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In this work, we study the phenomenon of disordered quenching in arrays of coupled Bautin oscillators, which are the normal form for bifurcation in the vicinity of the equilibrium point when the first Lyapunov coefficient vanishes and the second one is nonzero. For particular parameter values, the Bautin oscillator is in a bistable regime with two attractors—the equilibrium and the limit cycle—whose basins are separated by the unstable limit cycle. We consider arrays of coupled Bautin oscillators and study how they become quenched with increasing coupling strength. We analytically show the existence and stability of the dynamical regimes with amplitude disorder in a ring of coupled Bautin oscillators with identical natural frequencies. Next, we numerically provide evidence that disordered oscillation quenching holds for rings as well as chains with nonidentical natural frequencies and study the characteristics of this effect.

We study a new type of attractor, the so-called reversible core, which is a mathematical image of mixed dynamics, in a strongly dissipative time-irreversible system of two adaptively coupled phase oscillators. The existence of mixed dynamics in this system was proved in our previous article [A. A. Emelianova and V. I. Nekorkin, Chaos 29, 111102 (2019)]. In this paper, we attempt to identify the dynamic mechanisms underlying the existence of mixed dynamics. We give the region of the existence of mixed dynamics on the parameter plane and demonstrate in what way, when a type of attractor changes, its main characteristics, such as its fractal dimension and the sum of Lyapunov exponents, transform. We demonstrate that when mixed dynamics appear in the system, the average frequencies of the oscillations in forward and reverse time begin to almost coincide, and its spectra gradually approach each other with an increase in the parameter responsible for the presence of mixed dynamics.

We consider a case study of perturbing a system with a boundary crisis of a chaotic attractor by periodic forcing. In the static case, the system exhibits persistent chaos below the critical value of the control parameter but transient chaos above the critical value. We discuss what happens to the system and particularly to the transient chaotic dynamics if the control parameter periodically oscillates. We find a non-exponential decaying behavior of the survival probability function, study the impact of the forcing frequency and amplitude on the escape rate, analyze the phase-space image of the observed dynamics, and investigate the influence of initial conditions.

Pulse-mediated interactions are common in networks of different nature. Here we develop a general framework for simulation of networks with pulse delayed coupling. We introduce the discrete map governing the dynamics of such networks and describe the computation algorithm for its numerical simulation.

Effects of the slowly varying control parameters on bifurcations of the equilibria and limit cycles have been previously studied in detail. In this paper, the concept of dynamic bifurcations is extended to chaotic phenomena. We consider this problem for a Lorenz-type map. As the control parameter passes through a critical value, the dynamic boundary crisis of a chaotic attractor takes place. We discover and analyze the effects of delayed exit from the chaotic region and non-exponential decay of the number of surviving trajectory points. The property of the delay increase with increasing rate of the control parameter change has also been demonstrated and explained.

We have developed a new approach for the description of sequential dynamics of excitatory neural networks. Our approach is based on the dynamics of synapses possessing the short-term plasticity property. We suggest a model of such synapses in the form of a second-order system of nonlinear ODEs. In the framework of the model two types of responses are realized-the fast and the slow ones. Under some relations between their timescales a cellular automaton (CA) on the graph of connections is constructed. Such a CA has only a finite number of attractors and all of them are periodic orbits. The attractors of the CA determine the regimes of sequential dynamics of the original neural network, i.e., itineraries along the network and the times of successive firing of neurons in the form of bunches of spikes. We illustrate our approach on the example of a Morris-Lecar neural network.