Chimera states in two populations with heterogeneous phase-lag

Abstract

The simplest network of coupled phase-oscillators exhibiting chimera states is given by two populations with disparate intra-and inter-population coupling strengths. We explore the effects of heterogeneous coupling phase-lags between the two populations. Such heterogeneity arises naturally in various settings, for example as an approximation to transmission delays, excitatory-inhibitory interactions, or as amplitude and phase responses of oscillators with electrical or mechanical coupling. We find that breaking the phase-lag symmetry results in a variety of states with uniform and non-uniform synchronization, including in-phase and anti-phase synchrony, full incoherence (splay state), chimeras with phase separation of 0 or π between populations, and states where both populations remain desynchronized. These desynchronized states exhibit stable, oscillatory, and even chaotic dynamics. Moreover, we identify the bifurcations through which chimeras emerge. Stable chimera states and desynchronized solutions, which do not arise for homogeneous phase-lag parameters, emerge as a result of competition between synchronized in-phase, anti-phase equilibria, and fully incoherent states when the phase-lags are near ± π 2 (cosine coupling). These findings elucidate previous experimental results involving a network of mechanical oscillators and provide further insight into the breakdown of synchrony in biological systems. The synchronization of oscillators is a ubiquitous phenomenon that manifests itself in a wide range of biological and technological settings, including the beating of the heart 1 , flashing fireflies 2 , pedestrians on a bridge locking their gait 3 , circadian clocks in the brain 4 , superconducting Josephson junctions 5 , chemical oscillations 6,7 , metabolic oscillations in yeast cells 8,9 , and life cycles of phytoplankton 10 . Recent studies have reported the emergence of solutions where oscillators break into localized synchronized and desynchronized populations, commonly known as chimera states 11,12 . These solutions have been studied in the Kuramoto-Sakaguchi model with homogeneous coupling phase-lag 13-16 . Significant progress has been made understanding how chimera states emerge with respect to different topologies 17-20 , their robustness towards heterogeneity 21,22 , how they manifest in real-world experiments such as (electro-) chemical and mechanical oscillator systems 23-25 and laser systems 26 , and recently in explaining their basins of attraction 15 and controllability 15,27 . Here we generalize one of the simplest systems in which chimera states are known to occur, two populations of identical phase-oscillators with heterogeneous intra-and inter-population coupling, to account for effects of breaking the symmetry in the phase-lag parameters. Using syma) Electronic mail: erik.martens@ds.mpg.de; http://eam.webhop.net metry considerations, numerical methods and perturbative approaches, we explore and explain the emergence of dynamics which only occur for heterogeneous phase lags, including new...

Ott-Antonsen attractiveness for parameter-dependent oscillatory systems

“…A first urgent one is the case of δ-peaked frequency distributions. Numerical simulations 54 and heuristic arguments hint at convergence of the OA manifold, where a proper mathematical derivation is omitted under the pretence of "nearly identical oscillators" 45,55,56 . A thorough proof would render the OA ansatz rigorously applicable to "chimera states", a topic that is particularly en vogue; see, e.g., the recent review paper by Panaggio and Abramscite 57 .…”

Chaos in Kuramoto oscillator networks

“…Points on the attractor in close proximity to these invariant surfaces are highlighted in gray. unstable 12,13 . As α s is increased, the chaotic attractors are destroyed as they approach the invariant surfaces r σ = 1 where one of the populations is phase-synchronized.…”

“…The system was integrated numerically from the fixed initial condition (r1(0), r2(0), ψ(0)) = (0.8601, 0.4581, 1.1815). Panel (a) shows the maximal Lyapunov exponents overlaid with twoparameter bifurcation lines: the transcritical (TC), Hopf, and first period-doubling (PD1) lines emanate from (αs, αn) = ( π 2 , 0) and end in the degenerate bifurcation (Deg) where SS0 and SSπ swap stability12 . Panel (b) shows a magnification of the region highlighted in Panel (a) where positive Lyapunov exponents arise (red color); a dotted line indicates the parameter range shown in…”

Directed Flow of Information in Chimera States

“…4 in Appendix A. Subsequent studies demonstrated robustness of chimera states against non-uniformity of delays (α µν = α) [33], heterogeneity of oscillator frequencies (σ ω > 0) [56], and additive noise [57]. Although nonuniform phase lags lead to less degenerate dynamics and give room for more complex dynamics [33,53], we restrict ourselves to the case of uniform phase lags without compromising the essential phenomenology of chimera states.…”

“…Alongside efforts studying synchronization on networks, a new symmetry breaking regime coined chimera state has been observed. In a chimera state an oscillator population 'splits' into two parts, one being synchronized and the other being desynchronized [30][31][32], or more generally, different levels of synchronization [33]. This state is a striking manifestation of symmetry breaking, as it may occur even if oscillators are identical and coupled symmetrically; see [34,35] for recent reviews.…”