Transition from spiral wave chimeras to phase cluster states

Time-delay-induced spiral chimeras on a spherical surface of globally coupled oscillators

We consider globally coupled networks of identical oscillators, located on the surface of a sphere with interaction time delays, and show that the distance-dependent time delays play a key role for the spiral chimeras to occur as a generic state in different systems of coupled oscillators. For the phase oscillator system, we analyze the existence and stability of stationary solutions along the Ott-Antonsen invariant manifold to find the bifurcation structure of the spiral chimera state. We demonstrate via an extensive numerical experiment that the time-delay-induced spiral chimeras are also present for coupled networks of the Stuart-Landau and Van der Pol oscillators in the same parameter regime as that of phase oscillators, with a series of evenly spaced band-type regions. It is found that the spiral chimera state occurs as a consequence of a resonant-type interplay between the intrinsic period of an individual oscillator and the interaction time delay as a topological structure property.

From Turing patterns to chimera states in the 2D Brusselator model

The Brusselator has been used as a prototype model for autocatalytic reactions and, in particular, for the Belousov-Zhabotinsky reaction. When coupled at the diffusive limit, the Brusselator undergoes a Turing bifurcation resulting in the formation of classical Turing patterns, such as spots, stripes, and spirals in two spatial dimensions. In the present study, we use generic nonlocally coupled Brusselators and show that in the limit of the coupling range R→1 (diffusive limit), the classical Turing patterns are recovered, while for intermediate coupling ranges and appropriate parameter values, chimera states are produced. This study demonstrates how the parameters of a typical nonlinear oscillator can be tuned so that the coupled system passes from spatially stable Turing structures to dynamical spatiotemporal chimera states.

Asymmetric spiral chimeras on a spheric surface of nonlocally coupled phase oscillators

The spiral chimera state shows a remarkable spatiotemporal pattern in a two-dimensional array of oscillators for which the coherent spiral arms coexist with incoherent cores. In this work, we report on an asymmetric spiral chimera having incoherent cores of different sizes on the spherical surface of identical phase oscillators with nonlocal coupling. This asymmetric spiral chimera exhibits a strongly symmetry-broken state in the sense that not only the coherent and incoherent domains coexist, but also their incoherent cores are nonidentical, although the underlying coupling structure is symmetric. On the basis of analyses along the Ott-Antonsen invariant manifold, the bifurcation conditions of asymmetric spiral chimeras are derived, which reveals that the asymmetric spiral chimera state emerges via a supercritical symmetry-breaking bifurcation from the symmetric spiral chimera. For the coupling function composed of two Legendre modes, rigorous stability analyses are carried out to present a complete stability diagram for different types of spiral chimeras, including the stationary symmetric and asymmetric spiral chimeras as well as breathing asymmetric spiral chimera. For the general coupling scheme the asymmetric spiral chimera occurs as a result of competition between the odd and even Legendre modes of the coupling function. Our theoretical findings are verified by using extensive numerical simulations of the model system.

Intermittent regimes as a synchronization phenomenon in two sets of nonlinear chemical oscillators

Systems of nonlinear chemical oscillators can exhibit a large diversity of non-trivial states depending on the parameters that characterize them. Among these, a synchronization phenomenon is of special interest due to its direct link with chemical and biological processes in nature. We carry out numerical experiments for two different sets of chemical oscillators with different properties and immersed in a Belousov-Zhabotinsky solution. We document the emergence of different states of synchronization that depend on the parameters characterizing the solution. We also show that, in the interface regions, this system generates a stable dynamics of intermittency between the different synchronization states where interesting phenomena, such as the "devil's staircase," emerge. In general, the added complexity introduced with the additional set of oscillators results in more complex non-trivial synchronization states.

A new class of chimeras in locally coupled oscillators with small-amplitude, high-frequency asynchrony and large-amplitude, low-frequency synchrony

Chimeras are surprising yet important states in which domains of decoherent (asynchronous) and coherent (synchronous) oscillations co-exist. In this article, we report on the discovery of a new class of chimeras, called mixed-amplitude chimera states, in which the structures, amplitudes, and frequencies of the oscillations differ substantially in the decoherent and coherent regions. These mixed-amplitude chimeras exhibit domains of decoherent small-amplitude oscillations (phase waves) coexisting with domains of stable and coherent large-amplitude or mixed-mode oscillations (MMOs). They are observed in a prototypical bistable partial differential equation with oscillatory dynamics, spatially homogeneous kinetics, and purely local, isotropic diffusion. They are observed in parameter regimes immediately adjacent to regimes in which common large-amplitude solutions exist, such as trigger waves, spatially homogeneous MMOs, and sharp-interface solutions. Also, key singularities, folded nodes, and folded saddles arising commonly in multi-scale, bistable systems play important roles, and these have not previously been studied in systems with chimeras. The discovery of these mixed-amplitude chimeras is an important advance for understanding some processes in neuroscience, pattern formation, and physics, which involve both small-amplitude and large-amplitude oscillations. It may also be of use for understanding some aspects of electroencephalogram recordings from animals that exhibit unihemispheric slow-wave sleep.

Moving spiral wave chimeras

We consider a two-dimensional array of heterogeneous nonlocally coupled phase oscillators on a flat torus and study the bound states of two counter-rotating spiral chimeras, shortly two-core spiral chimeras, observed in this system. In contrast to other known spiral chimeras with motionless incoherent cores, the two-core spiral chimeras typically show a drift motion. Due to this drift, their incoherent cores become spatially modulated and develop specific fingerprint patterns of varying synchrony levels. In the continuum limit of infinitely many oscillators, the two-core spiral chimeras can be studied using the Ott-Antonsen equation. Numerical analysis of this equation allows us to reveal the stability region of different spiral chimeras, which we group into three main classes-symmetric, asymmetric, and meandering spiral chimeras.

Shooting solitaries due to small-world connectivity in leaky integrate-and-fire networks

We study the synchronization properties in a network of leaky integrate-and-fire oscillators with nonlocal connectivity under probabilistic small-world rewiring. We demonstrate that the random links lead to the emergence of chimera-like states where the coherent regions are interrupted by scattered, short-lived solitaries; these are termed "shooting solitaries." Moreover, we provide evidence that random links enhance the appearance of chimera-like states for values of the parameter space that otherwise support synchronization. This last effect is counter-intuitive because by adding random links to the synchronous state, the system locally organizes into coherent and incoherent domains.

Amplitude-mediated spiral chimera pattern in a nonlinear reaction-diffusion system

Formation of diverse patterns in spatially extended reaction-diffusion systems is an important aspect of study that is pertinent to many chemical and biological processes. Of special interest is the peculiar phenomenon of chimera state having spatial coexistence of coherent and incoherent dynamics in a system of identically interacting individuals. In the present article, we report the emergence of various collective dynamical patterns while considering a system of prey-predator dynamics in the presence of a two-dimensional diffusive environment. Particularly, we explore the observance of four distinct categories of spatial arrangements among the species, namely, spiral wave, spiral chimera, completely synchronized oscillations, and oscillation death states in a broad region of the diffusion-driven parameter space. Emergence of amplitude-mediated spiral chimera states displaying drifted amplitudes and phases in the incoherent subpopulation is detected for parameter values beyond both Turing and Hopf bifurcations. Transition scenarios among all these distinguishable patterns are numerically demonstrated for a wide range of the diffusion coefficients which reveal that the chimera states arise during the transition from oscillatory to steady-state dynamics. Furthermore, we characterize the occurrence of each of the recognizable patterns by estimating the strength of incoherent subpopulations in the two-dimensional space.