# 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: 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: 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: 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: 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: 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: Resonancelike emergence of chaos in complex networks of damped-driven nonlinear systems.

AA << solve a critical outstanding problem in this multidisciplinary research field: the emergence and persistence of spatiotemporal chaos in complex networks of damped-driven nonlinear oscillators in the significant weak-coupling regime, while they exhibit regular behavior when uncoupled. >>

They << uncover and characterize the basic physical mechanisms concerning both heterogeneity-induced and impulse-induced emergence, enhancement, and suppression of chaos in starlike and scale-free networks of periodically driven, dissipative nonlinear oscillators. >>️

Ricardo Chacon, Pedro J. Martínez. Resonancelike emergence of chaos in complex networks of damped-driven nonlinear systems. Phys. Rev. E 110, 014209. Jul 19, 2024. 

https://journals.aps.org/pre/abstract/10.1103/PhysRevE.110.014209

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

Keywords: gst, network, resonance, chaos

# gst: implementation of a generalized intermittency scenario in the Rossler dynamical system.

<< The realization of novel scenario involving transitions between different types of chaotic attractors is investigated for the Rossler system. Characteristic features indicative of the presence of generalized intermittency scenario in this system are identified. The properties of "chaos-chaos" transitions following the generalized intermittency scenario are analyzed in detail based on phase-parametric characteristics, Lyapunov characteristic exponents, phase portraits, and Poincare sections. >>

O.O. Horchakov, A.Yu. Shvets. Implementation of a generalized intermittency scenario in the Rossler dynamical system. arXiv: 2511.03364v1 [nlin.CD]. Nov 5, 2025.

https://arxiv.org/abs/2511.03364

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

Keywords: gst, intermittency, attractors, chaos, transitions, chaos-chaos transitions.

# gst: spiral defect chaos with intermittency increases mean termination time

<< ️Cardiac models are examples of excitable systems and can support stable spiral waves. For certain parameter values, however, these spiral waves can become unstable, resulting in spiral defect chaos (SDC), characterized by the continuous creation and annihilation of spiral waves and thought to underlie atrial fibrillation. During SDC, the number of spiral waves fluctuates and drops to zero at termination. >>

<< ️In this work, (AA) demonstrate that varying a single parameter allows the system to transition from SDC to a single spiral wave, passing through an intermediate regime of intermittency. In such intermittent dynamics, intervals of SDC are sandwiched between non-SDC intervals during which the number of spiral waves remains small and constant. (They) quantify this intermittency and show that the mean termination time increases significantly as the control parameter approaches values for which a single spiral wave is stable. >>

<<  In addition, (AA) observe that quasistable spiral waves may intermittently persist in part of the computational domain, while the rest of the domain exhibits SDC. >> 

Mahesh Kumar Mulimani, Wouter-Jan Rappel. Spiral defect chaos with intermittency increases mean termination time. Phys. Rev. E 112, 034203. Sep 3, 2025.

https://journals.aps.org/pre/abstract/10.1103/bwpl-qqyl

arXiv: 2505.06427v1 [nlin.CD]. May 9,  2025

https://arxiv.org/abs/2505.06427

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

Keywords: gst, waves, spiral & scroll waves, intermittency, vortex, chaos, spiral defect chaos.

# gst: transition to chaos with conical billiards.

<< ️(AA) adapt ideas from geometrical optics and classical billiard dynamics to consider particle trajectories with constant velocity on a cone with specular reflections off an elliptical boundary formed by the intersection with a tilted plane, with tilt angle γ. >>

<< ️(They) explore the dynamics as a function of γ and the cone deficit angle χ that controls the sharpness of the apex, where a point source of positive Gaussian curvature is concentrated. >>

<< ️(AA) find regions of the (γ,χ) plane where, depending on the initial conditions, either (A) the trajectories sample the entire cone base and avoid the apex region; (B) sample only a portion of the base region while again avoiding the apex; or (C) sample the entire cone surface much more uniformly, suggestive of ergodicity. >>

<< ️The special case of an untilted cone displays only type A trajectories which form a ring caustic at the distance of closest approach to the apex. However, (They) observe an intricate transition to chaotic dynamics dominated by Type (C) trajectories for sufficiently large χ and γ. A Poincaré map that summarizes trajectories decomposed into the geodesic segments interrupted by specular reflections provides a powerful method for visualizing the transition to chaos. (AA) then analyze the similarities and differences of the path to chaos for conical billiards with other area-preserving conservative maps. >>

Lara Braverman, David R. Nelson. Transition to chaos with conical billiards. arXiv: 2508.02786v1 [nlin.CD]. Aug 4, 2025.

https://arxiv.org/abs/2508.02786

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

Keywords: gst, billiard, particles, transitions, chaos. 

# gst: apropos of the transition of order from chaos, a universal behavior near a critical point.

<< As the Reynolds number is increased, a laminar fluid flow becomes turbulent, and the range of time and length scales associated with the flow increases. Yet, in a turbulent reactive flow system, as we increase the Reynolds number, (AA) observe the emergence of a single dominant timescale in the acoustic pressure fluctuations, as indicated by its loss of multifractality. >>️

AA << study the evolution of short-time correlated dynamics between the acoustic field and the flame in the spatiotemporal domain of the system.   >>️

<< the susceptibility of the order parameter, correlation length, and correlation time diverge at a critical point between chaos and order. (AA) results show that the observed emergence of order from chaos is a continuous phase transition (..) the critical exponents characterizing this transition fall in the universality class of directed percolation. >>️

The << paper demonstrates how a real-world complex, nonequilibrium turbulent reactive flow system exhibits universal behavior near a critical point. >>️

Sivakumar Sudarsanan, Amitesh Roy, et al. Emergence of order from chaos through a continuous phase transition in a turbulent reactive flow system. Phys. Rev. E 109, 064214. Jun 20, 2024. 

https://journals.aps.org/pre/abstract/10.1103/PhysRevE.109.064214

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

Keywords: gst, order, chaos, transition 

# gst: labyrinth chaos: revisiting the elegant, chaotic and hyperchaotic walks

<< Labyrinth chaos was discovered by Otto Rossler and Rene' Thomas in their endeavour to identify the necessary mathematical conditions for the appearance of chaotic and hyperchaotic motion in continuous flows. Here, (AA) celebrate their discovery by considering a single labyrinth walks system and an array of coupled labyrinth chaos systems that exhibit complex, chaotic behaviour, reminiscent of chimera-like states, a peculiar synchronisation phenomenon. >>

 << As all Rossler’s pioneering contributions, labyrinth chaos still holds promise for very interesting further developments. Its simplicity and elegance, both in terms of symmetries, topology and feedback-circuit structure, makes it a good candidate to compare it with other nonlinear, cyclically coupled systems, such as the Arabesques, the Lotka-Voltera system and its variants, and the Arnold-Beltrami-Childress  model. >> 

Vasileios Basios, Chris G. Antonopoulos, Anouchah Latifi. Labyrinth chaos: Revisiting the elegant, chaotic and hyperchaotic walks. arXiv: 2011.11009v1. Nov 22, 2020.

https://arxiv.org/abs/2011.11009

# gst: apropos of transverse instabilities, from chimeras to extensive chaos

<< Populations of coupled oscillators can exhibit a wide range of complex dynamical behavior, from complete synchronization to chimera and chaotic states. We can thus expect complex dynamics to arise in networks of such populations. >>️

Here AA << analyze the dynamics of networks of populations of heterogeneous mean-field coupled Kuramoto-Sakaguchi oscillators, and show that the instability that leads to chimera states in a simple two-population model also leads to extensive chaos in large networks of coupled populations. >>️

Pol Floriach, Jordi Garcia-Ojalvo, Pau Clusella. From chimeras to extensive chaos in networks of heterogeneous Kuramoto oscillator populations. arXiv: 2407.20408v2 [nlin.CD]. Oct 11, 2024.

https://arxiv.org/abs/2407.20408

Also: chimera, instability, chaos, network, in 

https://www.inkgmr.net/kwrds.html 

Keywords: gst, chimera, instability, chaos, network

# game: hypothesis of a geometric design of chaotic attractors, on demand

AA << propose a method using reservoir computing to generate chaos with a desired shape by providing a periodic orbit as a template, called a skeleton. (They) exploit a bifurcation of the reservoir to intentionally induce unsuccessful training of the skeleton, revealing inherent chaos. The emergence of this untrained attractor, resulting from the interaction between the skeleton and the reservoir's intrinsic dynamics, offers a novel semi-supervised framework for designing chaos. >>️

Tempei Kabayama, Yasuo Kuniyoshi, et al. Designing Chaotic Attractors: A Semi-supervised Approach. arXiv: 2407.09545v1 [cs.NE]. Jun 27, 2024.

https://arxiv.org/abs/2407.09545

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

Keywords: game, chaos, chaotic attractors

# gst: transition to chaos with conical billiards.

<< ️In this paper, (AA) introduced and extensively investigated dynamical billiards on the surface of a cone with a tilted base. Upon varying the cone angle β, corresponding to a deficit angle 

2πχ = 2π(1 − sin(β)), and tilt angle γ, (They) identified three distinct types of trajectories with associated Poincaré map for conical billiards: rim, hourglass, and mixed. >>

<< ️Region I, where Poincaré space consists of rim, hourglass, and mixed trajectories; Region IIB, where Poincaré space consists of only hourglass and mixed trajectories; and Region IIA, in which (They) find choices of γ and χ for which almost all trajectories are strongly mixing. (..) (AA) also developed a scheme for identifying strongly mixing trajectories. >>

<< ️Furthermore, (They) were able to show that a dynamical billiard on a surface with exclusively convex and positive Gaussian curvature in three dimensions can still exhibit ergodic behavior in certain parameter regimes. >>

<< ️A particularly intriguing feature of this system is that by tuning χ and γ, nearly all points in (θ,ϕ) Poincaré space describing conical line segments in between bounces can be placed at the edge between chaotic and integrable dynamics. Thus this work highlights the potential of conical billiards as a model system for exploring intriguing problems inspired by neural networks at the “edge of chaos”. >>

Lara Braverman, David R. Nelson. Transition to chaos with conical billiards. arXiv: 2508.02786v1 [nlin.CD]. Aug 4, 2025. 

https://arxiv.org/abs/2508.02786

Phys. Rev. E 112, 044221. Oct 21, 2025.

https://journals.aps.org/pre/abstract/10.1103/18f7-sqhz

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

Keywords: gst, billiards, conical billiard, chaos, transitions, neural networks