Cascades of Multiheaded Chimera States for Coupled Phase Oscillators

Abstract

Chimera state is a recently discovered dynamical phenomenon in arrays of nonlocally coupled oscillators, that displays a self-organized spatial pattern of co-existing coherence and incoherence. We discuss the appearance of the chimera states in networks of phase oscillators with attractive and with repulsive interactions, i.e. when the coupling respectively favors synchronization or works against it. By systematically analyzing the dependence of the spatiotemporal dynamics on the level of coupling attractivity/repulsivity and the range of coupling, we uncover that different types of chimera states exist in wide domains of the parameter space as cascades of the states with increasing number of intervals of irregularity, so-called chimera's heads. We report three scenarios for the chimera birth: 1) via saddle-node bifurcation on a resonant invariant circle, also known as SNIC or SNIPER, 2) via blue-sky catastrophe, when two periodic orbits, stable and saddle, approach each other creating a saddle-node periodic orbit, and 3) via homoclinic transition with complex multistable dynamics including an "eight-like" limit cycle resulting eventually in a chimera state.

Chimera states in population dynamics: Networks with fragmented and hierarchical connectivities

“…Impact of coupling range R Chimera states with multiple domains of incoherence and coherence have been reported in several works and are referred to as clustered chimera or multichimera states. It is known that they may be achieved through time delay [62] or by manipulating the range of the coupling between oscillators [15][16][17]63]. The range R of the coupling reflects the migration range of the different species in the system.…”

Chimera states in three dimensions

“…Since the pioneering works [1,2,14,15], chimera states have been reported and thoroughly studied with different models, network configurations, and coupling schemes. The main attention has been paid to a ring of oscillators, see [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37] and to the case of two and three groups of oscillators [18,[38][39][40]. Robustness of the chimera states was examined from different viewpoints [41][42][43] and confirmed by recent experiments in various fields including chemistry [44][45][46], optical and microelectronic systems [47][48][49][50], and mechanics [51,52].…”

Different types of chimera states: An interplay between spatial and dynamical chaos

“…First observed by Kuramoto [1], it was later studied by many scientists and today is one of the fastest growing branches of dynamical systems and networks theory. The works on chimeras relate to a variety of models like phase oscillators [2][3][4][5][6][7] (for which they were originally found), chemical oscillators [8,9], neuron models [10], planar oscillators [11], and many other different types of networks [12][13][14][15][16][17][18][19][20][21][22][23][24]. Chimera states are well known for nonlocally coupled systems, but recently they also have been found in feedback-delayed oscillators [25][26][27] and in globally coupled networks [18,19].…”

“…In the incoherent intervals of the first type (chimera type I) one observes space-temporal chaos characterized by hyperchaotic behavior with many positive Lyapunov exponents [33]. This type of chimera state is widely observed for networks of continuous time nodes (given by differential equations, e.g., complex Ginzburg-Landau equations or Kuramoto model [1][2][3][4][5][6][7][8]). In the second type (chimera type II) only spatial chaos is observed in the chimera's incoherent interval such that the temporal dynamics is very simple, in most cases periodic.…”

“…1 are divided by the dotted lines which determine the additional subregions signed by k = 1,2,3 numbers. These numbers refer to the number of the so-called chimeras heads (i.e., the regions of chaoticity) [4]; characteristic examples are illustrated in the insets. Further decrease of r yields additional higher order regions of multiheaded chimera states following the head-adding cascade k = 4,5, .…”