:1911.07871v1 [cond-mat.dis-nn]

Two populations of coupled quadratic maps exhibit a plentitude of symmetric and symmetry broken dynamics

“…22 and 25). Further work on chimera states in maps revealed temporal intermittency, 29,32 noise-induced inter-layer switching, 44 and relay synchronization. 43 By means of a linear change of variables, the logistic map can be transformed into the quadratic map f(z) = z 2 + c, which is a particularly simple map that can show chaos.…”

“…22 Nayak and Gupte 45 reported on chimera states in two populations of sine-circle maps. The first experimental observation of chimeras in electro-optical coupled map lattices in one and two dimensions was provided by Hagerstrom et al 24 Apart from coupled logistic maps, 22,23,[27][28][29][30][37][38][39][41][42][43][44] chimera states were illustrated for coupled Henon maps, 26,[32][33][34]36 sinecircle maps, 45,46 sine-squared maps, 35,44 cosine maps, 24 piecewise linear and logistic maps, 25 and cubic maps. 31,39,40 They were found for ring networks with nonlocal coupling, [22][23][24][25][26][28][29][30][31][32][33][34][35][36]39,[41][42]…”

Chimeras confined by fractal boundaries in the complex plane

“…Chimera states are characterized by the coexistence of synchronization and desynchronization in networks of coupled dynamics 19,20 . The set of initial conditions leading to chimera states can form riddled and fractal basins of attraction [21][22][23] . Initial conditions outside of these basins of attraction can result in fully synchronized or fully desynchronized dynamics, among others.…”

“…While chimera states were first described 19 and continue to be studied for networks of time-continuous oscillators, an increasing interest is paid to networks of time-discrete maps. Chimera states were found for networks of coupled logistic maps [4][5][6][7][8][9][23][24][25][26][27][28][29][30][31] , Henon maps [10][11][12][13][14][15]21 , cubic maps 7,16,17 , map-based neuron models 18,32 , sine-circle maps [33][34][35] , sine-squared maps 23,26,36 , cosine maps 37 , piecewise linear maps 38 , and piecewise logistic maps 38 . Recently, two populations of quadratic maps with real-valued parameter c were shown to yield a plentitude of dynamics including chimera states 39 .…”

“…The boundaries between these different states in the domain of the complex-valued c form fractal patterns. In contrast to the high-dimensional fractal basins of attraction [21][22][23] , these fractals reside in the complex plane and are therefore straightforward to visualize. In qualitative terms, the fractals generated by our network of coupled quadratic maps seem less filigree, more disordered, but even fuller of variety than the Mandelbrot set.…”