# gst: apropos of weakly coupled oscillators, a new approach to depict about their spontaneous stochastic activities

<<  Many systems in physics, chemistry and biology exhibit oscillations with a pronounced random component. Such stochastic oscillations can emerge via different mechanisms, for example linear dynamics of a stable focus with fluctuations, limit-cycle systems perturbed by noise, or excitable systems in which random inputs lead to a train of pulses. Despite their diverse origins, the phenomenology of random oscillations can be strikingly similar. >>

<< Here (AA) introduce a nonlinear transformation of stochastic oscillators to a new complex-valued function Q∗1(x) that greatly simplifies and unifies the mathematical description of the oscillator's spontaneous activity, its response to an external time-dependent perturbation, and the correlation statistics of different oscillators that are weakly coupled. >>

AA << approach makes qualitatively different stochastic oscillators comparable, provides simple characteristics for the coherence of the random oscillation, and gives a framework for the description of weakly coupled oscillators. >>️

Alberto Pérez-Cervera, Boris Gutkin, et al. A Universal Description of Stochastic Oscillators. arXiv: 2303.03198v1 [nlin.AO]. Feb 27, 2023. 

https://arxiv.org/abs/2303.03198

Also

'oscillations' in FonT

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

Keywords: gst, oscillations, random oscillations, weakly coupled oscillators

# brain: rhythm-regulation and plasticity through oscillations

<< During inattentive wakefulness and non-rapid eye movement (NREM) sleep, the neocortex and thalamus cooperatively engage in rhythmic activities >>

<< thalamic oscillations of low-vigilance states have a plasticity function that (..) can shape ongoing oscillations during inattention and NREM sleep and may potentially reconfigure thalamic networks for faithful information processing during attentive wakefulness >>

Vincenzo Crunelli, Magor L. Lorincz, et al. Dual function of thalamic low-vigilance state oscillations: rhythm-regulation and plasticity. Nature Reviews Neurosci doi: 10.1038/nrn.2017.151. Jan 11, 2018

https://www.nature.com/articles/nrn.2017.151

# gst: apropos of oscillations, viscous streaming around an immersed microfeature (e.g. a bubble)

<< Viscous streaming refers to the rectified, steady flows that emerge when a liquid oscillates around an immersed microfeature, typically a solid body or a bubble. The ability of such features to locally concentrate stresses produces strong inertial effects to which both fluid and immersed particles respond within short length (O(100) microns) and time (milliseconds) scales, rendering viscous streaming arguably the most efficient mechanism to exploit inertia at the microscale. >>️

(AA) << demonstrate that a multi-curvature approach in viscous streaming dramatically extends the range of accessible flow topologies. (They) show that numerically predicted, but never experimentally observed, streaming flows can be physically reproduced, computationally engineered, and in turn used to enhance particle manipulation, filtering and separation in compact, robust, tunable and inexpensive devices. >>️

Yashraj Bhosale, Giridar Vishwanathan, et al. Multi-curvature viscous streaming: flow topology and particle manipulation. arXiv: 2111.07184v1 [physics.flu-dyn]. Nov 13, 2021.

https://arxiv.org/abs/2111.07184

keywords: gst, viscosity, viscous streaming, bubble, oscillations, liquid oscillations, flanking vortex, particle manipulation 

# gst: the complex world of Bacteria; collective oscillations via discontinuous transitions, chemical quorum sensing and brainlike bursts of electricity to communicate.

<< collective oscillations in cell populations can emerge suddenly with nonzero amplitude via a discontinuous transition. >>

Rosa Martinez-Corral, Jintao Liu, et al. Bistable emergence of oscillations in growing Bacillus subtilis biofilms. PNAS Sep 4, 2018;  115 (36) E8333-E8340.  doi: 10.1073/pnas.1805004115

https://www.pnas.org/content/115/36/E8333 

<< As in all communities, cohabiting bacteria need ways to exchange messages. Biologists have known for decades that bacteria can use chemical cues to coordinate their behavior. >>

AA << are now finding that bacteria in biofilms can also talk to one another electrically. Biofilms appear to use electrically charged particles to organize and synchronize activities across large expanses. This electrical exchange has proved so powerful that biofilms even use it to recruit new bacteria from their surroundings, and to negotiate with neighboring biofilms for their mutual well-being. >>

<< Step by step we find that all the things we think bacteria don’t do, they actually do, [..] It’s displacing us from our pedestal. >> Ned Wingreen.

Gabriel Popkin. Bacteria Use Brainlike Bursts of Electricity to Communicate. With electrical signals, cells can organize themselves into complex societies and negotiate with other colonies. Sep 5, 2017

https://twitter.com/QuantaMagazine/status/1071417372500783104

https://www.quantamagazine.org/bacteria-use-brainlike-bursts-of-electricity-to-communicate-20170905/

Also

Gurol M. Suel publ.

http://labs.biology.ucsd.edu/suel/index.html

FonT

intrigante qui l'idea di una Intelligenza Artificiale che, dalle relativamente comode e veloci osservazioni sull'evoluzione di colonie batteriche sotto differenti condizioni, sia in grado di circoscrivere pattern comportamentali di interesse per la modellizzazione (e previsione)  di comunita' altre ...

# gst: reconfiguration and oscillations of sheets subject to vortex

<< The dynamics of a thin low-density polyethylene sheet subject to periodic forcing due to Bénard-Kàrmàn vortices in a long narrow water channel is investigated here. In particular, the time-averaged sheet deflection and its oscillation amplitude are considered. >>

<< The former is first illustrated to be well-approximated by the static equilibrium between the buoyancy force, the elastic restoring force, and the profile drag based on the depth-averaged water speed. >>

AA << observations also indicate that the presence of upstream vortices hinder the overall reconfiguration effect, well-known in an otherwise steady flow. For the sheet-tip oscillations, a simple model based on a torsional-spring-mounted flat plate correctly captures the measured tip amplitude δb over a wide range of sheet physical properties and flow conditions. >>

<< Furthermore, a rich phenomenology of structural dynamics including vortex-forced-vibration, lock-in with the sheet natural frequency, and flow-induced vibration due to the sheet wake, multiple-frequency, and modal response is reported. >>

️

J. John Soundar Jerome, Yohann Bachelier, et al. Reconfiguration and oscillations of a vertical, cantilevered sheet subject to vortex shedding behind a cylinder. Phys. Rev. Fluids 8, 093801. Sep 15, 2023. 

https://journals.aps.org/prfluids/abstract/10.1103/PhysRevFluids.8.093801

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

Keywords: gst, sheet, vortex, Benard-Karman vortices

# brain: arterial pressure pulsations could modulate neuronal activity.

<< Spontaneous slow oscillations have been described in the rat olfactory bulb local field potential, even in the absence of respiration. What is the origin of these oscillations? >>

AA << discovered a subpopulation of neurons within the olfactory bulb that can directly sense cardiovascular pressure pulsations (..). The modulation of their excitability is transduced by mechanosensitive ion channels. >>

<< Thus, there exists a fast pathway for the interoception of heartbeat whereby arterial pressure pulsations within the brain modulate neuronal activity. >> Peter Stern. ️

Luna Jammal Salameh, Sebastian H. Bitzenhofer, et al. Blood pressure pulsations modulate central neuronal activity via mechanosensitive ion channels. Science. Vol 383, Issue 6682. Feb 2, 2024. 

https://www.science.org/doi/10.1126/science.adk8511

img: https://www.science.org/cms/10.1126/science.adk8511/asset/76a1eaf8-61d3-4233-bc2a-be2276f367cd/assets/images/large/science.adk8511-fa.jpg

Also: brain, pnei, soliton, in https://www.inkgmr.net/kwrds.html  

Keywords: brain, pnei, olfactory bulb, spontaneous slow oscillations, wave, soliton

# geo: apropos of oscillations, a 'true polar wander' (TPW); Earth tip on its side 84 Ma ago.

<< It has been debated for the past few decades whether the outer, solid shell of the Earth can wobble about, or even tip over relative to the spin axis. Such a shift of Earth is called "true polar wander," (TPW) but the evidence for this process has been contentious. New research (..) provides some of the most convincing evidence to date that such planetary tipping has indeed occurred in Earth's past. >>️

<< The Earth is a stratified ball, with a solid metal inner core, a liquid metal outer core, and a solid mantle and overriding crust at the surface which we live on. All of this is spinning like a top, once per day. Because the Earth's outer core is liquid, the solid mantle and crust are able to slide around on top of it.  >>

️

Did the Earth tip on its side 84 million years ago? Tokyo Institute of Technology. Oct 18, 2021. 

https://phys.org/news/2021-10-earth-side-million-years.html

<< a new high-resolution palaeomagnetic record from two overlapping stratigraphic sections in Italy (..) provides evidence for a ~12° TPW oscillation from 86 to 78 Ma. This observation represents the most recent large-scale TPW documented and challenges the notion that the spin axis has been largely stable over the past 100 million years. >>️

Mitchell, R.N., Thissen, C.J.,  et al. A Late Cretaceous true polar wander oscillation. Nat Commun 12, 3629. doi: 10.1038/ s41467-021-23803-8. June 15, 2021. 

https://www.nature.com/articles/s41467-021-23803-8

keywords: gst, geo, geodynamics, geophysics, palaeomagnetism, oscillations, wobble, true polar wander, tpw

# gst: anomalous oscillation modes of (superfluid) pendant droplets; horizontal translation on a flat surface, bouncing off at the corner and vertical oscillations at the edge.

<< Droplets should exhibit various dynamical phenomena when adhered to a surface; not all of them are realized in classical fluids. Visualization of superfluid  4^He (helium-4) pendant droplets revealed that the droplets were horizontally translated on a flat surface, bouncing off at the corner, known as the Noether mode that reflects the translation symmetry. >>️

<< The droplets exhibited another mode in vertical oscillations with high amplitude that included oscillation of the droplet edge. The oscillation period remained constant even as the droplets grew, exhibiting an anomalously weak size dependence. The high mobility of the droplet edges owing to the superfluidity was a crucial factor for the appearance of these anomalous modes. >>️

Keita Onodera, Ryuma Nagatomo, et al. Anomalous Oscillation Modes of Superfluid Pendant Droplets. Phys. Rev. Lett. 133, 216001. Nov 19, 2024.

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.133.216001

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

Keywords: gst, drop, droplet, droploid, transition

# gst: chaotic dynamics in the transition between sleep and wakefulness

AA <<  have investigated how to describe one of the most significant state changes in the brain, namely the transition between sleep and wakefulness >>

<< The brain's neuronal activity depends on the overall state we are in (deep sleep, light sleep, sedentary awake or awake and active), and this is important for how sensory impressions are processed >>

Niels  Bohr  Institute. Chaos  in  the  transition from sleep to awake. Dec 15,  2017.

https://m.medicalxpress.com/news/2017-12-chaos-transition.html

AA << find that sleep is governed by stable, self-sustained oscillations in neuronal firing patterns, whereas the quiet awake state and active awake state are both governed by irregular oscillations and chaotic dynamics; transitions between these separable awake states are prompted by ionic changes. Although waking is indicative of a shift from stable to chaotic neuronal firing patterns, [AA]  illustrate that the properties of chaotic dynamics ensure that the transition between states is smooth and robust to noise >>

Rune Rasmussen, Mogens H. Jensen, Mathias L. Heltberg. Chaotic Dynamics Mediate Brain State Transitions, Driven by Changes in Extracellular Ion Concentrations.
Cell Systems. 2017; 5 (6): 591 - 603.e4. doi: 10.1016/j.cels.2017.11.011. Dec 13, 2017.

http://www.cell.com/cell-systems/fulltext/S2405-4712(17)30539-2

# 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: stochastic resonance mediated by optimal noise level values; transport of colloidal particles induced by substrate concentration oscillations.

AA << show the existence of a  stochastic resonant regime in the transport of active colloidal particles under confinement. The periodic addition of substrate to the system causes the spectral amplification to exhibit a maximum for an optimal noise level value. The consequence of this is that particles can travel longer distances with lower fuel consumption. The stochastic resonance phenomenon found allows the identification of optimal scenarios for the transport of active particles, enabling them to reach regions that are otherwise difficult to access, and may therefore find applications in transport. >>️

J. D. Torrenegra-Rico, A. Arango-Restrepo, J. M. Rubi. Optimal transport of active particles induced by substrate concentration oscillations. Phys. Rev. E 108, 014134. July 28, 2023.

https://journals.aps.org/pre/abstract/10.1103/PhysRevE.108.014134

Img: https://twitter.com/PhysRevE/status/1687119409096089602

Also: particle, colloids, noise, in:  https://www.inkgmr.net/kwrds.html

Keywords: gst, particle, colloids, noise, stochastic resonance, substrate oscillation

# gst: apropos of vibrating pivots, driving a damped coplanar double pendulum.

AA << present results of linear and nonlinear motions of a parametrically driven coplanar double pendulum with velocity-dependent damping. The equations of motion of a damped double pendulum of unequal masses with its pivot vibrated vertically are different from those obtained under gravity modulation. 

Linear stability analysis shows that tongue-shaped marginal stability curves divide the plane of driving parameters into multiple regions of subharmonic and harmonic instabilities. The instability zones for one normal mode overlap with those for the other. 

The double pendulum may oscillate or rotate about its pivot harmonically or subharmonically. The limit cycles corresponding to the normal mode oscillations of a double pendulum of equal masses are squeezed into a line in its configuration space. 

For unequal masses, two marginal curves for subharmonic instabilities merge to form a double-well shaped curve in the presence of damping, which is qualitatively new. The pendulum shows driving amplitude sensitive multi-period complex oscillations for driving parameters near the extrema of the merged instability zones and boundaries of the overlapping zones. 

For larger driving amplitude, the pendulum shows subharmonic, harmonic or chaotic rotations. >>

️

Rebeka Sarkar, Krishna Kumar, Sugata Pratik Khastgir. Parametrically driven damped coplanar double pendulum. arXiv:2208.03292v1 [physics.class-ph]. Aug 2, 2022. 

https://arxiv.org/abs/2208.03292

Also

keyword 'pendulum' in FonT

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

keyword 'pendolo' | 'pendola' in Notes

(quasi-stochastic poetry)

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

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

Keywords: gst, pendulum, double pendulum, instability, chaos, chaotic rotations

# gst: kirigami exposed to external flows.

<< Kirigami patterned materials have found several applications in recent years due to their ability to assume complicated shapes and exhibit emergent physical properties when exposed to external forces. >>️

<< Consisting of an array of cuts in a thin material, fabrication of these patterns can be quite simple. Here (AA) show that when they are placed in fluid flow, kirigami cut sheets with various patterns produce a verity of flow patterns in the wake. Through several sets of experiments, (AA) show that the kirigami sheets placed in flow can undergo static or dynamic flow-induced instabilities as a result of which they can buckle or undergo limit cycle oscillations, or they can remain stable while undergoing very large elongations. >>️

<< The ability to create controlled small-scale vortex shedding, induce desired flow-induced instabilities on structures, and form specifically-angled jets will enable several future applications in flow mixing (e.g., by producing small vortices in uniform flow at low Reynolds numbers), flow control (e.g., by controlling the direction and the number of jets that are produced downstream), and underwater soft robotics (e.g., by imposing desired flow-induced oscillations on structures). >>

️

Adrian G. Carleton, Yahya Modarres-Sadeghi. Kirigami Sheets in Fluid Flow.  arXiv: 2311.09381v1 [physics.flu-dyn]. Nov 15, 2023. 

https://arxiv.org/abs/2311.09381

Also: kirigami, origami, vortex, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, kirigami, origami, fluid flows, vortex

# gst: active drops: from steady to chaotic self-propulsion

<< Individual chemically active drops suspended in a surfactant solution were observed to self-propel spontaneously with straight, helical, or chaotic trajectories. (..) strong advection (e.g., large droplet size) may destabilize a steadily self-propelling drop; once destabilized, the droplet spontaneously stops and a symmetric extensile flow emerges. If advection is strengthened even further in comparison with molecular diffusion, the droplet may perform chaotic oscillations. >>

Matvey Morozov, Sebastien Michelin. Nonlinear dynamics of a chemically-active drop: From steady to chaotic self-propulsion. J. Chem. Phys. 150, 044110 (2019). doi: 10.1063/1.5080539. Jan 31, 2019.  https://aip.scitation.org/doi/10.1063/1.5080539

# gst: odd electrical circuits

<< Non-reciprocal interactions in elastic media give rise to rich non-equilibrium behaviors, but controllable experimental realizations of such odd elastic phenomena remain scarce. Building on recent breakthroughs in electrical analogs of non-Hermitian solid-state systems, (AA) design and analyze scalable odd electrical circuits (OECs) as exact analogs of an odd solid. >>

AA << show that electrical work can be extracted from OECs via cyclic excitations and trace the apparent energy gain back to active circuit elements. (They) show that OECs host oscillatory modes that resemble recent experimental observations in living chiral crystals and identify active resonances that reveal a perspective on odd elasticity as a mechanism for mechanical amplification. >>️

Harry Walden, Alexander Stegmaier, et al. Odd electrical circuits. arXiv: 2503.14383v1 [cond-mat.soft]. Mar 18, 2025.

https://arxiv.org/abs/2503.14383

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

Keywords: gst, chirality, chiral crystals, chiral oscillations, chiral waves, oddity, odd elastic materials

# gst: contactless manipulation of droplets levitating in an acoustic wave

<< A unique, versatile, and material-independent approach to manipulate contactlessly and merge two chemically distinct droplets suspended in an acoustic levitator is reported. Large-amplitude axial oscillations are induced in the top droplet by low-frequency amplitude modulation of the ultrasonic carrier wave, which causes the top sample to merge with the sample in the pressure minimum below. The levitator is enclosed within a pressure-compatible process chamber to enable control of the environmental conditions. The merging technique permits precise control of the substances affecting the chemical reactions, the sample temperature, the volumes of the liquid reactants down to the picoliter range, and the mixing locations in space and time. >>

Stephen J. Brotton, Ralf I. Kaiser. 

Controlled Chemistry via Contactless Manipulation and Merging of Droplets in an Acoustic Levitator. Anal. Chem. 2020, 92, 12, 8371–8377. doi: 10.1021/ acs.analchem.0c00929. Jun 1, 2020. 

https://pubs.acs.org/doi/10.1021/acs.analchem.0c00929#

Levitating droplets allow scientists to perform 'touchless' chemical reactions.  American Chemical Society. Jun 24, 2020.

https://phys.org/news/2020-06-levitating-droplets-scientists-touchless-chemical.html  

# gst: intricate transitions in elastoactive structures.

<< The interplay between activity and elasticity often found in active and living systems triggers a plethora of autonomous behaviors ranging from self-assembly and collective motion to actuation. Among these, spontaneous self-oscillations of mechanical structures is perhaps the simplest and most widespread type of nonequilibrium phenomenon. >>️

<< Here, (AA) introduce a centimeter-sized model system for one-dimensional elastoactive structures. >>️

<< such structures exhibit flagellar motion when pinned at one end, self-snapping when pinned at two ends, and synchronization when coupled together with a sufficiently stiff link. (..) these transitions can be described quantitatively by simple models of coupled pendula with follower forces. >>️

Ellen Zheng, Martin Brandenbourger, et al. Self-Oscillation and Synchronization Transitions in Elastoactive Structures. Phys. Rev. Lett. 130, 178202. April 25, 2023. 

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.130.178202

Also:  transition, particle, self-assembly, elastic, pendulum in https://www.inkgmr.net/kwrds.html

Keywords: gst, transition, particle, self-assembly, elastic, pendulum

# gst: vacuum fluctuations (even in complete darkness)

<< physics is increasingly discovering how our universe is shaped by fluctuations of physical fields, which not only lead to tiny shifts of spectral lines of atoms, but moreover may cause the evaporation of black holes, and are ultimately responsible for the large-scale structure of our universe,  >>

AA  << have now made a large leap toward controlling strongly enhanced vacuum fluctuations much faster than typical timescales of virtual photons. To this end, they created a specialized semiconductor structure in which electrons are extremely strongly coupled to the light fields of tiny antennas designed for the so-called terahertz spectral range. As a result, vacuum fluctuations of light and matter fields participate in the interaction, strongly increasing the presence of virtual photons—even in complete darkness. >>

Understanding vacuum fluctuations in space. University of Regensburg. Aug 10, 2020.

https://phys.org/news/2020-08-vacuum-fluctuations-space.html 

<< The abrupt modification of the vacuum ground state causes spectrally broadband polarization oscillations confirmed by (AA) quantum model. In the future, this subcycle shaping of hybrid quantum states may trigger cavity-induced quantum chemistry, vacuum-modified transport or cavity-controlled superconductivity, opening new scenarios >>

M. Halbhuber, J. Mornhinweg, et al. Non-adiabatic stripping of a cavity field from electrons in the deep-strong coupling regime. Nat. Photonics. doi: 10.1038/ s41566-020-0673-2. Aug 10, 2020.

https://www.nature.com/articles/s41566-020-0673-2

# gst: even a single bubble can produce creative musical outcomes

<< Producing original and arranging existing musical outcomes is an art that takes years of learning and practice to master. Yet, despite the constant advances in the field of AI-powered musical creativity, production of quality musical outcomes remains a prerogative of the humans. Here we demonstrate that a single bubble in water can be used to produce creative musical outcomes, when it nonlinearly oscillates under an acoustic pressure signal that encodes a piece of classical music. >>️

Ivan S. Maksymov. Musical creativity enabled by nonlinear oscillations of a bubble in water. arXiv:2304.00822v1 [cs.SD]. Apr 3, 2023. 

https://arxiv.org/abs/2304.00822

keyword "bubble" in FonT

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

Keywords: gst, ai, fluid dynamics, bubble, sound, music, audio processing

# game: balance exploration and exploitation, making decisions cooperatively without sharing information.

<< Multiagent reinforcement learning (MARL) studies crucial principles that are applicable to a variety of fields, including wireless networking and autonomous driving. (AA) propose a photonic-based decision-making algorithm to address one of the most fundamental problems in MARL, called the competitive multiarmed bandit (CMAB) problem. >>

AA << demonstrate that chaotic oscillations and cluster synchronization of optically coupled lasers, along with (their) proposed decentralized coupling adjustment, efficiently balance exploration and exploitation while facilitating cooperative decision making without explicitly sharing information among agents. >>

AA << study demonstrates how decentralized reinforcement learning can be achieved by exploiting complex physical processes controlled by simple algorithms. >>

Shun Kotoku, Takatomo Mihana, et al. Decentralized multiagent reinforcement learning algorithm using a cluster-synchronized laser network. Phys. Rev. E 110, 064212. Dec 11, 2024.

https://journals.aps.org/pre/abstract/10.1103/PhysRevE.110.064212

Img: https://x.com/PhysRevE/status/1866889144657465632

Also: game, chaos, ai (artificial intell), in https://www.inkgmr.net/kwrds.html 

Keywords: game, cooperation, chaos, exploration, exploitation, ai, artificial intelligence, MARL, CMAB.