Smallest Chimeras Under Repulsive Interactions

We present an exemplary system of three identical oscillators in a ring interacting repulsively to show up chimera patterns. The dynamics of individual oscillators is governed by the superconducting Josephson junction. Surprisingly, the repulsive interactions can only establish a symmetry of complete synchrony in the ring, which is broken with increasing repulsive interactions when the junctions pass through serials of asynchronous states (periodic and chaotic) but finally emerge into chimera states. The chimera pattern first appears in chaotic rotational motion of the three junctions when two junctions evolve coherently, while the third junction is incoherent. For larger repulsive coupling, the junctions evolve into another chimera pattern in a periodic state when two junctions remain coherent in rotational motion and one junction transits to incoherent librational motion. This chimera pattern is sensitive to initial conditions in the sense that the chimera state flips to another pattern when two junctions switch to coherent librational motion and the third junction remains in rotational motion, but incoherent. The chimera patterns are detected by using partial and global error functions of the junctions, while the librational and rotational motions are identified by a libration index. All the collective states, complete synchrony, desynchronization, and two chimera patterns are delineated in a parameter plane of the ring of junctions, where the boundaries of complete synchrony are demarcated by using the master stability function.

Chimera state in coupled map lattice of matrices

In recent years, a lot of research has focused on understanding the behavior of when synchronous and asynchronous phases occur, that is, the existence of chimera states in various networks. Chimera states have wide-range applications in many disciplines including biology, chemistry, physics, or engineering. The object of research in this paper is a coupled map lattice of matrices when each node is described by an iterative map of matrices of order two. A regular topology network of iterative maps of matrices was formed by replacing the scalar iterative map with the iterative map of matrices in each node. The coupled map of matrices is special in a way where we can observe the effect of divergence. This effect can be observed when the matrix of initial conditions is a nilpotent matrix. Also, the evolution of the derived network is investigated. It is found that the network of the supplementary variable $\mu$ can evolve into three different modes: the quiet state, the state of divergence, and the formation of divergence chimeras. The space of parameters of node coupling including coupling strength $\varepsilon$ and coupling range $r$ is also analyzed in this study. Image entropy is applied in order to identify chimera state parameter zones.

Impact of Nonlocal Interaction on Chimera States in Nonlocally Coupled Stuart–Landau Oscillators

We investigate the existence of collective dynamical states in nonlocally coupled Stuart–Landau oscillators with symmetry breaking included in the coupling term. We find that the radius of nonlocal interaction and nonisochronicity parameter play important roles in identifying the swing of synchronized states through amplitude chimera states. Collective dynamical states are distinguished with the help of strength of incoherence. Different transition routes to multi-chimera death states are analyzed with respect to the nonlocal coupling radius. In addition, we investigate the existence of collective dynamical states including traveling wave state, amplitude chimera state and multi-chimera death state by introducing higher-order nonlinear terms in the system. We also verify the robustness of the given notable properties for the coupled system.

Phase transition to chimera state in two populations of oscillators interacting via a common external environment

We consider two populations of coupled oscillators, interacting each other through a common external environment. The external environment is synthesized by the contributions from all oscillators of both populations. Such indirect coupling via an external medium arises naturally in many fields, e.g., dynamical quorum sensing in coupled biological and chemical systems. We analyze the existence and stability of a variety of stationary states on the basis of the OttAntonsen reduction method, which reveals that the interaction via an external environment gives rise to unusual collective behaviors such as the uniform drifting, non-uniform drifting and chimera states. We present a complete bifurcation diagram, which provides the underlying mechanism of the phase transition towards chimera state with the route of incoherence → uniform drift → non-uniform drift → chimera.

Emergence of Stripe-Core Mixed Spiral Chimera on a Spherical Surface of Nonlocally Coupled Oscillators

We report on a stripe-core mixed spiral chimera in a system of nonlocally coupled phase oscillators, located on the spherical surface, where the spiral wave consisting of phase-locked oscillators is separated by a stripe-type region of incoherent oscillators into two parts. We analyze the existence and stability of the stripe-core mixed spiral chimera state rigorously, on the basis of the Ott–Antonsen reduction theory. The stability diagram for the stationary states including the spiral chimeras as well as incoherent state is presented. Our stability analysis reveals that the stripe-core mixed spiral chimera state emerges as a unique attractor and loses its stability via the Hopf bifurcation. We verify our theoretical results using direct numerical simulations of the model system.

Chimera State in the Network of Fractional-Order FitzHugh–Nagumo Neurons

The fractional calculus in the neuronal models provides the memory properties. In the fractional-order neuronal model, the dynamics of the neuron depends on the derivative order, which can produce various types of memory-dependent dynamics. In this paper, the behaviors of the coupled fractional-order FitzHugh–Nagumo neurons are investigated. The effects of the coupling strength and the derivative order are under consideration. It is revealed that the level of the synchronization is decreased by decreasing the derivative order, and the chimera state emerges for stronger couplings. Furthermore, the patterns of the formed chimeras rely on the order of the derivatives.

Pattern selection in multi-layer network with adaptive coupling

Feed-forward effect modulates collective behavior of a multiple neuron network and facilitates strongly synchronization of their firing in deep layers. However, full synchronization of neuron system corresponds to functional disorder. In this work, we investigate coexistence of synchronized and incoherent neurons in deeper layer (called chimera state) in order to avoid the contradiction between facilitation of full synchronization and complete functional failure of neuron system. We focus on a multiple network containing two layers and confirm that chimera state in layer 1 could also induce that in layer 2 when the feed-forward effect is strong enough. Cluster also is discovered as a transient state which separates full synchronization and chimera state and occupy a narrow region. Both feed-forward and back-forward effect together emerge of chimera states in both layer 1 and 2 under same parameter in large range of parameters selection. Further, we introduce adaptive dynamics into inter-layer rather than intra-layer couplings. Under this circumstance chimera state still can be induced and coupling matrix will be self-organized under suitable phase parameter to guarantee chimera formation. Indeed, chimera states exist and transit to deeper layer in a regular multiple network with very strict parameter selection. The results helps understanding better the neuron firing propagating and encoding scheme in a multi-layer neuron network.