The hidden complexity of a double-scroll attractor: Analytic proofs from a piecewise-smooth system.

Double-scroll attractors are one of the pillars of modern chaos theory. However, rigorous computer-free analysis of their existence and global structure is often elusive. Here, we address this fundamental problem by constructing an analytically tractable piecewise-smooth system with a double-scroll attractor.

Partial synchronization in the second-order Kuramoto model: An auxiliary system method.

Partial synchronization emerges in an oscillator network when the network splits into clusters of coherent and incoherent oscillators. Here, we analyze the stability of partial synchronization in the second-order finite-dimensional Kuramoto model of heterogeneous oscillators with inertia. Toward this goal, we develop an auxiliary system method that is based on the analysis of a two-dimensional piecewise-smooth system whose trajectories govern oscillating dynamics of phase differences between oscillators in the coherent cluster.

Sliding homoclinic bifurcations in a Lorenz-type system: Analytic proofs.

Non-smooth systems can generate dynamics and bifurcations that are drastically different from their smooth counterparts. In this paper, we study such homoclinic bifurcations in a piecewise-smooth analytically tractable Lorenz-type system that was recently introduced by Belykh et al. [Chaos 29, 103108 (2019)].

Ghost attractors in blinking Lorenz and Hindmarsh-Rose systems.

In this paper, we consider blinking systems, i.e., non-autonomous systems generated by randomly switching between several autonomous continuous time subsystems in each sequential fixed period of time.

When three is a crowd: Chaos from clusters of Kuramoto oscillators with inertia.

Modeling cooperative dynamics using networks of phase oscillators is common practice for a wide spectrum of biological and technological networks, ranging from neuronal populations to power grids. In this paper we study the emergence of stable clusters of synchrony with complex intercluster dynamics in a three-population network of identical Kuramoto oscillators with inertia. The populations have different sizes and can split into clusters where the oscillators synchronize within a cluster, but notably, there is a phase shift between the dynamics of the clusters.

A Lorenz-type attractor in a piecewise-smooth system: Rigorous results.

Chaotic attractors appear in various physical and biological models; however, rigorous proofs of their existence and bifurcations are rare. In this paper, we construct a simple piecewise-smooth model which switches between three three-dimensional linear systems that yield a singular hyperbolic attractor whose structure and bifurcations are similar to those of the celebrated Lorenz attractor. Due to integrability of the linear systems composing the model, we derive a Poincaré return map to rigorously prove the existence of the Lorenz-type attractor and explicitly characterize bifurcations that lead to its birth, structural changes, and disappearance.

Bistable gaits and wobbling induced by pedestrian-bridge interactions.

Several modern footbridges around the world have experienced large lateral vibrations during crowd loading events. The onset of large-amplitude bridge wobbling has generally been attributed to crowd synchrony; although, its role in the initiation of wobbling has been challenged. To study the contribution of a single pedestrian into overall, possibly unsynchronized, crowd dynamics, we use a bio-mechanically inspired inverted pendulum model of human balance and analyze its bi-directional interaction with a lively bridge.

Bistability of patterns of synchrony in Kuramoto oscillators with inertia.

We study the co-existence of stable patterns of synchrony in two coupled populations of identical Kuramoto oscillators with inertia. The two populations have different sizes and can split into two clusters where the oscillators synchronize within a cluster while there is a phase shift between the dynamics of the two clusters. Due to the presence of inertia, which increases the dimensionality of the oscillator dynamics, this phase shift can oscillate, inducing a breathing cluster pattern.

Cluster synchronization in oscillatory networks.

Synchronous behavior in networks of coupled oscillators is a commonly observed phenomenon attracting a growing interest in physics, biology, communication, and other fields of science and technology. Besides global synchronization, one can also observe splitting of the full network into several clusters of mutually synchronized oscillators. In this paper, we study the conditions for such cluster partitioning into ensembles for the case of identical chaotic systems.