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We considered a network consisting of two populations of phase oscillators, the interaction of which is determined by different rules for the coupling adaptation. The introduction of various adaptation rules leads to the suppression of splay states and the emergence of each population complex non-stationary behavior called transient circulant clusters. In such states, each population contains a pair of anti-phase clusters whose size and composition slowly change over time as a result of successive transitions of oscillators between clusters. We show that an increase in the mismatch of the adaptation rules makes it possible to stop the process of rearrangement of clusters in one or both populations of the network. Transitions to such modes are always preceded by the appearance of solitary states in one of the populations.

We report on the phenomenon of the emergence of mixed dynamics in a system of two adaptively coupled phase oscillators under the action of a harmonic external force. We show that in the case of mixed dynamics, oscillations in forward and reverse time become similar, especially at some specific frequencies of the external force. We demonstrate that the mixed dynamics prevents forced synchronization of a chaotic attractor. We also show that if an external force is applied to a reversible core formed in an autonomous case, the fractal dimension of the reversible core decreases. In addition, with increasing amplitude of the external force, the average distance between the chaotic attractor and the chaotic repeller on the global Poincare secant decreases almost to zero. Therefore, at the maximum intersection, we see a trajectory belonging approximately to a reversible core in the numerical simulation.

We report on the phenomenon of intersection of a chaotic attractor and a chaotic repeller in a system of two adaptively coupled phase oscillators. This is a feature of the presence of the so-called mixed dynamics, which is a new type of chaos characterized by the fundamental inseparability of conservative and dissipative behavior. The considered system is the first example of a time-irreversible system in which this type of dynamics is observed. We show that a crucial factor in this effect is the detuning of the natural frequencies of phase oscillators.

In this paper, we study collective dynamics of the network of rate neurons which constitute a central element of a reservoir computing system. The main objective of the paper is to identify the dynamic behaviors inside the reservoir underlying the performance of basic machine learning tasks, such as generating patterns with specified characteristics. We build a reservoir computing system which includes a reservoir—a network of interacting rate neurons—and an output element that generates a target signal. We study individual activities of interacting rate neurons, while implementing the task and analyze the impact of the dynamic parameter—a time constant—on the quality of implementation.

In this work, we consider two-layer multiplex networks of coupled Stuart-Landau oscillators. The first layer contains oscillators with amplitude heterogeneity and all-to-all adaptive links, while the second layer contains identical oscillators all-to-all coupled by links with constant weights. The links between different layers are adaptive and organized in a one-to-one manner. We study the evolution of one-layer and two-layer networks depending on intra- and interlayer coupling strengths and show hierarchical transitions between oscillatory and quenched regimes.

In this work, a new type of relaxation systems is considered. Their prominent feature is that they comprise two distinct epochs, one is slow regular motion and another is fast chaotic motion. Unlike traditionally studied slow-fast systems that have smooth manifolds of slow motions in the phase space and fast trajectories between them, in this new type one observes, apart the same geometric objects, areas of transient chaos. Alternating periods of slow regular motions and fast chaotic ones as well as transitions between them result in a specific chaotic attractor with chaos on a fast time scale. We formulate basic properties of such attractors in the framework of discrete-time systems and consider several examples. Finally, we provide an important application of such systems, the neuronal electrical activity in the form of chaotic spike-burst oscillations.