# gst: shear v. stretching as drivers for mixing in 3-D porous media flows.

<< ️Solute mixing in porous media results from the interplay between molecular diffusion and the deformation of fluid parcels, which elongates solute elements and steepens concentration gradients in the pore space. >> 

<< ️In three-dimensional porous media, fluid deformation is asymptotically governed by exponential stretching, arising from the chaotic stretching and folding of fluid elements. However, early-time deformation may be dominated by shear, induced by no-slip boundary conditions at grain surfaces, leading to linear elongation. Yet, the persistence and relative contributions of linear shear and exponential stretching to fluid deformation and solute mixing remain poorly understood. >>

<< ️Here, (AA) address this question using numerical simulations of fluid deformation and mixing in body-centered cubic bead packs, where the rate of chaotic stretching can be varied with the direction of flow, while maintaining the same average shear rate. (They) quantify the mean deformation dynamics of elementary surfaces as a function of their initial orientation with respect to streamlines, and their consequences on mixing through Lagrangian methods. >>

<< ️(AA) show that surface elements with certain initial orientations relative to the streamline direction experience a transient phase of shear-dominated deformation at early times, leading to persistently larger deformation compared to those with initial orientations that yield purely exponential growth. By expressing the deformation components in streamline coordinates, (They) derive approximate analytical expressions linking the different components of fluid deformation to shear, helicity, and chaotic stretching. (They) discuss consequences for the mixing of solute sheets and blobs, highlighting generic behaviors as well as differences between these two representations. >>

Manuel Maeritz, Tanguy Le Borgne, Daniel R. Lester, et al. Shear versus stretching as drivers for mixing in three-dimensional porous media flows. Phys. Rev. Fluids 11, 094501. Sep 17, 2026.

https://journals.aps.org/prfluids/abstract/10.1103/4wyw-nkrb

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

Keywords: gst, chaotic advection, fluid deformation, flows in porous media, laminar flows, mixing enhancement, shear flows, chaotic stretching, porous media.

# 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: 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.

# gst: survival probability of particles inside the lemon billiard.

<< ️(AA) study the escape of particles in the lemon billiard, a two-parameter family of billiard systems defined by the intersection of two identical circles. Using numerical simulations, (They) explore how the survival probability depends on the position and size of the hole, as well as on the billiard shape parameter. >>

<< ️(AA) find that the survival probability exhibits a two-stage decay pattern: an initial exponential regime followed by a long-time power-law tail, a signature of the stickiness effect. (Their) results show that the short-time exponential decay rate follows a power-law dependence on the hole size, with different scaling exponents for holes placed in chaotic regions versus mixed phase-space regions. For holes located in mixed phase-space regions, the decay exponent of the long-time power-law tail remains approximately constant, while the amplitude follows a power-law scaling with hole size. >>

<< ️(They) also examine the dependence of short-time exponential decay rate on the billiard shape parameter and observe scaling behavior for small values of this parameter, which breaks down as the parameter increases. >>

Daniel Borin, Edson Denis Leonel, Diego Fregolent Mendes de Oliveira. Survival probability of particles inside the lemon billiard. Phys. Rev. E 114, 014224. Jul 28, 2026.

https://journals.aps.org/pre/abstract/10.1103/md5j-ggyd

arXiv: 2601.13000v1 [nlin.CD]. Jan 19, 2026.

https://arxiv.org/abs/2601.13000

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

Keywords: gst, billiard, particles, chaos, transitions, survival probability, two-stage decay pattern, stickiness effect, scaling behavior, scaling laws of complex systems. 

# gst: natural invariant measures for chaotic game dynamics; finding order in chaos.

<< ️(AA) study the long-term behavior of the Multiplicative Weights Update (MWU) algorithm in game settings where learning dynamics frequently fail to converge to Nash equilibria and instead exhibit Li-Yorke chaos. While such chaos precludes the prediction of specific long-term strategy profiles, it does not imply a lack of statistical structure. >>

<< ️(AA) demonstrate that natural invariant measures — a fundamental concept from ergodic theory — provide the rigorous framework necessary to find order within this chaos. Focusing on a two-strategy congestion game, (They) prove that these measures allow for a comprehensive statistical characterization of the dynamics. Crucially, (They) show that this framework extends beyond simple strategy frequencies to general observables, enabling the precise calculation of long-term time averages for broad classes of economic metrics — including payoffs, social cost, and regret — despite chaos. >>

<< ️(They) results reveal that this simple learning algorithm captures the full spectrum of behaviors found in one-dimensional dynamical systems, from unique or multiple absolutely continuous invariant measures to complex periodic attractors as well as coexisting chaotic and stable (periodic) behaviors. By bridging game theory and dynamical systems, (AA) show that statistical predictability is attainable even in the absence of pointwise convergence. >>

<< ️The intersection of game theory, dynamical systems, computer science, and statistics is rich with open questions, particularly when considering learning dynamics in games. >>

Jakub Bielawski, Thiparat Chotibut, Fryderyk Falniowski, et al. Natural Invariant Measures for Chaotic Game Dynamics: Finding Order in Chaos. arXiv: 2607.21805v1 [math.DS]. Jul 23, 2026.

https://arxiv.org/abs/2607.21805

Also: game, chaos, order, disorder,  in https://www.inkgmr.net/kwrds.html 

Keywords: gst, game, chaos, Multiplicative Weights Update (MWU) algorithm, statistical structure, statistical predictability, two-strategy congestion game, complex periodic attractors, coexisting chaotic and stable (periodic) behaviors.

# gst: emergence of minimal chimera in uncoupled oscillators under common frequency-modulated driving.

<< (AA) report the experimental realization of minimal chimera states in a system of three uncoupled oscillators driven solely by frequency-modulated forcing. Unlike conventional scenarios where chimera states emerge due to interactions among oscillators, here the coexistence of coherent and incoherent dynamics arises entirely from a common external modulation of a system parameter. By tuning the modulation amplitude and frequency, the system exhibits transitions between global synchronization, global incoherence, and minimal chimera states. >>

<< The stability of these regimes is quantified using the maximal Lyapunov exponent, while a synchronization order parameter is employed to characterize the degree of coherence. A systematic exploration of the parameter space reveals well-defined regions associated with distinct dynamical behaviors. To provide analytical understanding, (AA) employ a phase-reduction approach and derive the corresponding phase dynamics, which elucidate the mechanisms underlying phase locking and desynchronization. The robustness of the proposed mechanism is further demonstrated in a time-delayed chaotic system. >>

<< Finally, experimental results obtained from an electronic circuit realization confirm the emergence of minimal chimera states under frequency-modulated driving. These findings establish external modulation as a viable route to chimera formation without coupling, offering a new perspective on collective dynamics in driven nonlinear systems. >>

Debabrata Biswas, Tanmoy Banerjee. Emergence of minimal chimera in uncoupled oscillators under common frequency-modulated driving: theory and experiment. arXiv: 2607.25449v1 [nlin.CD]. Jul 28, 2026.

https://arxiv.org/abs/2607.25449

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

Keywords: gst, chimera, transition, chaos, three uncoupled oscillators, coherent- incoherent dynamics, frequency-modulated forcing, global synchronization, global incoherence, minimal chimera states.

# gst: dynamics of coupled stochastic van der Pol oscillators; bifurcations, synchronization and chaos.

<< ️This (AA) work presents a comprehensive analysis of coupled stochastic van der Pol oscillators, a paradigm for understanding synchronization, bifurcations, and chaos in nonlinear systems subject to random fluctuations. The system comprises two or more oscillators with nonlinear damping, linear diffusive coupling, and additive Gaussian white noise. >>

<< ️(AA) develop a unified framework that systematically connects global bifurcations, synchronization phenomena, and chaotic dynamics within a single coherent stochastic model. (They) explore the stochastic dynamics of coupled van der Pol oscillators by seamlessly blending theoretical principles with in-depth numerical simulations. This integrated approach forms a robust framework for analysis, with essential phenomena clearly depicted in the accompanying figures. (AA) then extend this framework to a comprehensive investigation of large networks, focusing on their continuum limit, emergent pattern formation, the role of noise, and the onset of collective chaos. >>

Shenglan Yuan, Xiang Zhou. Dynamics of Coupled Stochastic van der Pol Oscillators: Bifurcations, Synchronization and Chaos.  arXiv: 2606.31445v1 [nlin.CD]. Jun 30, 2026. 

https://arxiv.org/abs/2606.31445v1

Also: chaos, noise, random, disorder & fluctuations, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, chaos, noise, randomness, disorder, fluctuations, coupled stochastic van der Pol oscillators, synchronization, bifurcations, emergent pattern formation, collective chaos.

# brain: apropos of persistent exploratory predisposition, chaotic pigeons are helping redefine what we know about learning.

<< ️Pigeons seem to defy a century-old psychology law about how rewards and consequences help us learn. >>

<< ️New research suggests the birds themselves avoid stability in their decision-making, instead choosing to live “at the edge of chaos.” As model species for learning and behavior, these birds are helping researchers test a century-old law about how humans and other creatures learn. >>

<< ️When learning something new, people and animals alike tend to repeat behaviors that are rewarded. First proposed by Edward Thorndike in 1898, this principle is so well established in psychology that it's become known as the law of effect. But the law implies that beyond making a behavior more frequent, rewards also make it more consistent: reducing variability in the specific way behaviors are performed over time. >>

<< ️Edward A. Wasserman and his colleagues decided to put it to the test in pigeons—a species that has been integral to the study of learning at the university’s Comparative Cognition Laboratory for more than 50 years. And the study’s results, published in the Journal of Experimental Psychology: Animal Learning and Cognition, suggest these birds experience variability as the spice of life. >>

<< ️You could argue the birds are just utterly resistant to locking into anything stable. >> Edward A. Wasserman.

<< ️this paper leaves open many questions about the [neurological] mechanisms for future scientists to explore. >> Aaron Blaisdell.

K. R. Callaway, Sarah Lewin Frasier. Chaotic pigeons are helping redefine what we know about learning. SciAm. Jun 30, 2026. 

https://www.scientificamerican.com/article/chaotic-pigeons-are-helping-redefine-what-we-know-about-learning

Wasserman E.A., Orr O.R.P., Li S. Variability, stability, and the law of effect. J. of Exp. Psychol: Animal Learning and Cognition, 52(3), 129–138. Jul 2026. 

https://psycnet.apa.org/record/2027-47688-001

Also: brain, neuro, behav, curiosity, game, Nomads, in https://www.inkgmr.net/kwrds.html 

Keywords: brain, behavior, curiosity, games, Nomads.

# gst: uncertainty growth in stably stratified turbulence.

<< ️(AA) investigate uncertainty growth and chaotic dynamics in statistically steady, stably stratified three-dimensional turbulence. Using direct numerical simulations of the Boussinesq equations, (They) quantify the divergence of initially infinitesimal perturbations via twin simulations and decorrelator diagnostics. >>

<< ️At short times, perturbations exhibit exponential growth, allowing us to define a (largest) Lyapunov exponent. (AA) systematically examine how this exponent depends on stratification strength, quantified by the Brunt-Väisälä frequency and the Froude number, in a parameter regime relevant to oceanic flows. >>

<< (They) find that increasing stratification leads to a monotonic reduction of the Lyapunov exponent, indicating suppressed chaoticity. Despite this reduction, uncertainty growth retains the universal temporal sequence observed in homogeneous isotropic turbulence—initial decay, exponential growth, and saturation. The growth phase is characterized by self-similar decorrelator spectra, but exhibits strong anisotropy: uncertainty spreads much more slowly along the stratification direction than horizontally, with the disparity increasing with stratification strength. >>

<< ️An analysis of the decorrelator evolution equation reveals that the suppression of chaos arises primarily from strain-mediated alignment dynamics rather than direct buoyancy coupling. (AA) results provide a quantitative characterization of predictability and uncertainty growth in stratified turbulence and highlight the utility of decorrelator-based methods for anisotropic geophysical flows. >>

Mrinal Jyoti Powdel, Samriddhi Sankar Ray. Uncertainty growth in stably stratified turbulence. Phys. Rev. Fluids 11, 064615. Jun 23, 2026. 

https://journals.aps.org/prfluids/abstract/10.1103/xblr-vpl3

arXiv: 2512.05656v1 [physics.flu-dyn].

https://arxiv.org/abs/2512.05656

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

Keywords: gst, chaos, turbulence, transitions, uncertainty growth, three-dimensional turbulence, stratified turbulence, infinitesimal perturbations, stratification strength, anisotropy. 

# gst: describing a universal critical behavior in a transition from order to chaos.

<< ️(AA) present a comprehensive discussion of a transition from integrability to nonintegrability in an oval billiard with a static boundary. This transition is controlled by a deformation parameter 𝜀, which modifies the boundary shape from circular, corresponding to 𝜀=0 and an integrable dynamics, to oval for 𝜀≠0, where nonintegrability emerges. >>

<< ️The deformation of the circular billiard gives rise to a chaotic layer that develops along a well-defined stripe in phase space. By introducing a set of transformations that isolate this chaotic stripe, (They) characterize the diffusive spreading of ensembles of trajectories and identify an observable, 𝜔_(rms,sat), which plays the role of an order parameter for the transition. >>

<< For small deformations, the saturation value of the diffusion obeys the scaling law 𝜔_(rms,sat)∝𝜀^(˜𝛼), with a critical exponent ˜𝛼=0.507⁢(2), vanishing continuously as 𝜀→0. The associated susceptibility, 𝜒=𝑑⁢𝜔_(rms,sat)/𝑑⁢𝜀, diverges in the same limit, signaling the presence of critical behavior analogous to that observed in second-order (continuous) phase transitions in statistical mechanics. >>

Edson D. Leonel, Mayla A. M. de Almeida, Juan Pedro Tarigo, et al. Describing a universal critical behavior in a transition from order to chaos. Phys. Rev. E 113, 054220. May 28, 2026.

https://journals.aps.org/pre/abstract/10.1103/7231-j7zv

arXiv: 2602.17810v1 [nlin.CD]. Feb 19, 2026.

https://arxiv.org/abs/2602.17810

Also: billiard, transition, order, disorder, chaos, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, billiard, oval billiard, transitions, order, disorder, elementary excitations, small deformations, topological defects, criticality, chaotic stripes, chaos.

# gst: from chaos to synchrony in recurrent excitatory-inhibitory networks with target-specific inhibition.

<< ️Biological neural networks can operate in qualitatively distinct dynamical regimes, and transitions between these regimes are thought to underlie changes in computation and behavior. The seminal work of Sompolinsky, Crisanti, and Sommers (SCS) showed that random recurrent networks undergo a transition from quiescence to asynchronous chaos, establishing a paradigmatic link between random connectivity, dynamical instability, and internally generated fluctuations in neural circuits. >>

<< ️Here, (AA) extend this framework to two-population firing-rate networks with segregated excitatory and inhibitory neurons and target-specific inhibitory couplings that break excitation--inhibition balance. Using dynamical mean-field theory, (They) derive self-consistent equations for the macroscopic mean activities and autocorrelations, together with stability criteria distinguishing mean-driven and fluctuation-driven instabilities. (They) show that target-specific inhibition organizes the phase diagram into three qualitative classes: inhibition-dominated or strictly balanced networks display only quiescent activity and asynchronous chaos; excitation-dominated networks display persistent activity together with either synchronous chaos with non-vanishing mean activity or coherent oscillations, depending on the stability-matrix eigenvalues. >>

<< Crucially, coherent oscillations do not coexist with chaotic fluctuations around the periodic mean trajectory; rather, their onset suppresses the chaotic component, reminiscent of input-induced suppression of chaos. These results generalize SCS theory to recurrent networks with explicit excitatory--inhibitory structure and identify target-specific inhibition as a key control parameter for large-scale neural dynamics. >>

Carles Martorell, Rubén Calvo, Alessia Annibale, et al. From Chaos to Synchrony in Recurrent Excitatory-Inhibitory Networks with Target-Specific Inhibition. arXiv: 2605.14916v1 [cond-mat.dis-nn]. May 14, 2026.

https://arxiv.org/abs/2605.14916

Also: network, fluctuations, instability, transition, chaos, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, networks, fluctuations, instability, transitions, chaos, biological neural networks, random recurrent networks, asynchronous chaos, excitation--inhibition balance, target-specific inhibition.

# gst: critical parameters of an oval billiard with an elliptical component.

<< ️(AA) explore the critical parameters responsible for the transition from integrability to chaos in a family of billiards combining elliptical and oval deformations. Unlike standard oval billiards, where a known critical parameter governs the destruction of the last invariant curve, the introduction of an integrable elliptic component yields a second deformation axis. >>

<< (They) derive an analytical expression for the critical parameter in this combined system and validate it numerically using Slater's theorem, showing that increasing the elliptical component lowers the critical threshold for global chaos. >>

<< ️Moreover, (They) uncover a previously unexplored regime: when the two deformation components are in phase, the elliptic contribution progressively suppresses chaos, leading to the restoration of invariant curves and periodic orbits. A first-order analytical approximation confirms this behavior, supported by numerical validation. >>

<< ️(Their) results reveal how the interplay between distinct boundary deformations enriches phase-space organization and offers enhanced controllability of chaotic dynamics in billiard systems. >>

Anne Kétri P. da Fonseca, Joelson D. V. Hermes, Edson D. Leonel. Critical parameters of an oval billiard with an elliptical component. arXiv: 2605.00145v1 [nlin.CD]. Apr 30, 2026. 

https://arxiv.org/abs/2605.00145

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

Keywords: gst, billiard, transition, chaos, criticality, elliptical and oval deformations. 

# gst: chaotic billiard lasers.

<< ️This chapter provides an overview of chaotic billiard lasers as a prominent branch of quantum chaos. These lasers offer an ideal experimental platform for demonstrating the principles of quantum chaos within a physical system. >>

<< ️(AA) begin by introducing the fundamental principles of chaotic ray dynamics in optical microcavities, where the transition from regular to fully chaotic dynamics fundamentally alters the underlying wavefunctions and lasing properties. A central focus is placed on "chaos-assisted light emission," which serves as a practical manifestation of "chaos-assisted tunneling" -- a hallmark phenomenon in the study of quantum chaos. >>

<< ️(They) discuss both theoretical frameworks and experimental validations, demonstrating how chaotic orbits facilitate the coupling between evanescently localized modes and far-field emission. >>

<<️ Furthermore, exploring how the presence of a gain medium influences established results from quantum chaos research remains a fundamental and intriguing problem in physics. To address this, (They) establish a rigorous and comprehensive derivation of the Maxwell-Bloch equations for two-dimensional microcavity lasers, specifically examining their application to fully chaotic, stadium-shaped billiard lasers. >>

<< ️By bridging the gap between nonlinear lasing processes and chaotic wavefunctions, this chapter highlights the unique potential of chaotic billiards for controlling light-matter interactions and shaping the next generation of unconventional coherent light sources. >>

Takahisa Harayama. Chaotic Billiard Lasers. arXiv: 2604.23614v1 [quant-ph]. 26 Apr 26, 2026.

https://arxiv.org/abs/2604.23614

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

Keywords: gst, billiard, billiard laser, waves, transitions, chaos, quantum chaos, chaotic billiard, chaos-assisted light emission, chaos-assisted tunneling. 

# gst: chaotic ghosts in systems with parameter drift; delay and control critical transitions.

<< ️In dynamical systems with a time-dependent parameter, i.e., parameter drift, after crossing a saddle-node bifurcation, the so-called ghost state formed by the disappeared equilibria or periodic orbit can influence transient dynamics, causing a delayed transition. >>

<< ️This phenomenon has been investigated previously. However, the effect of chaotic ghosts on the critical transition in drifting systems has been less studied. In this paper, (AA) explore how chaotic ghosts and drifting rates influence critical transitions from the perspective of the ensemble. >> 

<< ️The (AA) results reveal the mechanism of the delayed transition related to chaos and how trajectories on the initial ensemble composed of a chaotic attractor transition to a qualitatively different object during the drift. In addition, (They) quantify the delayed transition and further find that the delay follows a power-law scaling with respect to the drifting rate. Finally, (AA) show that the critical transition is fully avoided as long as the reversal rate of the parameter exceeds a certain critical rate, even though the bifurcation point has been crossed. >>

Han Su, Denghui Li, Jicheng Duan, et al. Chaotic ghosts in systems with parameter drift: Delay and control critical transitions. Phys. Rev. E 113, 044207. April 13, 2026.

https://journals.aps.org/pre/abstract/10.1103/bdf8-wkpm

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

Keywords: gst, chaos, attractor, transitions, chaotic attractor transition, chaotic ghosts, criticality, critical transitions, bifurcation point, saddle-node bifurcation, ghost state, transient dynamics, delay, delayed transition, drifting rate.

# gst: quantum kicked top; a paradigmatic model

<< ️The quantum kicked top (QKT) is one of the most widely studied models in quantum chaos, providing a minimal yet powerful framework for exploring the relationship between classical nonlinear dynamics and quantum behavior. Unlike many chaotic systems with infinite-dimensional Hilbert spaces, the QKT possesses a finite-dimensional Hilbert space, making it analytically and numerically controllable while still showing a rich dynamical phenomena. >>

<< ️(AA) present a comprehensive introduction to the QKT as a paradigmatic model of quantum chaos. Starting from the classical kicked top, (They) derive the discrete nonlinear map governing the dynamics on the unit sphere and analyze its phase space structure through fixed points, stability analysis, bifurcations and Lyapunov exponents. (They) then discuss the role of symmetries, including rotational and time-reversal symmetry, and how their breaking modifies the dynamics. >>

<< ️By linking classical phase space structures with quantum dynamical indicators, the QKT provides a clear setting to investigate the emergence of chaotic behavior in the semiclassical limit. The chapter, therefore, highlights the quantum kicked top as a bridge between nonlinear classical dynamics, quantum chaos and modern quantum information science. >>

Avadhut V. Purohit, Udaysinh T. Bhosale. Quantum Kicked Top: A Paradigmatic Model. arXiv: 2604.12345v1 [quant-ph]. Apr 14, 2026.

https://arxiv.org/abs/2604.12345

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

Keywords: gst, chaos, entropy, quantum kicked top, quantum chaos, bifurcations, rotational symmetry, time-reversal symmetry, entanglement, qubits, quantum information, entanglement entropy.

# gst: chaos and quantum tunneling.

<< ️In generic Hamiltonian systems that are neither completely integrable nor fully chaotic, phase space consists of a mixture of regular and chaotic components. In classical dynamics, transitions between different invariant sets in phase space are strictly forbidden, and these sets act as dynamical barriers to one another. In quantum mechanics, in contrast, wave effects allow transitions through such dynamical barriers. This process, known as dynamical tunneling, refers to penetration through dynamical barriers in phase space and was first recognized in the early 1980s. Since then, various aspects of dynamical tunneling have been elucidated, significantly advancing our understanding of such a novel quantum phenomenon. >>

<< ️In this article, (AA) provide an overview of several phenomenological perspectives of dynamical tunneling, including chaos-assisted and resonance-assisted tunneling, and also introduce approaches based on classical mechanics extended into the complex domain. In particular, (They) seek to clarify what is meant by the common claim that "chaos leads to an enhancement of the tunneling probability", which is often made when dynamical tunneling is dressed. (They) discuss what regime this refers to and, if such an enhancement occurs, what its likely origin is. >>

Akira Shudo. Chaos and Quantum Tunneling. arXiv: 2604.12926v1 [nlin.CD]. Apr 14, 2026.

https://arxiv.org/abs/2604.12926

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

Keywords: gst, waves, chaos, transitions, dynamical tunneling, chaos-assisted tunneling, resonance-assisted tunneling.

# gst: apropos of escape, Stochastic Web Map; survival probability and escape frequency.

<< ️(AA) study transport and escape in the Stochastic Web Map (SWM), an area-preserving system with phase-space structure controlled by a symmetry parameter q and nonlinearity K. By analyzing the survival probability P_S(n) and escape frequency P_E(lnn), (They) show that in the chaotic regime escape dynamics is governed by a single time scale n_(typ) ∝ K^(−2)h^(2); here h is the size of the escape horizon. Deviations at large K and small h indicate a breakdown of the quasilinear approximation. Then, upon rescaling the time by n_(typ), escape statistics becomes universal, independent of q. These results demonstrate that escape is controlled by global transport rather than symmetry. >>

K. B. Hidalgo-Castro, J. A. Méndez-Bermúdez, Edson D. Leonel. Stochastic Web Map: Survival probability and escape frequency. arXiv: 2603.20888v1 [nlin.CD]. Mar 21, 2026.

https://arxiv.org/abs/2603.20888

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

Keywords: gst, escape, escape horizon, chaos, global transport.

# gst: phase-space organization of the elastic pendulum; chaotic fraction, energy exchanges, and the order-chaos-order transition.

<< ️(AA) study the phase-space organization of the planar elastic pendulum as a function of its two dimensionless control parameters: the reduced energy R and the squared frequency ratio µ. By randomly sampling the isoenergetic volume to classify trajectories as oscillatory, rotational, or chaotic across the (µ,R) parameter plane, (They) obtain a global portrait of the coexistence and competition between dynamical regimes. >>

<< ️The chaotic fraction is not uniformly distributed across the parameter plane but concentrates in a well-defined central cloud whose ridge follows a linear relation in the (µ,R) plane and whose maximum does not exceed 70% of the available phase space. The order-chaos-order transition is not a global property of the parameter plane but occurs specifically in the central region surrounding this cloud: along paths that traverse it, oscillatory orbits progressively give way to chaotic trajectories, which in turn yield to rotational orbits as the energy grows, revealing a clear sequential mechanism underlying the transition. >> 

<< ️The onset of rotational motion is gradual rather than sharp, reflecting a strong dependence on initial conditions. By decomposing the total energy into spring-like, pendulum-like, and coupling contributions, (They) establish a direct correspondence between the coupling power and the abundance of chaotic trajectories, showing that enhanced inter-mode energy exchange is a reliable indicator of dynamical complexity. >>

Juan P. Tarigo, Cecilia Stari, Edson D. Leonel, et al. Phase-space organization of the elastic pendulum: chaotic fraction, energy exchanges, and the order-chaos-order transition. arXiv: 2604.01503v1 [nlin.CD]. Apr 2, 2026.

https://arxiv.org/abs/2604.01503

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

Keywords: gst, pendulum, planar elastic pendulum, rotation, rotational motion, chaos, transitions, order-chaos-order transition. 

# gst: localization of information driven by stochastic resetting.

<< ️The dynamics of extended many-body systems are generically chaotic. Classically, a hallmark of chaos is the exponential sensitivity to initial conditions captured by positive Lyapunov exponents. Supplementing chaotic dynamics with stochastic resetting drives a sharp dynamical phase transition: (AA) show that the Lyapunov spectrum, i.e., the complete set of Lyapunov exponents, abruptly collapses to zero above a critical resetting rate. >>

<< ️At criticality, (They) find a sudden loss of analyticity of the velocity-dependent Lyapunov exponent, which (They) relate to the transition from ballistic scrambling of information to an arrested regime where information becomes exponentially localized over a characteristic length diverging at criticality with an exponent 𝜈=1/2 and a dynamical exponent 𝑧=2. (They) illustrate (Their) analytical results on generic chaotic dynamics by numerical simulations of coupled map lattices. >>

Camille Aron, Manas Kulkarni. Localization of information driven by stochastic resetting. Phys. Rev. E 113, L022101. Feb 23, 2026.

https://journals.aps.org/pre/abstract/10.1103/pkgb-mp8s

arXiv:2510.07394v2 [cond-mat.stat-mech]. Feb 24, 2026.

https://arxiv.org/abs/2510.07394

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

Keywords: gst, chaos, transition, collapse, randomness, stochasticity, stochastic resetting, phase transition, criticality, critical resetting rate, ballistic scrambling of information.