{"title":"Quantifying the likelihood of learning collusive strategy equilibria.","authors":"Janusz M Meylahn","doi":"10.1063/5.0281443","DOIUrl":"https://doi.org/10.1063/5.0281443","url":null,"abstract":"<p><p>We develop a method for quantifying the likelihood of observing collusive strategies among provably convergent decentralized multiagent reinforcement learning algorithms in a pricing setting. This is necessary to accurately assess the threat that colluding algorithms pose for society. The tools are, however, more generally applicable. Specifically, we obtain conditions for the weak acyclicity of families of two-player, symmetric Markov games in which best responses are unique. In this case, the individual best-response graphs (a concept we introduce in the article) belong to the class of functional relations. Using the structural properties of this class of graphs, we provide conditions on the individual best-response graphs for the game being weakly acyclic. In addition, we characterize the stationary distribution of the best-response strategy adjustment process in such games. Using these results, we show that Decentralized Q-learning is provably convergent in three two-player, two-action games with a memory of one period, analyze its probability of converging to different equilibria, and interpret the results in the context of algorithmic collusion.</p>","PeriodicalId":9974,"journal":{"name":"Chaos","volume":"35 8","pages":""},"PeriodicalIF":3.2,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144764643","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Dynamical analysis and solutions of coupled 4D fractional differential systems with applications to predictability limit quantification in ocean-atmosphere models.","authors":"Xiaoyu Chen, Hongtao Fan, Yajing Li","doi":"10.1063/5.0283492","DOIUrl":"https://doi.org/10.1063/5.0283492","url":null,"abstract":"<p><p>For coupled two-dimensional fractional differential systems, a new two-step fractional-order Runge-Kutta method is proposed in this paper, which can reach a convergence order of 2α, with α being the fractional-order number. Further, we extend the two-step fractional-order Runge-Kutta algorithm to any coupled n-dimensional fractional differential system while maintaining convergence and consistency. To demonstrate the validity of the proposed method, numerical experiments are given for a four-dimensional fractional Lorenz system, and the dynamics of the four-dimensional fractional Lorenz system is analyzed using Lyapunov characteristic exponents, bifurcation diagrams, chaos diagrams, and C0 complexity. The results demonstrate that the system exhibits a diverse dynamical behavior and a broader range of fractional orders [0.43, 1] is accessible to the periodic orbit at the same parameter, compared to previous findings [He et al., Math. Methods Appl. Sci. 39, 2965-2973 (2016)]. Finally, we use global attractor radius and attractor radius to investigate the predictability of the coupled fractional-order ocean-atmosphere system [Li et al., Clim. Dyn. 51, 2359-2374 (2018)]. The results show that both global attractor radius and attractor radius decrease with decreasing fractional order, and the smaller the initial perturbation, the longer it takes to reach the attractor radius, but the attractor radius to the global attractor radius is not significantly correlated with the initial perturbation. These findings suggest that the predictability of the coupled fractional ocean-atmosphere system is limited by the presence of long-range memory effects captured in the fractional-order differential equations. By offering quantitative assessments of predictability, these approaches enhance our understanding of the intricate dynamics within such systems and can support informed decision-making in addressing and mitigating the effects of climate change on the global atmosphere and oceans.</p>","PeriodicalId":9974,"journal":{"name":"Chaos","volume":"35 8","pages":""},"PeriodicalIF":3.2,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144764641","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
L Moysis, M Lawnik, K F Kollias, M S Baptista, S Goudos, G Fragulis
{"title":"Dynamic analysis of a generalized attention deficit disorder model with Soboleva activation functions.","authors":"L Moysis, M Lawnik, K F Kollias, M S Baptista, S Goudos, G Fragulis","doi":"10.1063/5.0280557","DOIUrl":"https://doi.org/10.1063/5.0280557","url":null,"abstract":"<p><p>This work studies a modified chaotic neural network model consisting of two neurons for modeling attention deficit disorder. Considering an existing one-dimensional model from the literature, its two activation functions are replaced by the Soboleva hyperbolic tangent function. This change introduces four new control parameters to the system. The effect of these parameters on the system is extensively studied through a collection of phase, bifurcation, and Lyapunov exponent diagrams. Changing each of these parameters brings changes to the model's behavior, so the modified model is a significant generalization of the original one. Many phenomena are observed, including period doubling route to chaos, period halving route to period-1, crisis, antimonotonicity, coexisting attractors, and shrimps. The newly introduced degrees of freedom could provide a new direction toward modeling behavioral disorders using different activation functions.</p>","PeriodicalId":9974,"journal":{"name":"Chaos","volume":"35 8","pages":""},"PeriodicalIF":3.2,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144764640","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Manaswini Jella, Induja Pavithran, Vishnu R Unni, Norbert Marwan, Jürgen Kurths, R I Sujith
{"title":"Recurrence condensation during critical transitions in complex systems.","authors":"Manaswini Jella, Induja Pavithran, Vishnu R Unni, Norbert Marwan, Jürgen Kurths, R I Sujith","doi":"10.1063/5.0267157","DOIUrl":"https://doi.org/10.1063/5.0267157","url":null,"abstract":"<p><p>Critical transitions in complex systems pose challenges for the healthy functioning of natural and engineered systems, sometimes with catastrophic outcomes. These critical points, where small changes cause large regime shifts, are difficult to detect-especially in noisy, high-dimensional settings. We investigate such a transition from chaotic to periodic oscillations via intermittency in a turbulent fluid mechanical system by using recurrence analysis. Recurrence plots (RPs) constructed from the time series of a state variable reveal a distinct progression from disordered, short broken diagonal lines to patches of ordered short diagonal lines and, ultimately, to a pattern of long continuous diagonal lines. This evolution in the recurrence patterns captures a transition from dynamics involving multiple time scales to a dominant single time scale; we term this phenomenon \"recurrence condensation.\" We quantify recurrence condensation using recurrence quantification measures, such as the recurrence time, determinism, entropy, laminarity, and trapping time, all of which show collapse to a single dominant time scale. Furthermore, these recurrence measures exhibit power-law scaling with the deviation of the control parameter from the critical point. Optimizing for the best power law reveals the critical value of the parameter. We apply this method to the synthetic data from a basic noisy Hopf bifurcation model and confirm that the detected critical point coincides with the bifurcation point. Our findings offer insights into identifying the critical points in noisy systems with gradual transitions, where the transition point is not well defined.</p>","PeriodicalId":9974,"journal":{"name":"Chaos","volume":"35 8","pages":""},"PeriodicalIF":3.2,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144764644","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Approximate packing of independent transversals in locally sparse graphs","authors":"Debsoumya Chakraborti , Tuan Tran","doi":"10.1016/j.jctb.2025.07.005","DOIUrl":"10.1016/j.jctb.2025.07.005","url":null,"abstract":"<div><div>Fix <span><math><mi>ε</mi><mo>></mo><mn>0</mn></math></span> and consider a multipartite graph <em>G</em> with maximum degree at most <span><math><mo>(</mo><mn>1</mn><mo>−</mo><mi>ε</mi><mo>)</mo><mi>n</mi></math></span>, parts <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> of the same size <em>n</em>, and where every vertex has at most <span><math><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> neighbors in any part <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>. Loh and Sudakov proved that any such <em>G</em> has an independent transversal. They further conjectured that the vertex set of <em>G</em> can be decomposed into pairwise disjoint independent transversals. In the present paper, we resolve this conjecture approximately by showing that <em>G</em> contains <span><math><mo>(</mo><mn>1</mn><mo>−</mo><mi>ε</mi><mo>)</mo><mi>n</mi></math></span> pairwise disjoint independent transversals. As applications, we give approximate answers to questions of Yuster, and of Fischer, Kühn, and Osthus.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"175 ","pages":"Pages 187-212"},"PeriodicalIF":1.2,"publicationDate":"2025-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144738353","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"An \u0000 \u0000 \u0000 L\u0000 ∞\u0000 \u0000 $L_infty$\u0000 structure for Legendrian contact homology","authors":"Lenhard Ng","doi":"10.1112/topo.70034","DOIUrl":"https://doi.org/10.1112/topo.70034","url":null,"abstract":"<p>For any Legendrian knot or link in <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mn>3</mn>\u0000 </msup>\u0000 <annotation>$mathbb {R}^3$</annotation>\u0000 </semantics></math>, we construct an <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>L</mi>\u0000 <mi>∞</mi>\u0000 </msub>\u0000 <annotation>$L_infty$</annotation>\u0000 </semantics></math> algebra that can be viewed as an extension of the Chekanov–Eliashberg differential graded algebra. The <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>L</mi>\u0000 <mi>∞</mi>\u0000 </msub>\u0000 <annotation>$L_infty$</annotation>\u0000 </semantics></math> structure incorporates information from rational symplectic field theory and can be formulated combinatorially. One consequence is the construction of a Poisson bracket on commutative Legendrian contact homology, and we show that the resulting Poisson algebra is an invariant of Legendrian links under isotopy.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"18 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144740506","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}