一个全局耦合不连续映像格子中的奇异态

文章研究了一类由既不可逆又不连续映像构成的全局耦合映像格子系统中的奇异态行为,计算了系统的同步序参量和空间振幅变化图.结果表明,在某些特定的参数区间内,耦合映像格子系统会出现奇异态或团簇态,并且敏感地依赖于耦合强度的选择.上述丰富的动力学现象是由于单映像中不连续、不可逆性以及空间耦合相互作用的结果.通过数值模拟找到了奇异态或团簇态出现的特定参数区域.     

Chimera states in bipartite networks of FitzHugh–Nagumo oscillators

Chimera states consisting of spatially coherent and incoherent domains have been observed in different topologies such as rings, spheres, and complex networks. In this paper, we investigate bipartite networks of nonlocally coupled FitzHugh–Nagumo (FHN) oscillators in which the units are allocated evenly to two layers, and FHN units interact with each other only when they are in different layers. We report the existence of chimera states in bipartite networks. Owing to the interplay between chimera states in the two layers, many types of chimera states such as in-phase chimera states, antiphase chimera states, and out-of-phase chimera states are classified. Stability diagrams of several typical chimera states in the coupling strength–coupling radius plane, which show strong multistability of chimera states, are explored.     

Chimera States mediated by nonlocally attractive-repulsive coupling in FitzHugh–Nagumo neural networks

The spontaneous occurrence of heterogeneous behaviors in homogeneous systems is an intriguing phenomenon. Recently, a remarkable heterogeneous behavior, called “chimera states”, which consists of spatially coherent and incoherent domains, has been studied in a great variety of systems including physical, chemical, biological, or optical. In this paper, chimera states in FitzHugh–Nagumo (FHN) neural networks are investigated. The identical FHN neurons are assigned in a ring and nonlocally coupled by attractive and repulsive couplings. We show that, the chimera states can be induced by the cooperation of nonlocally attractive and repulsive interactions between these neurons. Moreover, depending on the strength and range of attractive or repulsive couplings, the neural networks display different spatiotemporal behaviors, including chimera states, multi-cluster (MC) chimera states, traveling waves, traveling coherent states, solitary states, bursting synchronizations, and synchronizations. These results suggest that attractive and repulsive couplings may play a crucial role in mediating dynamic behavior of neural networks, and these results could be useful in understanding and predicting the rich dynamics of neural networks.     

Entangled chimeras in nonlocally coupled bicomponent phase oscillators: From synchronous to asynchronous chimeras

Chimera states, a symmetry-breaking spatiotemporal pattern in nonlocally coupled identical dynamical units, have been identified in various systems and generalized to coupled nonidentical oscillators. It has been shown that strong heterogeneity in the frequencies of nonidentical oscillators might be harmful to chimera states. In this work, we consider a ring of nonlocally coupled bicomponent phase oscillators in which two types of oscillators are randomly distributed along the ring: some oscillators with natural frequency ω₁ and others with ω₂ . In this model, the heterogeneity in frequency is measured by frequency mismatch |ω₁−ω₂| between the oscillators in these two subpopulations. We report that the nonlocally coupled bicomponent phase oscillators allow for chimera states no matter how large the frequency mismatch is. The bicomponent oscillators are composed of two chimera states, one supported by oscillators with natural frequency ω₁ and the other by oscillators with natural frequency ω₂. The two chimera states in two subpopulations are synchronized at weak frequency mismatch, in which the coherent oscillators in them share similar mean phase velocity, and are desynchronized at large frequency mismatch, in which the coherent oscillators in different subpopulations have distinct mean phase velocities. The synchronization–desynchronization transition between chimera states in these two subpopulations is observed with the increase in the frequency mismatch. The observed phenomena are theoretically analyzed by passing to the continuum limit and using the Ott-Antonsen approach.     

Chimeras in random non-complete networks of phase oscillators

We consider the simplest network of coupled non-identical phase oscillators capable of displaying a "chimera" state (namely, two subnetworks with strong coupling within the subnetworks and weaker coupling between them) and systematically investigate the effects of gradually removing connections within the network, in a random but systematically specified way. We average over ensembles of networks with the same random connectivity but different intrinsic oscillator frequencies and derive ordinary differential equations (ODEs), whose fixed points describe a typical chimera state in a representative network of phase oscillators. Following these fixed points as parameters are varied we find that chimera states are quite sensitive to such random removals of connections, and that oscillations of chimera states can be either created or suppressed in apparent bifurcation points, depending on exactly how the connections are gradually removed.     

Synchronization patterns and chimera states in complex networks: Interplay of topology and dynamics

We review chimera patterns, which consist of coexisting spatial domains of coherent (synchronized) and incoherent (desynchronized) dynamics in networks of identical oscillators. We focus on chimera states involving amplitude as well as phase dynamics, complex topologies like small-world or hierarchical (fractal), noise, and delay. We show that a plethora of novel chimera patterns arise if one goes beyond the Kuramoto phase oscillator model. For the FitzHugh-Nagumo system, the Van der Pol oscillator, and the Stuart-Landau oscillator with symmetry-breaking coupling various multi-chimera patterns including amplitude chimeras and chimera death occur. To test the robustness of chimera patterns with respect to changes in the structure of the network, regular rings with coupling range R, small-world, and fractal topologies are studied. We also address the robustness of amplitude chimera states in the presence of noise. If delay is added, the lifetime of transient chimeras can be drastically increased.     

Chimera dynamics in nonlocally coupled moving phase oscillators

Chimera states, a symmetry-breaking spatiotemporal pattern in nonlocally coupled dynamical units, prevail in a variety of systems. However, the interaction structures among oscillators are static in most of studies on chimera state. In this work, we consider a population of agents. Each agent carries a phase oscillator. We assume that agents perform Brownian motions on a ring and interact with each other with a kernel function dependent on the distance between them. When agents are motionless, the model allows for several dynamical states including two different chimera states (the type-I and the type-II chimeras). The movement of agents changes the relative positions among them and produces perpetual noise to impact on the model dynamics. We find that the response of the coupled phase oscillators to the movement of agents depends on both the phase lag α, determining the stabilities of chimera states, and the agent mobility D. For low mobility, the synchronous state transits to the type-I chimera state for α close to π/2 and attracts other initial states otherwise. For intermediate mobility, the coupled oscillators randomly jump among different dynamical states and the jump dynamics depends on α. We investigate the statistical properties in these different dynamical regimes and present the scaling laws between the transient time and the mobility for low mobility and relations between the mean lifetimes of different dynamical states and the mobility for intermediate mobility.     

Chimera states in networks of logistic maps with hierarchical connectivities

Chimera states are complex spatiotemporal patterns consisting of coexisting domains of coherence and incoherence. We study networks of nonlocally coupled logistic maps and analyze systematically how the dilution of the network links influences the appearance of chimera patterns. The network connectivities are constructed using an iterative Cantor algorithm to generate fractal (hierarchical) connectivities. Increasing the hierarchical level of iteration, we compare the resulting spatiotemporal patterns. We demonstrate that a high clustering coefficient and symmetry of the base pattern promotes chimera states, and asymmetric connectivities result in complex nested chimera patterns.     

Chimera states and synchronization in magnetically driven SQUID metamaterials

One-dimensional arrays of Superconducting QUantum Interference Devices (SQUIDs) form magnetic metamaterials exhibiting extraordinary properties, including tunability, dynamic multistability, negative magnetic permeability, and broadband transparency. The SQUIDs in a metamaterial interact through non-local, magnetic dipole-dipole forces, that makes it possible for multiheaded chimera states and coexisting patterns, including solitary states, to appear. The spontaneous emergence of chimera states and the role of multistability is demonstrated numerically for a SQUID metamaterial driven by an alternating magnetic field. The spatial synchronization and temporal complexity are discussed and the parameter space for the global synchronization reveals the areas of coherence-incoherence transition. Given that both one- and two-dimensional SQUID metamaterials have been already fabricated and investigated in the lab, the presence of a chimera state could in principle be detected with presently available experimental set-ups.     

Chimera states for coupled oscillators

Arrays of identical oscillators can display a remarkable spatiotemporal pattern in which phase-locked oscillators coexist with drifting ones. Discovered two years ago, such "chimera states" are believed to be impossible for locally or globally coupled systems; they are peculiar to the intermediate case of nonlocal coupling. Here we present an exact solution for this state, for a ring of phase oscillators coupled by a cosine kernel. We show that the stable chimera state bifurcates from a spatially modulated drift state, and dies in a saddle-node bifurcation with an unstable chimera state.     

The effect of topology on organization of synchronous behavior in dynamical networks with adaptive couplings

We study the influence of the initial topology of connections on the organization of synchronous behavior in networks of phase oscillators with adaptive couplings. We found that networks with a random sparse structure of connections predominantly demonstrate the scenario as a result of which chimera states are formed. The formation of chimera states retains the features of the hierarchical organization observed in networks with global connections [D.V. Kasatkin, S. Yanchuk, E. Schöll, V.I. Nekorkin, Phys. Rev. E 96, 062211 (2017)], and also demonstrates a number of new properties due to the presence of a random structure of network topology. In this case, the formation of coherent groups takes a much longer time interval, and the sets of elements that form these groups can be significantly rearranged during the evolution of the network. We also found chimera states, in which along with the coherent and incoherent groups, there are subsets, whose different elements can be synchronized with each other for sufficiently long periods of time.     

Clustered chimera states in delay-coupled oscillator systems

We investigate chimera states in a ring of identical phase oscillators coupled in a time-delayed and spatially nonlocal fashion. We find novel clustered chimera states that have spatially distributed phase coherence separated by incoherence with adjacent coherent regions in antiphase. The existence of such time-delay induced phase clustering is further supported through solutions of a generalized functional self-consistency equation of the mean field. Our results highlight an additional mechanism for cluster formation that may find wider practical applications.     

Solvable model for chimera states of coupled oscillators

Networks of identical, symmetrically coupled oscillators can spontaneously split into synchronized and desynchronized subpopulations. Such chimera states were discovered in 2002, but are not well understood theoretically. Here we obtain the first exact results about the stability, dynamics, and bifurcations of chimera states by analyzing a minimal model consisting of two interacting populations of oscillators. Along with a completely synchronous state, the system displays stable chimeras, breathing chimeras, and saddle-node, Hopf, and homoclinic bifurcations of chimeras.     

Spectral properties of chimera states

Chimera states are particular trajectories in systems of phase oscillators with nonlocal coupling that display a spatiotemporal pattern of coherent and incoherent motion. We present here a detailed analysis of the spectral properties for such trajectories. First, we study numerically their Lyapunov spectrum and its behavior for an increasing number of oscillators. The spectra demonstrate the hyperchaotic nature of the chimera states and show a correspondence of the Lyapunov dimension with the number of incoherent oscillators. Then, we pass to the thermodynamic limit equation and present an analytic approach to the spectrum of a corresponding linearized evolution operator. We show that, in this setting, the chimera state is neutrally stable and that the continuous spectrum coincides with the limit of the hyperchaotic Lyapunov spectrum obtained for the finite size systems.     

Cloning of Chimera States in a Multiplex Network of Two-Frequency Oscillators with Linear Local Couplings

Cloning of chimera states, which is a new effect caused by the short-term interaction in a multiplex network, has been described. This effect is observed when two ring networks of linearly coupled two-frequency (bistable) oscillators are combined into the multiplex network. At certain values of the strength and duration of the inter-ring (multiplex) interaction, a copy of a chimera state with accuracy to phases in the incoherent part is formed in the ring with an initially random phase distribution. It has been shown that the effect is structurally stable and is due to the competition of self-sustained oscillations in individual rings.     

Synchronization scenarios of chimeras in multiplex networks

Complex networks consisting of several interacting layers allow for remote synchronization of distant layers via an intermediate relay layer. We extend the notion of relay synchronization to chimera states, and study the scenarios of relay synchronization in a three-layer network of FitzHugh–Nagumo (FHN) oscillators, where each layer has a nonlocal coupling topology. Varying the coupling strength and time delay in the inter-layer connections, we observe relay synchronization between chimera states, i.e., complex spatio-temporal patterns of coexisting coherent and incoherent domains, in the outer network layers. Special regimes where only the coherent domains of chimeras are synchronized, and the incoherent domains remain desynchronized, as well as transitions between different synchronization regimes are analyzed.     

Chimeras in a network of three oscillator populations with varying network topology

We study a network of three populations of coupled phase oscillators with identical frequencies. The populations interact nonlocally, in the sense that all oscillators are coupled to one another, but more weakly to those in neighboring populations than to those in their own population. Using this system as a model system, we discuss for the first time the influence of network topology on the existence of so-called chimera states. In this context, the network with three populations represents an interesting case because the populations may either be connected as a triangle, or as a chain, thereby representing the simplest discrete network of either a ring or a line segment of oscillator populations. We introduce a special parameter that allows us to study the effect of breaking the triangular network structure, and to vary the network symmetry continuously such that it becomes more and more chain-like. By showing that chimera states only exist for a bounded set of parameter values, we demonstrate that their existence depends strongly on the underlying network structures, and conclude that chimeras exist on networks with a chain-like character.     

Controlling chaos and supressing chimeras in a fractional-order discrete phase-locked loop using impulse control

A fractional-order difference equation model of a third-order discrete phase-locked loop (FODPLL) is discussed and the dynamical behavior of the model is demonstrated using bifurcation plots and a basin of attraction. We show a narrow region of loop gain where the FODPLL exhibits quasi-periodic oscillations, which were not identified in the integer-order model. We propose a simple impulse control algorithm to suppress chaos and discuss the effect of the control step. A network of FODPLL oscillators is constructed and investigated for synchronization behavior. We show the existence of chimera states while transiting from an asynchronous to a synchronous state. The same impulse control method is applied to a lattice array of FODPLL, and the chimera states are then synchronized using the impulse control algorithm. We show that the lower control steps can achieve better control over the higher control steps.     

Multi-chimera states and transitions in the Leaky Integrate-and-Fire model with nonlocal and hierarchical connectivity

The effects of nonlocal and fractal connectivity are investigated in a network of Leaky Integrate-and-Fire (LIF) elements. The idea of fractal coupling originates from the hierarchical topology of networks formed by neuronal axons, which transmit the electrical signals in the brain. If a number of LIF elements with finite refractory period are nonlocally coupled, multi-chimera states emerge whose multiplicity depends both on the coupling strength and on the refractory period. We provide evidence that the introduction of a hierarchical topology in the coupling induces novel complex spatial and temporal structures, such as nested chimera states and transitions between multi-chimera states with different multiplicities. These results demonstrate new complex patterns, as well as transitions between different multi-chimera states arising from the combination of nonlinear dynamics with the hierarchical coupling.     

Chimera Structures in the Ensembles of Nonlocally Coupled Chaotic Oscillators

Radiophysics and Quantum Electronics - We study the structure and properties of the chimera states in the ensembles of chaotic oscillators with nonlocal coupling. It is shown that the phase and...