Chimerapedia: coherence–incoherence patterns in one, two and three dimensions

Abstract

Chimera states, or coherence-incoherence patterns in systems of symmetrically coupled identical oscillators, have been the subject of intensive study for the last two decades. In particular it is now known that the continuum limit of phase-coupled oscillators allows an elegant mathematical description of these states based on a nonlinear integro-differential equation known as the Ott-Antonsen equation. However, a systematic study of this equation usually requires a substantial computational effort. In this paper, we consider a special class of nonlocally coupled phase oscillator models where the above analytical approach simplifies significantly, leading to a semi-analytical description of both chimera states and of their linear stability properties. We apply this approach to phase oscillators on a onedimensional lattice, on a two-dimensional square lattice and on a three-dimensional cubic lattice, all three with periodic boundary conditions. For each of these systems we identify multiple symmetric coherence-incoherence patterns and compute their linear stability properties. In addition, we describe how chimera states in higher-dimensional models are inherited from lower-dimensional models and explain how they can be grouped according to their symmetry properties and global order parameter.

Collective states in a ring network of theta neurons

“…If we find a pair false(w^0,w^1false)T∈double-struckR2 that satisfies (4.7) and (4.8), then formula (4.6) yields the corresponding solution of equation (4.5), while the formula afalse(xfalse)=Uγfalse(wfalse(xfalse)false) yields the corresponding equilibrium of equation (2.1). Note that similar self-consistency equations have been found in similar systems [31–33] but those authors were studying chimera states.…”

Chimeras and solitary states in 3D oscillator networks with inertia

“…Among other factors, a new class of chimera states called the spiral wave chimeras has been introduced in [3]. This kind of space-temporal behavior is characterized by the standard 2D spiraling and, moreover, by a finite-size incoherent core (see, e.g., [4][5][6][7][8][9][10][11][12]).…”

“…As for the 3D case, we mention a few papers about chimera states in the model of coupled phase oscillators in a three-dimensional grid topology [13][14][15][16][17][18].…”

Effect of topology upon relay synchronization in triplex neuronal networks

“…Chimera states were found in a plethora of networks of coupled oscillators. [35][36][37][38][39][40][41] The coexistence of synchrony and asynchrony in the brain is reminiscent of the phenomenon of unihemispheric sleep. 17,[42][43][44] Similarly, during perceptual organization, 45 behavioral sensation, 46,47 and epileptic seizures, 16,48,49 partially synchronized patterns arise, which can be associated with chimera states.…”