A chimera state is a spatio-temporal pattern in a network of identical coupled oscillators in which synchronous and asynchronous oscillation coexist. This state of broken symmetry, which usually coexists with a stable spatially symmetric state, has intrigued the nonlinear dynamics community since its discovery in the early 2000s. Recent experiments have led to increasing interest in the origin and dynamics of these states. Here we review the history of research on chimera states and highlight major advances in understanding their behaviour.

http://dmabrams.esam.northwestern.edu/pubs/Panaggio%20and%20Abrams%20-%20Chimera%20states%20(review)%20-%20Nonlinearity%2028,%20R67-R87%20(2015).pdf

Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators

Conventional evolutionary game theory predicts that natural selection favours the selfish and strong even though cooperative interactions thrive at all levels of organization in living systems. Recent investigations demonstrated that a limiting factor for the evolution of cooperative interactions is the way in which they are organized, cooperators becoming evolutionarily competitive whenever individuals are constrained to interact with few others along the edges of networks with low average connectivity. Despite this insight, the conundrum of cooperation remains since recent empirical data shows that real networks exhibit typically high average connectivity and associated single-to-broad–scale heterogeneity. Here, a computational model is constructed in which individuals are able to self-organize both their strategy and their social ties throughout evolution, based exclusively on their self-interest. We show that the entangled evolution of individual strategy and network structure constitutes a key mechanism for the sustainability of cooperation in social networks. For a given average connectivity of the population, there is a critical value for the ratio W between the time scales associated with the evolution of strategy and of structure above which cooperators wipe out defectors. Moreover, the emerging social networks exhibit an overall heterogeneity that accounts very well for the diversity of patterns recently found in acquired data on social networks. Finally, heterogeneity is found to become maximal when W reaches its critical value. These results show that simple topological dynamics reflecting the individual capacity for self-organization of social ties can produce realistic networks of high average connectivity with associated single-to-broad–scale heterogeneity. On the other hand, they show that cooperation cannot evolve as a result of “social viscosity” alone in heterogeneous networks with high average connectivity, requiring the additional mechanism of topological co-evolution to ensure the survival of cooperative behaviour.

Cooperation prevails when individuals adjust their social ties

Oscillation quenching mechanisms: Amplitude vs. oscillation death

Chimera states in neuronal networks: A review.

The model of nonlocally coupled identical phase oscillators on complex networks is investigated. We find the existence of chimera states in which identical oscillators evolve into distinct coherent and incoherent groups. We find that the coherent group of chimera states always contains the same oscillators no matter what the initial conditions are. The properties of chimera states and their dependence on parameters are investigated on both scale-free networks and Erdos-Renyi networks.

Chimera states on complex networks.

In reality, migration of an individual usually correlates with the individual's financial or social status. Here, we consider the situation where migration of a player depends on the player's payoff to an evolutionary prisoner's dilemma game. When the mobility of a player is positively correlated with the player's normalized payoff, Pi, where the mobility of player i is defined as ?i=Pi?, we found that cooperation could be promoted strongly in the case of a high density of players because of the introduction of this kind of migration. Moreover, the system could reach a state of complete cooperation in a large region on the given parameter space. Interestingly, enhancement of cooperation shows a non-monotonic behavior with an increase in ?. We also found that the positive effects of this kind of migration on cooperation are robust in the face of changes to the network structure and the strategy-updating rule. In addition, we consider another situation where the mobility of a player is anti-correlated with the player's normalized payoff, and we observe that cooperation enhancement still exists.

https://iopscience.iop.org/article/10.1088/1367-2630/13/4/043032/pdf

Payoff-related migration enhances cooperation in the prisoner's dilemma game

In the evolution of cooperation, the motion of players plays an important role. In this paper, we incorporate, into an evolutionary prisoner dilemma's game on networks, a new factor that cooperators and defectors move with different probabilities. By investigating the dependence of the cooperator frequency on the moving probabilities of cooperators and defectors, µc and µd, we find that cooperation is greatly enhanced in the parameter regime of µc < µd. The snapshots of strategy pattern and the evolutions of cooperator clusters and defector clusters reveal that either the fast motion of defectors or the slow motion of cooperators always favors the formation of large cooperator clusters. The model is investigated on different types of networks such as square lattices, Erdos-Renyi networks and scale-free networks and with different types of strategy-updating rules such as the richest-following rule and the Fermi rule. The numerical results show that the observed phenomena are robust to different networks and to different strategy-updating rules.

https://iopscience.iop.org/article/10.1088/1367-2630/12/12/123014/pdf

Motion depending on the strategies of players enhances cooperation in a co-evolutionary prisoner's dilemma game

The phenomenon of phase death in coupled multistable oscillators which have stable equilibria as one of their attractors when isolated is investigated. We find that the phase death state can be induced by the coupling interaction which may have the attraction basin of the existing stable equilibria expanded. The mechanism for the onset of the phase death is proposed. The phase death is further classified into complete and incomplete phase deaths depending on whether the arbitrary initial condition can lead the coupled system to the phase death state.

https://iopscience.iop.org/article/10.1209/0295-5075/82/40001/pdf

A study of phase death states in a coupled system with stable equilibria

The effects of amplitude-dependent coupling on oscillator death (OD) are investigated for two repulsively coupled Lorenz oscillators. Based on numerical simulations, it is shown that as constraint strengths on the amplitude-dependent coupling change, an oscillatory statemay undergo a transition to an OD state. The parameter regimes of the OD domain are theoretically determined, which coincide well with the numerical results. An electronic circuit is set up to exhibit the transition process to the OD state with an amplitude-dependent coupling. These findings may have practical importance on chaos control and oscillation depression.

Yun Zhu

Papers

Chimera states on complex networks.

Payoff-related migration enhances cooperation in the prisoner's dilemma game

Motion depending on the strategies of players enhances cooperation in a co-evolutionary prisoner's dilemma game

A study of phase death states in a coupled system with stable equilibria

Oscillator death induced by amplitude-dependent coupling in repulsively coupled oscillators.