Bumps, chimera states, and Turing patterns in systems of coupled active rotators

Chimera states in a chain of superdiffusively coupled neurons

Two- and three-component systems of superdiffusion equations describing the dynamics of action potential propagation in a chain of non-locally interacting neurons with Hindmarsh-Rose nonlinear functions have been considered. Non-local couplings based on the fractional Laplace operator describing superdiffusion kinetics are found to support chimeras. In turn, the system with local couplings, based on the classical Laplace operator, shows synchronous behavior. For several parameters responsible for the activation properties of neurons, it is shown that the structure and evolution of chimera states depend significantly on the fractional Laplacian exponent, reflecting non-local properties of the couplings. For two-component systems, an anisotropic transition to full incoherence in the parameter space responsible for non-locality of the first and second variables is established. Introducing a third slow variable induces a gradual transition to incoherence via additional chimera states formation. We also discuss the possible causes of chimera states formation in such a system of non-locally interacting neurons and relate them with the properties of the fractional Laplace operator in a system with global coupling.

Transient chimera states emerging from dynamical trapping in chaotic saddles

Nonlinear systems possessing nonattracting chaotic sets, such as chaotic saddles, embedded in their state space may oscillate chaotically for a transient time before eventually transitioning into some stable attractor. We show that these systems, when networked with nonlocal coupling in a ring, are capable of forming chimera states, in which one subset of the units oscillates periodically in a synchronized state forming the coherent domain, while the complementary subset oscillates chaotically in the neighborhood of the chaotic saddle constituting the incoherent domain. We find two distinct transient chimera states distinguished by their abrupt or gradual termination. We analyze the lifetime of both chimera states, unraveling their dependence on coupling range and size. We find an optimal value for the coupling range yielding the longest lifetime for the chimera states. Moreover, we implement transversal stability analysis to demonstrate that the synchronized state is asymptotically stable for network configurations studied here.

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.

Solitary routes to chimera states

We show how solitary states in a system of globally coupled FitzHugh-Nagumo oscillators can lead to the emergence of chimera states. By a numerical bifurcation analysis of a suitable reduced system in the thermodynamic limit we demonstrate how solitary states, after emerging from the synchronous state, become chaotic in a period-doubling cascade. Subsequently, states with a single chaotic oscillator give rise to states with an increasing number of incoherent chaotic oscillators. In large systems, these chimera states show extensive chaos. We demonstrate the coexistence of many of such chaotic attractors with different Lyapunov dimensions, due to different numbers of incoherent oscillators.

Patched patterns and emergence of chaotic interfaces in arrays of nonlocally coupled excitable systems

We disclose a new class of patterns, called patched patterns, in arrays of non-locally coupled excitable units with attractive and repulsive interactions. The self-organization process involves the formation of two types of patches, majority and minority ones, characterized by uniform average spiking frequencies. Patched patterns may be temporally periodic, quasiperiodic, or chaotic, whereby chaotic patterns may further develop interfaces comprised of units with average frequencies in between those of majority and minority patches. Using chaos and bifurcation theory, we demonstrate that chaos typically emerges via a torus breakup and identify the secondary bifurcation that gives rise to chaotic interfaces. It is shown that the maximal Lyapunov exponent of chaotic patched patterns does not decay, but rather converges to a finite value with system size. Patched patterns with a smaller wavenumber may exhibit diffusive motion of chaotic interfaces, similar to that of the incoherent part of chimeras.

Exotic states induced by coevolving connection weights and phases in complex networks

We consider an adaptive network, whose connection weights coevolve in congruence with the dynamical states of the local nodes that are under the influence of an external stimulus. The adaptive dynamical system mimics the adaptive synaptic connections common in neuronal networks. The adaptive network under external forcing displays exotic dynamical states such as itinerant chimeras whose population density of coherent and incoherent domains coevolves with the synaptic connection, bump states, and bump frequency cluster states, which do not exist in adaptive networks without forcing. In addition, the adaptive network also exhibits partial synchronization patterns such as phase and frequency clusters, forced entrained, and incoherent states. We introduce two measures for the strength of incoherence based on the standard deviation of the temporally averaged (mean) frequency and on the mean frequency to classify the emergent dynamical states as well as their transitions. We provide a two-parameter phase diagram showing the wealth of dynamical states. We additionally deduce the stability condition for the frequency-entrained state. We use the paradigmatic Kuramoto model of phase oscillators, which is a simple generic model that has been widely employed in unraveling a plethora of cooperative phenomena in natural and man-made systems.