Mathematical Framework for Breathing Chimera States

Abstract

About two decades ago it was discovered that systems of nonlocally coupled oscillators can exhibit unusual symmetry-breaking patterns composed of coherent and incoherent regions. Since then such patterns, called chimera states, have been the subject of intensive study but mostly in the stationary case when the coarse-grained system dynamics remains unchanged over time. Nonstationary coherence–incoherence patterns, in particular periodically breathing chimera states, were also reported, however not investigated systematically because of their complexity. In this paper we suggest a semi-analytic solution to the above problem providing a mathematical framework for the analysis of breathing chimera states in a ring of nonlocally coupled phase oscillators. Our approach relies on the consideration of an integro-differential equation describing the long-term coarse-grained dynamics of the oscillator system. For this equation we specify a class of solutions relevant to breathing chimera states. We derive a self-consistency equation for these solutions and carry out their stability analysis. We show that our approach correctly predicts macroscopic features of breathing chimera states. Moreover, we point out its potential application to other models which can be studied using the Ott–Antonsen reduction technique.

Nontrivial Twisted States in Nonlocally Coupled Stuart-Landau Oscillators

“…In order to validate that the NTS is in fact a coherent traveling wave, as a TTS is, we determined the Kuramoto local order parameter z(x, t) defined as [29,30] z(x, t) = lim…”

Travelling chimeras in oscillator lattices with advective–diffusive coupling

“…Substituting the current approximation for hfalse(ξfalse) to equation (4.2 a ), one gets an equation that can be interpreted as the Adler equation with a periodic forcing in the nonlinear term. Transforming to a variable z=eifalse(ϕ+αfalse), equation (4.2 a ) can be written in a complex form dzdξ=12v false(h∗ false(ξfalse)z2+2iΩz−hfalse(ξfalse)false), which is the complex Riccati equation with periodic coefficients [34–37]. Here and below, symbol ∗ denotes complex conjugate.…”

“…It is noteworthy that this equation is partially reminiscent of the Ott–Antonsen equation for a coarse-grained complex order parameter (e.g. see [16–18,34]). Formally, equation (A 1) can be considered not only on the unit circle |z|=1 but also inside the unit disc |z|<1 of the complex plane.…”

“…Formally, equation (A 1) can be considered not only on the unit circle |z|=1 but also inside the unit disc |z|<1 of the complex plane. In the paper [34], it is shown that, in general, there exists a unique stable solution to equation (A 1) starting from the initial condition zfalse(ξ=0false)=z0, where |z0|≤1, and lying entirely in such a closure of the corresponding domain. Moreover, if |z0|<1 or |z0|=1, then |zfalse(ξfalse)|<1 or |zfalse(ξfalse)|=1 for all ξ>0, respectively, i. e. every solution zfalse(ξfalse) of equation (A 1) satisfying |z0|<1 remains trapped inside the domain |z|<…”

“…Moreover, if |z0|<1 or |z0|=1, then |zfalse(ξfalse)|<1 or |zfalse(ξfalse)|=1 for all ξ>0, respectively, i. e. every solution zfalse(ξfalse) of equation (A 1) satisfying |z0|<1 remains trapped inside the domain |z|<1, and each trajectory starting from the initial point |z0|=1 on the unit circle of the complex plane stays on the given circle.It is well-known (e.g. see [34–37]) that the Poincaré map of the periodic complex Riccati equation coincides with the Möbius transformation. In our case, this Möbius transformation maps the closed unit disk |z|≤1 onto itself, thus it can be written in the canonical form [35,38] scriptMq,ψfalse(zfalse)...…”