Blue sky catastrophe in the phenomenological model of neuron-astrocyte interaction.

We study a bifurcation scenario that corresponds to the transition from bursting activity to spiking in a phenomenological model of neuron-astrocyte interaction in neuronal populations. In order to do this, we numerically obtain one-dimensional Poincaré return map and highlight its bifurcation structure using an analytically constructed piecewise smooth model map. This map reveals the existence of a cascade of period-adding bifurcations, leading to a bursting-spiking transition via blue sky catastrophe.

Dynamics of large oscillator populations with random interactions.

We explore large populations of phase oscillators interacting via random coupling functions. Two types of coupling terms, the Kuramoto-Daido coupling and the Winfree coupling, are considered. Under the assumption of statistical independence of the phases and the couplings, we derive reduced averaged equations with effective non-random coupling terms.

Dynamics of Oscillator Populations Globally Coupled with Distributed Phase Shifts.

We consider a population of globally coupled oscillators in which phase shifts in the coupling are random. We show that in the maximally disordered case, where the pairwise shifts are independent identically distributed random variables, the dynamics of a large population reduces to one without randomness in the shifts but with an effective coupling function, which is a convolution of the original coupling function with the distribution of the phase shifts. This result is valid for noisy oscillators and/or in the presence of a distribution of natural frequencies.

Cyclops States in Repulsive Kuramoto Networks: The Role of Higher-Order Coupling.

Repulsive oscillator networks can exhibit multiple cooperative rhythms, including chimera and cluster splay states. Yet, understanding which rhythm prevails remains challenging. Here, we address this fundamental question in the context of Kuramoto-Sakaguchi networks of rotators with higher-order Fourier modes in the coupling.

Stability of rotatory solitary states in Kuramoto networks with inertia.

Solitary states emerge in oscillator networks when one oscillator separates from the fully synchronized cluster and oscillates with a different frequency. Such chimera-type patterns with an incoherent state formed by a single oscillator were observed in various oscillator networks; however, there is still a lack of understanding of how such states can stably appear. Here, we study the stability of solitary states in Kuramoto networks of identical two-dimensional phase oscillators with inertia and a phase-lagged coupling.

Appearance of chaos and hyperchaos in evolving pendulum network.

The study of deterministic chaos continues to be one of the important problems in the field of nonlinear dynamics. Interest in the study of chaos exists both in low-dimensional dynamical systems and in large ensembles of coupled oscillators. In this paper, we study the emergence of chaos in chains of locally coupled identical pendulums with constant torque.

Locking and regularization of chimeras by periodic forcing.

We study how a chimera state in a one-dimensional medium of nonlocally coupled oscillators responds to a homogeneous in space periodic in time external force. On a macroscopic level, where a chimera can be considered as an oscillating object, forcing leads to entrainment of the chimera's basic frequency inside an Arnold tongue. On a mesoscopic level, where a chimera can be viewed as an inhomogeneous, stationary, or nonstationary pattern, strong forcing can lead to regularization of an unstationary chimera.

Variety of rotation modes in a small chain of coupled pendulums.

This article studies the rotational dynamics of three identical coupled pendulums. There exist two parameter areas where the in-phase rotational motion is unstable and out-of-phase rotations are realized. Asymptotic theory is developed that allows us to analytically identify borders of instability areas of in-phase rotation motion.