Chimeras in Two-Dimensional Domains: Heterogeneity and the Continuum Limit

Abstract

We consider three different two-dimensional networks of nonlocally coupled heterogeneous phase oscillators. These networks were previously studied with identical oscillators and a number of spatiotemporal patterns found, mostly as a result of direct numerical simulation. Here we take the continuum limit of an infinite number of oscillators and use the Ott/Antonsen ansatz to derive continuum level evolution equations for order parameter-like quantities. Most of the patterns previously found in these networks correspond to relative fixed points of these evolution equations, and we show here the results of extensive numerical investigations of these fixed points: their existence and stability, and the bifurcations involved in their loss of stability as parameters are varied. Our results answer a number of questions posed by previous authors who studied these networks, and provide a better understanding of these networks' dynamics.

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]).…”

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

“…Moreover, a systematic exploitation of our techniques enables us to uncover a number of new chimera states in higher-dimensional systems, and to group them into families distinguished by their symmetry properties and values of their global order parameter. These results are obtained using a continuum analysis valid in the limit of infinitely many oscillators [22][23][24][25][26].…”

Collective in-plane magnetization in a two-dimensional XY macrospin system within the framework of generalized Ott–Antonsen theory

“…It is noteworthy that the OA theory can be used for the description of the macroscopic dynamics of ensembles with non-local interaction, as one can see for one-dimensional systems in [23][24][25][26][27][28] and for 2D systems in [29,30]. The formation of domains can be described on the basis of circular cumulant reductions in a similar way, but this issue is beyond the scope of our paper.…”

“…The opportunity to have an exact closed equation for the dynamics of the order parameter resulted in an eruption of works employing this mathematical tool. The OA theory proved itself to be a useful tool for studies on Josephson junction arrays [13], neuronal networks [14][15][16][17][18], populations of active rotators [19,20], fundamental studies on collective phenomena [21][22][23][24][25][26][27][28][29][30][31][32], etc. Below we will show that the in-plane dynamics of magnetic moments in a system, where the interaction between elements is mediated by the magnetic field [1,2], is also governed by equations identical or similar to (1.1).…”