# gst: braids of three strands and geodesics shooting in SL_2(R).

<< ️(AA) describe in detail and provide computer code for constructing optimal geometric braids of three strands from algebraic data encoding the braiding pattern. >>

<< ️The exposition is complete and multi-pronged, aimed at a broad spectrum of readers. (Numerous figures are the backbone of the narrative and should be viewed in color.) >>

Jaroslaw (Jarek) Kwapisz. Braids of Three Strands and Geodesics Shooting in SL_2(R). arXiv: 2608.21484v1 [math.DG]. Aug 21, 2026.

https://arxiv.org/abs/2608.21484

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

Keywords: gst, perception, metamorphosis, transitions. 

# gst: the ultimate state of elastic turbulence.

<< The asymptotic properties, with vanishing viscosity, of inertial turbulence in Newtonian fluids have been of intense scrutiny over the years. Much less is known about elastic turbulence - a distinct spatio-temporally chaotic state exhibited by viscoelastic fluids. >>

<< In this work, using direct numerical simulations of turbulence in a triperiodic box, (AA) show that elastic turbulence achieves an ultimate state with increasing polymer relaxation time, i.e., the Deborah number, for two different constitutive models (Oldroyd-B and FENE-P). >>

<< In the limiting state, various bulk quantities - the fluid dissipation, the polymeric energy transfer, as well as the elastic stresses - are shown to become independent of the polymer relaxation time scale, notwithstanding the microstructure deformation scales distinctly depending on the model. >>

Piyush Garg, Marco Edoardo Rosti. The ultimate state of elastic turbulence. arXiv: 2608.26479v1 [physics.flu-dyn]. Aug 27, 2026.

https://arxiv.org/abs/2608.26479

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

Keywords: gst, elasticity, turbulence, chaos, vanishing viscosity, inertial turbulence, elastic turbulence, elastic stress, viscoelastic fluids, polymer relaxation.

# gst: self-organizing chimera states in adaptive networks.

<< ️(AA) propose a minimal adaptive network model with product-form coupling that reduces dimensionality while capturing key features of synaptic plasticity. The system spontaneously self-organizes into adaptive chimera states, where identical oscillators separate into synchronized and desynchronized groups through adaptive weight dynamics. >>

<< These adaptive chimeras organize into branches with fixed coherent cluster fraction and exhibit transitions between stationary, breathing, and chaotic collective dynamics, revealing a collective bifurcation structure. Crucially, the resulting attractor landscape is highly multistable: repeated cluster reorganizations generate distinct dynamical pathways that coexist within the same parameter regime and depend sensitively on initial conditions. >>

Felix Augustsson, Rok Cestnik, Matthias Wolfrum, et al. Self-organizing Chimera States in Adaptive Networks. arXiv: 2608.20517v1 [nlin.AO]. Aug 20, 2026.

https://arxiv.org/abs/2608.20517

Also: chimera, self-assembly, network, chaos, transition, brain, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, chimera, self-assembly, networks, transitions, brain, oscillators, adaptive chimera, synaptic plasticity, synchronized- desynchronized stationary states, breathing, chaotic collective dynamics, collective bifurcations, multistable attractors.

# 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.