# gst: transient and steady-state chaos in dissipative quantum systems.

<< Dissipative quantum chaos plays a central role in the characterization and control of information scrambling, non-unitary evolution, and thermalization, but it still lacks a precise definition. >>

AA << properly restore the quantum-classical correspondence through a dynamical approach based on entanglement entropy and out-of-time-order correlators (OTOCs), which reveal signatures of chaos beyond spectral statistics. Focusing on the open anisotropic Dicke model, (They) identify two distinct regimes: transient chaos, marked by rapid early-time growth of entanglement and OTOCs followed by low saturation values, and steady-state chaos, characterized by high long-time values. >>

AA << introduce a random matrix toy model and show that Ginibre spectral statistics signals short-time chaos rather than steady-state chaos. (Their) results establish entanglement dynamics and OTOCs as reliable diagnostics of dissipative quantum chaos across different timescales. >>

Debabrata Mondal, Lea F. Santos, S. Sinha. Transient and steady-state chaos in dissipative quantum systems. arXiv: 2506.05475v1 [quant-ph]. Jun 5, 2025. 

https://arxiv.org/abs/2506.05475

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

Keywords: gst, information scrambling, entropy, chaos, transient chaos, steady-state chaos.

# 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: apropos of transitions, a perpetual dance between states of meta-stability and chaos (in brain).

img: https://wp.technologyreview.com/wp-content/uploads/2023/02/Radio_image_15.jpeg?fit=2160,1214

<< Hello! Today: new research is shining a light on how our brains flit between states of stability and chaos, depending on what we’re doing. >>

<< Our brains exist in a state somewhere between stability and chaos as they help us make sense of the world, according to recordings of brain activity taken from volunteers over the course of a week. >>

<< As we go from reading a book to chatting with a friend, for example, our brains shift from one semi-stable state to another—but only after chaotically zipping through multiple other states in a pattern that looks completely random. >>

<< Understanding how our brains restore some degree of stability after chaos could help us work out how to treat disorders at either end of this spectrum. Too much chaos is probably what happens when a person has a seizure, whereas too much stability might leave a person comatose. >>

Jessica Hamzelou. Neuroscientists listened in on people’s brains for a week. They found order and chaos. Rhiannon Williams. MIT Download. Feb 8, 2023.

https://mailchi.mp/technologyreview.com/mass-market-military-drones-have-changed-the-way-wars-are-fought-839014

<< The team (Avniel Ghuman, Maxwell Wang, et al.) found some surprising patterns in brain activity over the course of the week. Specific brain networks seemed to communicate with each other in what looked like a “dance,” with one region appearing to “listen” while the other “spoke,” say the researchers, who presented their findings at the Society for Neuroscience annual meeting in San Diego last year. >>

Jessica Hamzelou. MIT Tech Rev. Feb 7, 2023. 

https://www.technologyreview.com/2023/02/07/1067951/brains-week-order-chaos

Also 

keyword 'danza' in Notes

(quasi-stochastic poetry)

https://inkpi.blogspot.com/search?q=danza

keyword 'dance' in FonT

https://flashontrack.blogspot.com/search?q=dance

keyword 'cervello' | 'brain' in Notes

(quasi-stochastic poetry)

https://inkpi.blogspot.com/search?q=cervello

https://inkpi.blogspot.com/search?q=brain

keyword 'brain' in FonT

https://flashontrack.blogspot.com/search?q=brain

keyword 'chaos' | 'chaotic' in Font

https://flashontrack.blogspot.com/search?q=chaos

https://flashontrack.blogspot.com/search?q=chaotic

keyword 'caos' | 'caotico' in Notes (quasi-stochastic poetry)

https://inkpi.blogspot.com/search?q=caos

https://inkpi.blogspot.com/search?q=caotico

<< Amico, qualunque  cosa suonerai . . . >>  Jelly Roll Morton. cit.: 2113 - soniche a ramulo. Jan 28, 2007

http://inkpi.blogspot.it/2007/01/2113-soniche-ramulo.html

https://inkpi.blogspot.com/search?q=jelly+roll

Keywords: gst, brain, transition, chaos, dance

# 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: 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: nonreciprocity induced spatiotemporal chaos: reactive vs dissipative routes.

<< ️Nonreciprocal interactions fundamentally alter the collective dynamics of nonlinear oscillator networks. Here (AA) investigate Stuart-Landau oscillators on a ring with nonreciprocal reactive or dissipative couplings combined with Kerr-type or dissipative nonlinearities. >>

<< ️Through numerical simulations and linear analysis, (They) uncover two distinct and universal pathways by which enhanced nonreciprocity drives spatiotemporal chaos. Nonreciprocal reactive coupling with Kerr-type nonlinearity amplifies instabilities through growth-rate variations, while nonreciprocal dissipative coupling with Kerr-type nonlinearity broadens eigenfrequency distributions and destroys coherence, which, upon nonlinear saturation, evolve into fully developed chaos. In contrast, dissipative nonlinearities universally suppress chaos, enforcing bounded periodic states. >>

<< ️(AA) findings establish a minimal yet general framework that goes beyond case-specific models and demonstrate that nonreciprocity provides a universal organizing principle for the onset and control of spatiotemporal chaos in oscillator networks and related complex systems. >>

Jung-Wan Ryu. Nonreciprocity induced spatiotemporal chaos: Reactive vs dissipative routes. arXiv: 2509.20992v1 [nlin.CD]. Sep 25, 2025

https://arxiv.org/abs/2509.20992

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

Keywords: gst, networks, instability, chaos, nonreciprocity, nonreciprocal interactions, nonreciprocal reactive-- dissipative couplings.

# gst: energy transport and chaos in a one-dimensional disordered nonlinear stub lattice

<< ️(AA) investigate energy propagation in a one-dimensional stub lattice in the presence of both disorder and nonlinearity. In the periodic case, the stub lattice hosts two dispersive bands separated by a flat band; however, (They) show that sufficiently strong disorder fills all intermediate band gaps. By mapping the two-dimensional parameter space of disorder and nonlinearity, (AA) identify three distinct dynamical regimes (weak chaos, strong chaos, and self-trapping) through numerical simulations of initially localized wave packets. >>

<< ️When disorder is strong enough to close the frequency gaps, the results closely resemble those obtained in the one-dimensional disordered discrete nonlinear Schrödinger equation and Klein-Gordon lattice model. In particular, subdiffusive spreading is observed in both the weak and strong chaos regimes, with the second moment m_2 of the norm distribution scaling as m_2 ∝ t^0.33 and m_2 ∝ t^0.5, respectively. The system’s chaotic behavior follows a similar trend, with the finite-time maximum Lyapunov exponent Λ decaying as Λ ∝ t^−0.25 and Λ ∝ t^−0.3. For moderate disorder strengths, i.e., near the point of gap closing, (They) find that the presence of small frequency gaps does not exert any noticeable influence on the spreading behavior. >>

<< ️(AA) findings extend the characterization of nonlinear disordered lattices in both weak and strong chaos regimes to other network geometries, such as the stub lattice, which serves as a representative flat-band system. >>

Su Ho Cheong, Arnold Ngapasare, et al. Energy transport and chaos in a one-dimensional disordered nonlinear stub lattice. arXiv: 2511.04159v1 [nlin.CD].  Nov 6, 2025.

https://arxiv.org/abs/2511.04159

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

Keywords: gst, networks, waves, disorder, chaos, stub lattice, subdiffusive spreading.

# gst: protected chaos in a topological lattice.

<< The erratic nature of chaotic behavior is thought to erode the stability of periodic behavior, including topological oscillations. However, (AA) discover that in the presence of chaos, non-trivial topology not only endures but also provides robust protection to chaotic dynamics within a topological lattice hosting non-linear oscillators. >>

<< Despite the difficulty in defining topological invariants in non-linear settings, non-trivial topological robustness still persists in the parametric state of chaotic boundary oscillations. (AA) demonstrate this interplay between chaos and topology by incorporating chaotic Chua's circuits into a topological Su-Schrieffer-Heeger (SSH) circuit. >>

<< By extrapolating from the linear limit to deep into the non-linear regime, (AA) find that distinctive correlations in the bulk and edge scroll dynamics effectively capture the topological origin of the protected chaos. (Their)  findings suggest that topologically protected chaos can be robustly achieved across a broad spectrum of periodically-driven systems, thereby offering new avenues for the design of resilient and adaptable non-linear networks. >>️

Haydar Sahin, Hakan Akgün, et al. Protected chaos in a topological lattice. arXiv: 2411.07522v1 [cond-mat.mes-hall]. Nov 12, 2024.

https://arxiv.org/abs/2411.07522

Also: chaos, random, instability, transition, network, ai (artificial intell), in https://www.inkgmr.net/kwrds.html 

Keywords: gst, chaos, random,  instability, transition, network, AI, Artificial Intelligence

# gst: apropos of ab.normal criticalities, a hypothetical scenario of non-normal route to chaos.

<< ️Deterministic chaos is commonly associated with spectral criticality: exponential sensitivity is expected when Jacobian eigenvalues exceed unity in parts of the attractor, producing the local expansion that offsets contraction elsewhere. (AA) show that this paradigm is incomplete in dimensions d>1.  >>

<< ️(They) construct a bounded 3D dynamical system whose Jacobian is pointwise spectrally contracting, namely all instantaneous eigenvalues remain strictly inside the stability region, yet the system develops a positive maximal Lyapunov exponent and undergoes a transition to chaos as a non-normality index increases at fixed spectral radius. The mechanism relies on the repeated regeneration of transient non-normal amplification through endogenous switching that reinjects trajectories into amplifying non-orthogonal directions. >>

<< ️Although demonstrated here for a discrete-time map, the mechanism is geometric and applies more broadly to deterministic dynamical systems. These results show that chaos can emerge without spectral criticality and identify non-normality as an independent route to deterministic chaos. >>

D. Sornette, V.R. Saiprasad, V. Troude. Non-Normal Route to Chaos. arXiv: 2603.08191v1 [nlin.CD]. Mar 9, 2026.

https://arxiv.org/abs/2603.08191

Also:  Virgile Troude, Sandro Claudio Lera, Ke Wu, Didier Sornette. Illusions of Criticality: Crises Without Tipping Points. arXiv: 2412.01833v5 [nlin.CD]. Oct 3, 2025. https://arxiv.org/abs/2412.01833

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

Keywords: gst, chaos, criticality, transitions, non-normality, transient non-normality, reinjection via endogenous switch.

# 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: a note on spinning billiards and chaos

AA << investigate the impact of internal degrees of freedom - specifically spin - on the classical dynamics of billiard systems. While traditional studies model billiards as point particles undergoing specular reflection, (AA) extend the paradigm by incorporating finite-size effects and angular momentum, introducing a dimensionless spin parameter that characterizes the moment of inertia. Using numerical simulations across circular, rectangular, stadium, and Sinai geometries, (AA) analyze the resulting trajectories and quantify chaos via the leading Lyapunov exponent. >>

<< Strikingly, (They) find that spin regularizes the dynamics even in geometries that are classically chaotic: for a wide range of α, the Lyapunov exponent vanishes at late times in the stadium and Sinai tables, signaling suppression of chaos. This effect is corroborated by phase space analysis showing non-exponential divergence of nearby trajectories. >>

AA << results suggest that internal structure can qualitatively alter the dynamical landscape of a system, potentially serving as a mechanism for chaos suppression in broader contexts. >>

Jacob S. Lund, Jeff Murugan, Jonathan P. Shock. A Note on Spinning Billiards and Chaos. arXiv: 2505.15335v1 [nlin.CD]. May 21, 2025.

https://arxiv.org/abs/2505.15335

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

Keywords: gst, billiard, spinning billiards, chaos.

# gst: chaos and regularity in an anisotropic soft squircle billiard.

<< ️A hard-wall billiard is a mathematical model describing the confinement of a free particle that collides specularly and instantaneously with boundaries and discontinuities. Soft billiards are a generalization that includes a smooth boundary whose dynamics are governed by Hamiltonian equations and overcome overly simplistic representations. >>

<< ️(AA) study the dynamical features of an anisotropic soft-wall squircle billiard. This curve is a geometric figure that seamlessly blends the angularity of a square with the smooth curves of a circle. (They) characterize the billiard's emerging trajectories, exhibiting the onset of chaos and its alternation with regularity in the parameter space. (They) characterize the transition to chaos and the stabilization of the dynamics by revealing the nonlinearity of the parameters (squareness, ellipticity, and hardness) via the computation of Poincaré surfaces of section and the Lyapunov exponent across the parameter space. (They) expect (Their) work to introduce a valuable tool to increase understanding of the onset of chaos in soft billiards. >>

A. González Andrade, H. N. Núñez-Yépez, M. A. Bastarrachea-Magnani. Chaos and regularity in an anisotropic soft squircle billiard. Phys. Rev. E 112, 064213. Dec 22, 2025.

https://journals.aps.org/pre/abstract/10.1103/4xhm-r64f

arXiv: 2504.20270v2 [nlin.CD]. 28 Apr 2025

https://arxiv.org/abs/2504.20270

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

Keywords: gst, chaos, billiard, hard-wall billiards, soft billiards, particles, transitions.

# gst: a scenario in which System Theory meets Poetry, bird's-eye vistas into a primitive chaos

<< The notion of primitive chaos was proposed [J. Phys. Soc. Jpn. 79, 15002 (2010)] as a notion closely related to the fundamental problems of physics itself such as determinism, causality, free will, predictability, and irreversibility. In this letter, (AA) introduce the notion of bird's-eye view into the primitive chaos, and (they) find a new hierarchic structure of the primitive chaos. This means that if we find a chaos in a real phenomenon or a computer simulation, behind it, we can clearly realize the possibility of tremendous varieties of chaos in the hierarchic structure unless we can see them visually. >>

<< This fact provides a totally new method of viewing our world. >>️️

Yoshihito Ogasawara. Bird's-Eye View of Primitive Chaos. arXiv:2105.04796v2 [nlin.CD]. May 17, 2021. 

https://arxiv.org/abs/2105.04796

Also

Ludwig von Bertalanffy  (gst)  

https://en.m.wikipedia.org/wiki/Ludwig_von_Bertalanffy

keyword 'caos' | 'caotico' in Notes (quasi-stochastic poetry)

https://inkpi.blogspot.com/search?q=caos

https://inkpi.blogspot.com/search?q=caotico

# gst: overcoming overly simplistic representations, chaos and regularity in an anisotropic soft squircle billiard.

<< A hard-wall billiard is a mathematical model describing the confinement of a free particle that collides specularly and instantaneously with boundaries and discontinuities. >>

<< Soft billiards are a generalization that includes a smooth boundary whose dynamics are governed by Hamiltonian equations and overcome overly simplistic representations. >>

AA << study the dynamical features of an anisotropic soft-wall squircle billiard. This curve is a geometric figure that seamlessly blends the angularity of a square with the smooth curves of a circle. (AA) characterize the billiard's emerging trajectories, exhibiting the onset of chaos and its alternation with regularity in the parameter space. (They) characterize the transition to chaos and the stabilization of the dynamics by revealing the nonlinearity of the parameters (squarness, ellipticity, and hardness) via the computation of Poincaré surfaces of section and the Lyapunov exponent across the parameter space. >>

AA << expect (Their) work to introduce a valuable tool to increase understanding of the onset of chaos in soft billiards. >>

A. González-Andrade, H. N. Núñez-Yépez, M. A. Bastarrachea-Magnani. Chaos and Regularity in an Anisotropic Soft Squircle Billiard. arXiv: 2504.20270v1 [nlin.CD]. Apr 28, 2025.

https://arxiv.org/abs/2504.20270

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

Keywords: gst, billiard, soft billiard, soft-wall squircle billiard, particles, smooth boundary,  specular collisions, transitions, chaos

# 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: 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.

# gst: discontinuous transition to chaos in a canonical random neural network

AA << study a paradigmatic random recurrent neural network introduced by Sompolinsky, Crisanti, and Sommers (SCS). In the infinite size limit, this system exhibits a direct transition from a homogeneous rest state to chaotic behavior, with the Lyapunov exponent gradually increasing from zero. (AA)  generalize the SCS model considering odd saturating nonlinear transfer functions, beyond the usual choice 𝜙⁡(𝑥)=tanh⁡𝑥. A discontinuous transition to chaos occurs whenever the slope of 𝜙 at 0 is a local minimum [i.e., for 𝜙′′′⁢(0)>0]. Chaos appears out of the blue, by an attractor-repeller fold. Accordingly, the Lyapunov exponent stays away from zero at the birth of chaos. >>

In the figure 7 << the pink square is located at the doubly degenerate point (𝑔,𝜀)=(1,1/3). >>️️

Diego Pazó. Discontinuous transition to chaos in a canonical random neural network. Phys. Rev. E 110, 014201. July 1, 2024.

https://journals.aps.org/pre/abstract/10.1103/PhysRevE.110.014201

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

Keywords: gst, chaos, random, network, transition, neuro

# gst: apropos of Parrondo paradox, controlling quantum chaos via Parrondo strategies on noisy intermediate-scale quantum hardware

<< ️Advancements in noisy intermediate-scale quantum (NISQ) computing are steadily pushing these systems toward outperforming classical supercomputers on specific well-defined computational tasks. In this work (AA) explore and control quantum chaos in NISQ systems using discrete-time quantum walks (DTQWs) on cyclic graphs. To efficiently implement quantum walks on NISQ hardware, (They) employ the quantum Fourier transform to diagonalize the conditional shift operator, optimizing circuit depth and fidelity. >>

<< ️(AA) experimentally realize the transition from quantum chaos to order via DTQW dynamics on both odd and even cyclic graphs, specifically 3- and 4-cycle graphs, using the counterintuitive Parrondo paradox strategy across three different NISQ devices. >>

<< ️While the 4-cycle graphs exhibit high-fidelity quantum evolution, the 3-cycle implementation shows significant fidelity improvement when augmented with dynamical decoupling pulses. (Their) results demonstrate a practical approach to probing and harnessing controlled chaotic dynamics on real quantum hardware, laying the groundwork for future quantum algorithms and cryptographic protocols based on quantum walks. >>

Aditi Rath, Dinesh Kumar Panda, Colin Benjamin. Controlling quantum chaos via Parrondo strategies on noisy intermediate-scale quantum hardware. Phys. Rev. E 112, 054219. Nov 18, 2025.

https://journals.aps.org/pre/abstract/10.1103/m89r-2dy5

arXiv: 2506.11225v2 [quant-ph]. Nov 4, 2025.

https://arxiv.org/abs/2506.11225

Also: parrondo, noise, walk, walking, order, chaos, transition, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, parrondo, noise, walk, walking, quantum walk, order, chaos, quantum chaos, transition, dynamical decoupling pulses, cryptography.

# gst: period-doubling route to chaos in viscoelastic flows

<< Polymer solutions can develop chaotic flows, even at low inertia. This purely elastic turbulence is well studied, but little is known about the transition to chaos. In two-dimensional (2D) channel flow and parallel shear flow, traveling wave solutions involving coherent structures are present for sufficiently large fluid elasticity. >>

AA << numerically study 2D periodic parallel shear flow in viscoelastic fluids, and (They) show that these traveling waves become oscillatory and undergo a series of period-doubling bifurcations en-route to chaos. >>

Jeffrey Nichols, Robert D. Guy, Becca Thomases. Period-doubling route to chaos in viscoelastic Kolmogorov flow. Phys. Rev. Fluids 10, L041301. Apr 17, 2025.

https://journals.aps.org/prfluids/abstract/10.1103/PhysRevFluids.10.L041301

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

Keywords: gst, chaos, waves, traveling waves, elasticity, viscoelastic fluids, turbulence, elastic turbulence, period-doubling bifurcations, transitions

# 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.