Cyclops states - three-cluster configurations consisting of two synchronous groups and a solitary oscillator - dominate in ensembles of phase oscillators with inertia and multiple coupling harmonics [Phys. Rev. E 109, 054202 (2024)]. In this work, for the first time, we systematically study nonstationary cyclops states that preserve the three-cluster structure: breathing and rotobreathing cyclops states. We identify two scenarios for their destabilization: period doubling, leading to quasicyclops states while preserving frequency synchronization within the clusters, and the destruction of one or two clusters, resulting in the emergence of switching cyclops or multicluster states. We show that breathing and rotobreathing cyclops states occupy vast parameter regions of the second coupling harmonic and are key elements of the dynamics. The results are important for predicting and controlling complex collective states in ensembles with higher-order interaction harmonics of various natures.

Swarmalators are a class of coupled oscillators that simultaneously synchronize in both space and phase, providing a minimal model for systems ranging from biological microswimmers to robotic swarms. Time delay is ubiquitous in such systems, arising from finite signal propagation speeds and sensory processing lags, yet its structural form, whether symmetric or asymmetric, has received little attention. Here, we study a one-dimensional swarmalator model with asymmetric time delay, in which the delay enters only the self-interaction terms of the spatial and phase dynamics, breaking the symmetry assumed in prior work. We identify various collective states such as async, static phase wave, static {\pi}, and active {\pi}, and derive analytical stability boundaries for each as a function of the coupling parameters and delay. Our analysis reveals that the asymmetric delay structure fundamentally reshapes the collective phase diagram: in particular, for the asymmetric delay models, increasing the delay systematically expands the active {\pi} state at the expense of other ordered states, in contrast to the symmetric delay model, which more strongly promotes the presence of unsteady states that are generally not well ordered. By providing closed-form stability conditions validated against numerical simulations, our work establishes that the internal structure of the delay, not merely its magnitude, is a decisive factor in determining the emergent collective behavior of swarmalator populations.

Billiards tables - a minimal model for particles moving in a confined region - are known to present classical (and quantum) different features according to their shape, ranging from strongly chaotic to integrable dynamics. Here we consider the role of a stochastic perturbation of the elastic reflection law, and show that while chaotic billiards maintain their key statistical feature, the behaviour for integrable billiard tables is completely different: it can be linked, for tiny perturbations, to Evans stochastic billiard, where at each collision the reflected angle is a uniformly distributed stochastic variable on $(-\pi/2,\pi/2$). The resulting spatial stationary measure has peculiar aspects, like being typically non uniform along the boundary, differently from any chaotic billiard table.

Properties of spatially dependent relaxation oscillations near a SNIPER bifurcation are described. A SNIPER bifurcation creates a large-amplitude long-period periodic orbit via the annihilation of a pair of fixed points in a saddle-node bifurcation. We show that in spatially extended media, this orbit may undergo a long-wavelength instability, leading to spatially modulated oscillations that persist on both sides of the SNIPER. The oscillations take different forms depending on the system: a chimera state in a theta-reaction-diffusion model, and chaotic spiking in an activator-inhibitor-substrate model. The results are expected to have applications in a number of physical systems exhibiting SNIPER bifurcations, ranging from models of the nervous system through chemical reactions to nonlinear optics.

We experimentally generate sine-Gordon-like solitons in a spin-1 spinor Bose-Einstein condensate (BEC) utilizing a robust and reproducible local phase-imprinting scheme. We find that the soliton velocity can be tuned by the effective quadratic Zeeman shift. This enables the investigation of controlled soliton interactions, in which we observe the characteristic elastic collision behavior of the integrable sine-Gordon model. The spatial displacement -- the so-called phase shift -- between incoming and outgoing solitons, the signature of their pairwise interaction, is found to be in quantitative agreement with numerical spin-1 simulations within the error bars. These results establish spinor BECs as a highly controllable experimental platform for studying aspects of the dynamics of sine-Gordon-like models.

We present the first part of an efficient framework for nonlinear beam dynamics, termed Approximate Invariant Analysis (AIA). The framework is based on the construction of approximate invariants~[Y.~Li, D.~Xu, and Y.~Hao, Phys.\ Rev.\ Accel.\ Beams \textbf{28}, 074001 (2025)] and on the extraction of the betatron frequency with the geometric foundations of Poincaré rotation number~[S.~Nagaitsev and T.~Zolkin, Phys.\ Rev.\ Accel.\ Beams \textbf{23}, 054001 (2020)]. The method is demonstrated using the National Synchrotron Light Source~II (NSLS-II) storage ring as an illustrative example.

Ball lightning is one of the most mysterious atmospheric phenomena, whose long lifetime, penetrative ability, and stability are difficult to explain with traditional physical models. This paper proposes a novel theoretical framework, interpreting ball lightning as a projection of a high-dimensional topological soliton into three-dimensional space. Its essence is described by a nonlinear Schrödinger equation with attractive interaction, protected by a non-zero topological charge. Through numerical simulation of the three-dimensional Gross-Pitaevskii equation, we verify the core predictions of this model: in a Bose-Einstein condensate with attractive interactions, solitons carrying topological charge exhibit: (1)long-lived stability (topological charge conserved under perturbations); (2)low transmission probability (due to minimal overlap integral resulting from orthogonality with the ground state wavefunction); (3)energy and size scales consistent with actual observations. Theoretical analysis indicates that the soliton lifetime is governed by the system's decoherence rate, providing a natural explanation for the observed second-scale lifetimes. This work not only offers a self-consistent physical explanation for ball lightning but also provides a concrete scheme for the experimental preparation and observation of three-dimensional topological solitons.