Stability of amplitude chimeras in oscillator networks

Abstract

-We show that amplitude chimeras in ring networks of Stuart-Landau oscillators with symmetry-breaking nonlocal coupling represent saddle-states in the underlying phase space of the network. Chimera states are composed of coexisting spatial domains of coherent and of incoherent oscillations. We calculate the Floquet exponents and the corresponding eigenvectors in dependence upon the coupling strength and range, and discuss the implications for the phase space structure. The existence of at least one positive real part of the Floquet exponents indicates an unstable manifold in phase space, which explains the nature of these states as long-living transients. Additionally, we find a Stuart-Landau network of minimum size N = 12 exhibiting amplitude chimeras.Introduction. -The dynamical state of networks of homogeneously coupled identical elements can show a peculiar behavior by self-organizing into two spatially separated domains with dramatically different behavior, e.g. a spatially coherent and a spatially incoherent region. This phenomenon was named chimera state by Abrams and Strogatz [1] after the Greek fire-breathing monster, whose body consists of different animals. These hybrid states were discovered for phase oscillators in the early 2000's by Kuramoto and Battogtokh [2]. They observed a spontaneous breakup of the system into spatially coexisting synchronized and desynchronized domains with respect to the phase.Chimera states have possible applications to neural activity [3,4], heart fibrillation [5] and social systems [6]. Chimera states were also associated with such phenomena as epileptic seizure [7] and unihemispheric sleep, which has been detected for some sea mammals and birds [8]. These creatures can sleep with only one half of their brain while the other half remains awake. For instance, this enables sleeping dolphins to detect predators and migrant birds can travel for hundreds of kilometers without having a break [9]. Recently, unihemispheric sleep has been found also for humans [10].Chimera states were initially found for coupled phase oscillators, where coherence is related to phase-and frequency-locked oscillators and incoherence is associated with drifting oscillators. Since then numerous chimera

Filtering Suppresses Amplitude Chimeras

“…Throughout this paper we consider ω = 2 and use the following initial conditions [29]: x i = 1 and y i = −1 for 1 ≤ i ≤ N 2 and x i = −1 and y i = 1 for N 2 < i ≤ N. Figure 1A shows the phase diagram in the P − ε space: we can see that the amplitude chimera (AC) state is interspersed in between the completely synchronized oscillation zone (Sync) and the oscillation death (OD) zone. This is in accordance with the results of Zakharova et al [29], Schneider et al [52], Zakharova et al [53], and Tumash et al [54] where this system was studied in detail. Figures 1B-D illustrate the spatiotemporal evolution of the synchronized state (ε = 5), AC pattern (ε = 20) and multicluster OD state (ε = 30) at P = 10.…”

Stable amplitude chimera states in a network of locally coupled Stuart-Landau oscillators

“…These authors have also found that for a random distribution without symmetries amplitude chimera states appear to be short transients towards in-phase synchronized region, while their lifetime may significantly increase for symmetric initial conditions. Further Tumash et al 35 have noted that the existence of at least one positive real part of the Floquet exponents indicates an unstable manifold in phase space, which explains the nature of these states as long-living transients. In the present study, we investigate the existence of transient amplitude chimera states and also analyze how such amplitude chimera states become stable with respect to an increase of nonisochronicity parameter.…”

Networks of coupled oscillators: From phase to amplitude chimeras

“…The detailed study of the lifetime of AC states in Stuart-Landau oscillators is reported in Refs. 46 and 52. Note that in case of Stuart-Landau oscillators with symmetry-breaking nonlocal coupling large lifetimes can arise for certain values of the coupling range and strength due to the phase space structure, and they have been explained by a Floquet stability analysis 46 . In the present case, although the center of mass-shifted limit cycles are unstable, however, they are always trapped in between two symmetry-breaking bifurcation points PB1 and PB2.…”

“…In amplitude-mediated phase chimeras (AMC) incoherent fluctuations occur in both the phase and the amplitude in the incoherent domain; also, in the incoherent domain the temporal evolution of the oscillators is chaotic. On the other hand, amplitude chimeras (AC) were discovered by Zakharova et al [44][45][46] where all the oscillators have the same phase velocity but they have uncorrelated amplitude fluctuations in the incoherent domain; Also, unlike AMC (or classical phase chimera), the dynamics of all the oscillators in the AC state is periodic.…”