# gst: hypothesis of stable rotating vortex clusters in three-dimensional quantum droplets

<< ️(AA) predict a new type of stable three-dimensional (3D) vortex quantum droplets in binary Bose-Einstein condensate arranged into ring clusters that per-sistently rotate in the external potential. In contrast to clusters composed from localized quantum droplets, the states introduced here represent interacting vortex lines with identical topological charges nested in common 3D envelope and existing in much broader parameter range in comparison with local-ized quantum droplets. >>

<< ️The intricate interplay between mean-field nonlinearity of Bose-Einstein condensate, quantum fluctuations described by Lee-Huang-Yang correction, Coriolis force arising due to rotation, and external potential leads to substantial variations of cluster shape upon increase of its rotation frequency. >>

<< ️Rotating vortex clusters bifurcate from single-vortex quantum droplets and exist only above critical value of the rotation frequency that decreases with increase of chemical potential and depends on the number of vortex lines in the cluster. The radius of the cluster decreases with increase of rotation frequency and weakly varies with chemical potential, which determines mostly localization of the individual vortices in cluster and overall width of its envelope. >>

<< ️Vortex droplet clusters are very robust objects existing in stable form in wide intervals of the number of particles for any number (odd or even) of vortex lines forming them. (AA) results may open the route to observation of stable arrays of vortex lines of different configurations per-forming regular collective motion in condensate. >>

Liangwei Dong, Yaroslav V. Kartashov. Stable rotating vortex clusters in three-dimensional quantum droplets. arXiv: 2608.24036v1 [nlin.PS]. Aug 25, 2026.

https://arxiv.org/abs/2608.24036

Also: Vortex solitons in twisted circular waveguide arrays. Ago 22, 2022. https://flashontrack.blogspot.com/2022/08/gst-vortex-solitons-in-twisted-circular.html

Also: vortex, drop, droplet, droploid, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, vortex, drops, droplets.

# gst: non-normal route to chaos

<< Deterministic chaos is usually associated with local spectral expansion: Jacobian eigenvalues are expected to exceed unity somewhere on the attractor. (AA) show that this view is incomplete in dimensions 𝑑>1. For non-normal Jacobians, pointwise spectral stability can suggest everywhere local contraction, while nonorthogonal eigenvectors still allow transient singular-vector amplification. >>

<< (AA) construct four low-dimensional deterministic maps realizing this mechanism: partition-reinjected, phase-prescribed, feedback-driven, and affine-reinjected non-normal routes to chaos. In all cases, the sampled one-step Jacobian remains spectrally stable at every point on the attractor away from switching boundaries, with the eigenvalues of the common planar core fixed inside the unit disk. Nevertheless, increasing non-normality raises the maximal Lyapunov exponent from negative to positive values, corresponding to sustained asymptotic chaos, not transient chaos. Across the four classes, the common signature is spectral radius 𝜌max traj<1, singular value 𝜎max traj>1, maximal Lyapunov exponent 𝜆1>0, and an increase of attractor dimension. >>

<< These examples identify non-normality and recurrent reinjection of transiently amplified directions as a deterministic route to chaos distinct from eigenvalue instability. >>

D. Sornette, V.R. Saiprasad, V. Troude. Non-normal route to chaos. Phys. Rev. E 114, 024222. Aug 27, 2026.

https://journals.aps.org/pre/abstract/10.1103/z9qv-51yj

arXiv: 2603.08191v2 [nlin.CD]. 

https://arxiv.org/abs/2603.08191

Also: attractor, chaos, transition, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, attractor, chaos, transitions

# gst: foliated by invariant quadrics, outer contact billiards.

<< ️(AA) introduce outer contact billiards as an odd dimensional counterpart to outer symplectic billiards. Until now, outer billiards have only been considered in even dimensional symplectic vector spaces. By projectivizing outer symplectic billiards, (They) obtain outer contact billiards, where the affine midpoint condition descends to its projective analog, namely harmonic conjugation. (They) prove that outer contact billiards generate contactomorphisms. >>

<< ️For quadratic surfaces in RP^3, (AA) show that the correspondence is completely integrable: its domain is foliated by invariant quadrics and, on each leaf, the dynamics is determined by the iteration of an explicit linear transformation. (They) also establish two rigidity results for periodic trajectories: outer contact billiards admit no 3-periodic orbits and, among quadratic tables, only one admits 4-periodic orbits. >>

Ana Chavez Caliz, Connor Jackman. Outer Contact Billiards. arXiv: 2608.19393v1 [math.DS]. Aug 19, 2026.

https://arxiv.org/abs/2608.19393

Also: billiard, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, billiard, outer contact billiards, symplectic billiards, harmonic conjugation, contactomorphisms, foliated domain. 

# gst: dissipative self-assembly of colloidal suspensions.

<< Suspensions of paramagnetic colloids exhibit kinetic arrest in strong magnetic fields. Through a dissipative process of toggling the field on and off, suspensions self-assemble into dense and dynamic steady-state phases. >>

<< (AA) construct a phase diagram using 𝑘-means clustering analysis based on domain elongation, α-shapes, contour metrics, and the degree of phase separation. (They) identify six characteristic structural regimes: a structureless phase, an arrested structure, sheets, ribbons, a spiky phase, and a transient fluid-fluid regime. >>

<< (AA) further report the distribution and alignment of domains and the generality of the results. (They) model self-assembled domain shapes using an equilibrium mean-field magnetostatic energy calculation, which predicts the surprising emergence of highly anisotropic structures driven by the sample's confinement. >>

Jason Conradt, Eric M. Furst. Dissipative self-assembly of colloidal suspensions. Phys. Rev. E 114, 025416. Aug 21, 2026.

https://journals.aps.org/pre/abstract/10.1103/8h2w-vhcn

arXiv: 2603.14637v1 [cond-mat.soft]. 

https://arxiv.org/abs/2603.14637

Also: dissipation, self-assembly, colloids, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, dissipation, self-assembly, colloids, anisotropic structures, steady-state phases. 

# gst: skimming transition in flexible granular sweeping.

<< ️A flexible body placed in a steady flow bends to reduce drag. This self-streamlining is a hallmark of fluid-structure interaction (FSI). Granular-structure interaction is equally ubiquitous in nature. However, it remains poorly understood. >>

<< ️Thus, (AA) investigate inertial granular-structure interaction (IGSI). Specifically, ejection induced by a flexible plate sweeping a granular bed is experimentally examined. (They) find that a faster sweep results in less ejection, particularly for a flexible plate. >> 

<< ️To understand the underlying physics of this behavior, a dimensionless number Sk is introduced as the ratio of the plate elastic timescale to the sweep timescale. At Sk≲1, the plate deflection follows the self-streamlining law of FSI and induces substantial ejection. At Sk≳1, on the other hand, the plate skims the bed and the mass of ejected grains decreases sharply. Sk organizes IGSI as the granular counterpart of FSI. >>

Y. Ochi, H. Katsuragi. Skimming transition in flexible granular sweeping. arXiv: 2608.15069v1 [cond-mat.soft]. Aug 15, 2026.

https://arxiv.org/abs/2608.15069

Also: grain, transition, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, grain, transitions, fluid-structure interaction (FSI), inertial granular-structure interaction (IGSI)

# behav: amoeboid swimming of active vesicles.

<< ️(AA) investigate the shape dynamics and migration of weakly deflated active vesicles driven by processes acting either directly in the membrane or transmitted by the cytoskeleton. For a force-free vesicle, local membrane incompressibility suppresses rigid-body translation, so that migration arises from time-dependent shape deformations. Assuming small excess area enables a systematic analysis of the coupled deformation and migration dynamics in free space, i.e. in the absence of substrate adhesion or confinement. >>

<< ️Depending on the strength and frequency of the activity, the vesicle exhibits several dynamical regimes, including synchronized oscillations, quasiperiodic shape changes, transitions between non-propelling and propelling states, and intermittent motion. >>

Reiner Kree, Annette Zippelius. Amoeboid swimming of active vesicles. arXiv: 2607.14714v1 [cond-mat.soft]. Jul 16, 2026.

https://arxiv.org/abs/2607.14714

Also: behav, transition, swim, microswimmers, in https://www.inkgmr.net/kwrds.html 

Keywords: behavior, transitions, 

swim, vesicles, weakly deflated active vesicles, migration, time-dependent shape deformations, synchronized oscillations, quasiperiodic shape changes, transitions between non-propelling, propelling states, intermittent motion. 

# gst: flip rate prediction in the double pendulum.

<< ️The intermediate-energy double pendulum is a prototypical chaotic system. Despite its irregular motion, it exhibits recurrent flips - events in which one of the arms passes over the top. >>

<< ️(AA) develop a statistical prediction for the mean flip rate in terms of phase space flux. Applying this flux-based approach to the equal-mass, equal arm ("egalitarian") double pendulum, (They) find excellent agreement between the statistical prediction and numerical simulations: ensemble-averaged flip rates agree at about the 1% level, while even the statistics of individual chaotic trajectories agree at the few-percent level, quantifying the validity of the ergodic approximation. This agreement holds for flips of either arm and over a broad range of energies. This flux-based approach is closely related to that used for the egalitarian three-body system. >>

<< ️A central role is played by the saddle orbits: periodic orbits that tend to the stable saddle eigenmode as the energy approaches the saddle energy from above and approximately follow the ridge of the potential at higher energies. These orbits provide a natural dividing surface for defining flips while avoiding recrossings. Indeed, the resulting distribution of crossing times exhibits a distinct gap. >>

<< ️(AA) also present two alternative simplified dividing surfaces, one of which is based on an accurate analytic approximation to the saddle orbits. >>

Peleg Haham, Barak Kol. Flip rate prediction in the double pendulum. arXiv: 2608.20276v1 [nlin.CD]. Aug 20, 2026.

https://arxiv.org/abs/2608.20276

Also: pendulum, chaos, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, pendulum, double pendulum, chaos, recurrent flips, egalitarian three-body system, saddle orbits, crossing times.