Chimera states are spatiotemporal patterns in which coherence and incoherence coexist. We observe the coexistence of synchronous (coherent) and desynchronous (incoherent) domains in a neuronal network. The network is composed of coupled adaptive exponential integrate-and-fire neurons that are connected by means of chemical synapses. In our neuronal network, the chimera states exhibit spatial structures both with spike and burst activities. Furthermore, those desynchronized domains not only have either spike or burst activity, but we show that the structures switch between spikes and bursts as the time evolves. Moreover, we verify the existence of multicluster chimera states.

The stickiness effect is a fundamental feature of quasi-integrable Hamiltonian systems. We propose the use of an entropy-based measure of the recurrence plots (RPs), namely, the entropy of the distribution of the recurrence times (estimated from the RP), to characterize the dynamics of a typical quasi-integrable Hamiltonian system with coexisting regular and chaotic regions. We show that the recurrence time entropy (RTE) is positively correlated to the largest Lyapunov exponent, with a high correlation coefficient. We obtain a multi-modal distribution of the finite-time RTE and find that each mode corresponds to the motion around islands of different hierarchical levels.

We investigate the basin of attraction properties and its boundaries for chimera states in a circulant network of Hénon maps. It is known that coexisting basins of attraction lead to a hysteretic behaviour in the diagrams of the density of states as a function of a varying parameter. Chimera states, for which coherent and incoherent domains occur simultaneously, emerge as a consequence of the coexistence of basin of attractions for each state. Consequently, the distribution of chimera states can remain invariant by a parameter change, and it can also suffer subtle changes when one of the basins ceases to exist. A similar phenomenon is observed when perturbations are applied in the initial conditions. By means of the uncertainty exponent, we characterise the basin boundaries between the coherent and chimera states, and between the incoherent and chimera states. This way, we show that the density of chimera states can be not only moderately sensitive but also highly sensitive to initial conditions. This chimera’s dilemma is a consequence of the fractal and riddled nature of the basin boundaries.

A chaotic saddle is a common nonattracting chaotic set well known for generating finite-time chaotic behavior in low and high-dimensional systems. In general, dynamical systems possessing chaotic saddles in their state-space exhibit irregular behavior with duration lengths following an exponential distribution. However, when these systems are coupled into networks the chaotic saddle plays a role in the long-term dynamics by trapping network trajectories for times that are indefinitely long. This process transforms the network’s high-dimensional state-space by creating an alternative persistent desynchronized state coexisting with the completely synchronized one. Such coexistence threatens the synchronized state with vulnerability to external perturbations. We demonstrate the onset of this phenomenon in complex networks of discrete-time units in which the synchronization manifold is perturbed either in the initial instant of time or in arbitrary states of its asymptotic dynamics. The role of topological asymmetries of Erdös–Rényi and Barabási–Albert graphs are investigated. Besides, the required coupling strength for the occurrence of trapping in the chaotic saddle is unveiled.