Spiral wave chimeras in nonlocally coupled bicomponent oscillators.

Chimera states in nonidentical oscillators have received extensive attention in recent years. Previous studies have demonstrated that chimera states can exist in a ring of nonlocally coupled bicomponent oscillators even in the presence of strong parameter heterogeneity. In this study, we investigate spiral wave chimeras in two-dimensional nonlocally coupled bicomponent oscillators where oscillators are randomly divided into two groups, with identical oscillators in the same group. Using phase oscillators and FitzHugh-Nagumo oscillators as examples, we numerically demonstrate that each group of oscillators supports its own spiral wave chimera and two spiral wave chimeras coexist with each other. We find that there exist three heterogeneity regimes: the synchronous regime at weak heterogeneity, the asynchronous regime at strong heterogeneity, and the transition regime in between. In the synchronous regime, spiral wave chimeras supported by different groups are synchronized with each other by sharing a same rotating frequency and a same incoherent core. In the asynchronous regime, the two spiral wave chimeras rotate at different frequencies and their incoherent cores are far away from each other. These phenomena are also observed in a nonrandom distribution of the two group oscillators and the continuum limit of infinitely many phase oscillators. The transition from synchronous to asynchronous spiral wave chimeras depends on the component oscillators. Specifically, it is a discontinuous transition for phase oscillators but a continuous one for FitzHugh-Nagumo oscillators. We also find that, in the asynchronous regime, increasing heterogeneity leads irregularly meandering spiral wave chimeras to rigidly rotating ones.

Dynamics in the Sakaguchi-Kuramoto model with bimodal frequency distribution.

In this work, we study the Sakaguchi-Kuramoto model with natural frequency following a bimodal distribution. By using Ott-Antonsen ansatz, we reduce the globally coupled phase oscillators to low dimensional coupled ordinary differential equations. For symmetrical bimodal frequency distribution, we analyze the stabilities of the incoherent state and different partial synchronous states. Different types of bifurcations are identified and the effect of the phase lag on the dynamics is investigated. For asymmetrical bimodal frequency distribution, we observe the revival of the incoherent state, and then the conditions for the revival are specified.

Twisted states in nonlocally coupled phase oscillators with frequency distribution consisting of two Lorentzian distributions with the same mean frequency and different widths.

In globally coupled phase oscillators, the distribution of natural frequency has strong effects on both synchronization transition and synchronous dynamics. In this work, we study a ring of nonlocally coupled phase oscillators with the frequency distribution made up of two Lorentzians with the same center frequency but with different half widths. Using the Ott-Antonsen ansatz, we derive a reduced model in the continuum limit. Based on the reduced model, we analyze the stability of the incoherent state and find the existence of multiple stability islands for the incoherent state depending on the parameters. Furthermore, we numerically simulate the reduced model and find a large number of twisted states resulting from the instabilities of the incoherent state with respect to different spatial modes. For some winding numbers, the stability region of the corresponding twisted state consists of two disjoint parameter regions, one for the intermediate coupling strength and the other for the strong coupling strength.

Two-frequency chimera state in a ring of nonlocally coupled Brusselators.

Chimera states, which consist of coexisting domains of spatially coherent and incoherent dynamics, have been intensively investigated in the past decade. In this work, we report a special chimera state, 2-frequency chimera state, in one-dimensional ring of nonlocally coupled Brusselators. In a 2-frequency chimera state, there exist two types of coherent domains and oscillators in different types of coherent domains have different mean phase velocities. We present the stability diagram of 2-frequency chimera state and study the transition between the 2-frequency chimera state and an ordinary 2-cluster chimera state.

Deffuant model on a ring with repelling mechanism and circular opinions.

We investigate a Deffuant model on a ring by introducing two modifications: the repelling mechanism and the circular opinions. The repelling mechanism drives the opinions of two individuals away from each other and the circular opinions are defined on a circle. We find that the repelling mechanism tends to polarize the opinions of adjacent individuals and the circular opinions bring up a spatiotemporal pattern in which all individuals take different opinions but the opinion difference between two neighboring individuals tends to zero in the limit of the population size. In the Deffuant model with both repelling mechanism and the circular opinions, opinion dynamics depends on both the bounded confidence and the convergence rate. The interplay between the repelling mechanism and the circular opinion may give rise to time-dependent opinion dynamics.

Dynamics in the Sakaguchi-Kuramoto model with two subpopulations [corrected].

The dynamics in a variant of globally coupled Sakaguchi-Kuramoto [corrected]. phase oscillators is studied. The model consists of two subpopulations, each with a different phase lag and interaction strength. Using Ott-Antonson ansatz, we analyze the dynamics in the model and present the numerical results. There exist stationary synchronous states which are generalized π states and two types of traveling wave states. We find that the traveling wave states are the dominant dynamics in comparison with the stationary states. Particularly, we find that the stationary and traveling wave states can be smoothly connected through the properly chosen parameter paths.

Conditional imitation might promote cooperation under high temptations to defect.

In this paper we introduce a conditional imitation rule into an evolutionary game, in which the imitation probabilities of individuals are determined by a function of payoff difference and two crucial parameters µ and σ. The parameter µ characterizes the most adequate goal for individuals and the parameter σ characterizes the tolerance of individuals. By using the pair approximation method and numerical simulations, we find an anomalous cooperation enhancement in which the cooperation level shows a nonmonotonic variation with the increase of temptation. The parameter µ affects the regime of the payoff parameter which supports the anomalous cooperation enhancement, whereas the parameter σ plays a decisive role on the appearance of the nonmonotonic variation of the cooperation level. Furthermore, to give explicit implications for the parameters µ and σ we present an alterative form of the conditional imitation rule based on the benefit and the cost incurred to individuals during strategy updates. In this way, we also provide a phenomenological interpretation for the nonmonotonic behavior of cooperation with the increase of temptation. The results give a clue that a higher cooperation level could be obtained under adverse environments for cooperation by applying the conditional imitation rule, which is possible to be manipulated in real life. More generally, the results in this work might point out an efficient way to maintain cooperation in the risky environments to cooperators.

Crossover between structured and well-mixed networks in an evolutionary prisoner's dilemma game.

In a spatial evolutionary prisoner's dilemma game (PDG), individuals interact with their neighbors and update their strategies according to some rules. As is well known, cooperators are destined to become extinct in a well-mixed population, whereas they could emerge and be sustained on a structured network. In this work, we introduce a simple model to investigate the crossover between a structured network and a well-mixed one in an evolutionary PDG. In the model, each link j is designated a rewiring parameter τ(j), which defines the time interval between two successive rewiring events for link j. By adjusting the rewiring parameter τ (the mean time interval for any link in the network), we could change a structured network into a well-mixed one. For the link rewiring events, three situations are considered: one synchronous situation and two asynchronous situations. Simulation results show that there are three regimes of τ: large τ where the density of cooperators ρ(c) rises to ρ(c,∞) (the value of ρ(c) for the case without link rewiring), small τ where the mean-field description for a well-mixed network is applicable, and moderate τ where the crossover between a structured network and a well-mixed one happens.