More Like this

1. Evolutionary Kuramoto dynamics

   https://doi.org/10.1098/rspb.2022.0999  

   Tripp, Elizabeth A.; Fu, Feng; Pauls, Scott D. (November 2022, Proceedings of the Royal Society B: Biological Sciences)

   Biological systems have a variety of time-keeping mechanisms ranging from molecular clocks within cells to a complex interconnected unit across an entire organism. The suprachiasmatic nucleus, comprising interconnected oscillatory neurons, serves as a master-clock in mammals. The ubiquity of such systems indicates an evolutionary benefit that outweighs the cost of establishing and maintaining them, but little is known about the process of evolutionary development. To begin to address this shortfall, we introduce and analyse a new evolutionary game theoretic framework modelling the behaviour and evolution of systems of coupled oscillators. Each oscillator is characterized by a pair of dynamic behavioural dimensions, a phase and a communication strategy, along which evolution occurs. We measure success of mutations by comparing the benefit of synchronization balanced against the cost of connections between the oscillators. Despite the simple set-up, this model exhibits non-trivial behaviours mimicking several different classical games—the Prisoner’s Dilemma, snowdrift games, coordination games—as the landscape of the oscillators changes over time. Across many situations, we find a surprisingly simple characterization of synchronization through connectivity and communication: if the benefit of synchronization is greater than twice the cost, the system will evolve towards complete communication and phase synchronization. 

   more » « less
2. Synchronization in a Kuramoto mean field game

   https://doi.org/10.1080/03605302.2023.2264611  

   Carmona, Rene; Cormier, Quentin; Soner, H Mete (September 2023, Communications in Partial Differential Equations)

   The classical Kuramoto model is studied in the setting of an infinite horizon mean field game. The system is shown to exhibit both syn- chronization and phase transition. Incoherence below a critical value of the interaction parameter is demonstrated by the stability of the uni- form distribution. Above this value, the game bifurcates and develops self-organizing time homogeneous Nash equilibria. As interactions get stronger, these stationary solutions become fully synchronized. Results are proved by an amalgam of techniques from nonlinear partial dif- ferential equations, viscosity solutions, stochastic optimal control and stochastic processes. ARTICLE HISTORY Received 23 October 2022 Accepted 17 June 2023 KEYWORDS Mean field games; Kuramoto model; synchronization; viscosity solutions 2020 MATHEMATICS SUBJECT CLASSIFICATION 35Q89; 35D40; 39N80; 91A16; 92B25 1. Introduction Originally motivated by systems of chemical and biological oscillators, the classical Kuramoto model [1] has found an amazing range of applications from neuroscience to Josephson junctions in superconductors, and has become a key mathematical model to describe self organization in complex systems. These autonomous oscillators are coupled through a nonlinear interaction term which plays a central role in the long time behavior of the system. While the system is unsynchronized when this term is not sufficiently strong, fascinatingly they exhibit an abrupt transition to self organization above a critical value of the interaction parameter. Synchronization is an emergent property that occurs in a broad range of complex systems such as neural signals, heart beats, fire-fly lights and circadian rhythms, and the Kuramoto dynamical system is widely used as the main phenomenological model. Expository papers [2, 3] and the references therein provide an excellent introduction to the model and its applications. The analysis of the coupled Kuramoto oscillators through a mean field game formalism is first explored by [4, 5] proving bifurcation from incoherence to coordination by a formal linearization and a spectral argument. [6] further develops this analysis in their application to a jet-lag recovery model. We follow these pioneering studies and analyze the Kuramoto model as a discounted infinite horizon stochastic game in the limit when the number of oscillators goes to infinity. We treat the system of oscillators as an infinite particle system, but instead of positing the dynamics of the particles, we let the individual particles endogenously determine their behaviors by minimizing a cost functional and hopefully, settling in a Nash equilibrium. Once the search for equilibrium is recast in this way, equilibria are given by solutions of nonlinear systems. Analytically, they are characterized by a backward dynamic CONTACT H. Mete Soner soner@princeton.edu Department of Operations Research and Financial Engineering, Prince- ton University, Princeton, NJ, 08540, USA. © 2023 Taylor & Francis Group, LLC 

   more » « less
3. Cluster Synchronization in Networks of Kuramoto Oscillators

   Favaretto, C; Cenedese, A; Pasqualetti, F (January 2017, IFAC World Congress)

   A broad class of natural and man-made systems exhibits rich patterns of cluster synchronization in healthy and diseased states, where different groups of interconnected oscillators converge to cohesive yet distinct behaviors. To provide a rigorous characterization of cluster synchronization, we study networks of heterogeneous Kuramoto oscillators and we quantify how the intrinsic features of the oscillators and their interconnection parameters affect the formation and the stability of clustered configurations. Our analysis shows that cluster synchronization depends on a graded combination of strong intra-cluster and weak inter-cluster connections, similarity of the natural frequencies of the oscillators within each cluster, and heterogeneity of the natural frequencies of coupled oscillators belonging to different groups. The analysis leverages linear and nonlinear control theoretic tools, and it is numerically validated. 

   more » « less
4. Bifurcations in the Kuramoto model on graphs

   https://doi.org/10.1063/1.5039609  

   Chiba, Hayato; Medvedev, Georgi S.; Mizuhara, Matthew S. (July 2018, Chaos: An Interdisciplinary Journal of Nonlinear Science)
5. Equilibria in Kuramoto Oscillator Networks: An Algebraic Approach

   https://doi.org/10.1137/21M1457321  

   Nguyen, Tung T.; Budzinski, Roberto C.; Ðoàn, Jacqueline; Pasini, Federico W.; Mináč, Ján; Muller, Lyle E. (June 2023, SIAM Journal on Applied Dynamical Systems)