Showing papers by "Steven H. Strogatz published in 2021"

The Kuramoto model on a sphere: Explaining its low-dimensional dynamics with group theory and hyperbolic geometry

Abstract: We study a system of N identical interacting particles moving on the unit sphere in d-dimensional space. The particles are self-propelled and coupled all to all, and their motion is heavily overdamped. For d=2, the system reduces to the classic Kuramoto model of coupled oscillators; for d=3, it has been proposed to describe the orientation dynamics of swarms of drones or other entities moving about in three-dimensional space. Here, we use group theory to explain the recent discovery that the model shows low-dimensional dynamics for all N≥3 and to clarify why it admits the analog of the Ott–Antonsen ansatz in the continuum limit N→∞. The underlying reason is that the system is intimately connected to the natural hyperbolic geometry on the unit ball Bd. In this geometry, the isometries form a Lie group consisting of higher-dimensional generalizations of the Mobius transformations used in complex analysis. Once these connections are realized, the reduced dynamics and the generalized Ott–Antonsen ansatz follow immediately. This framework also reveals the seamless connection between the finite and infinite- N cases. Finally, we show that special forms of coupling yield gradient dynamics with respect to the hyperbolic metric and use that fact to obtain global stability results about convergence to the synchronized state.

Designing temporal networks that synchronize under resource constraints

Abstract: Being fundamentally a non-equilibrium process, synchronization comes with unavoidable energy costs and has to be maintained under the constraint of limited resources. Such resource constraints are often reflected as a finite coupling budget available in a network to facilitate interaction and communication. Here, we show that introducing temporal variation in the network structure can lead to efficient synchronization even when stable synchrony is impossible in any static network under the given budget, thereby demonstrating a fundamental advantage of temporal networks. The temporal networks generated by our open-loop design are versatile in the sense of promoting synchronization for systems with vastly different dynamics, including periodic and chaotic dynamics in both discrete-time and continuous-time models. Furthermore, we link the dynamic stabilization effect of the changing topology to the curvature of the master stability function, which provides analytical insights into synchronization on temporal networks in general. In particular, our results shed light on the effect of network switching rate and explain why certain temporal networks synchronize only for intermediate switching rate. The ability of complex networks to synchronize themselves is limited by available coupling resources. Zhang and Strogatz show that allowing temporal variation in the network structure can lead to synchronization even when stable synchrony is impossible in any static network under the fixed budget.

Sufficiently dense Kuramoto networks are globally synchronizing.

Abstract: Consider any network of n identical Kuramoto oscillators in which each oscillator is coupled bidirectionally with unit strength to at least μ ( n − 1 ) other oscillators. There is a critical value of the connectivity, μ c, such that whenever μ > μ c, the system is guaranteed to converge to the all-in-phase synchronous state for almost all initial conditions, but when μ 0.6838. This paper proves that μ c ≤ 0.75 and explain why this is the best upper bound that one can obtain by a purely linear stability analysis.

Basins with Tentacles.

Abstract: To explore basin geometry in high-dimensional dynamical systems, we consider a ring of identical Kuramoto oscillators. Many attractors coexist in this system; each is a twisted periodic orbit characterized by a winding number $q$, with basin size proportional to ${e}^{\ensuremath{-}k{q}^{2}}$. We uncover the geometry behind this size distribution and find the basins are octopuslike, with nearly all their volume in the tentacles, not the head of the octopus (the ball-like region close to the attractor). We present a simple geometrical reason why basins with tentacles should be common in high-dimensional systems.

Sufficiently dense Kuramoto networks are globally synchronizing.

Abstract: Consider any network of $n$ identical Kuramoto oscillators in which each oscillator is coupled bidirectionally with unit strength to at least $\mu (n-1)$ other oscillators. There is a critical value of the connectivity, $\mu_c$, such that whenever $\mu>\mu_c$, the system is guaranteed to converge to the all-in-phase synchronous state for almost all initial conditions, but when $\mu 0.6838$. In this paper, we prove that $\mu_c\leq 0.75$ and explain why this is the best upper bound that one can obtain by a purely linear stability analysis.

Synchronization of clocks and metronomes: A perturbation analysis based on multiple timescales.

Abstract: In 1665, Huygens observed that two pendulum clocks hanging from the same board became synchronized in antiphase after hundreds of swings On the other hand, modern experiments with metronomes placed on a movable platform show that they often tend to synchronize in phase, not antiphase Here, we study both in-phase and antiphase synchronization in a model of pendulum clocks and metronomes and analyze their long-term dynamics with the tools of perturbation theory Specifically, we exploit the separation of timescales between the fast oscillations of the individual pendulums and the much slower adjustments of their amplitudes and phases By scaling the equations appropriately and applying the method of multiple timescales, we derive explicit formulas for the regimes in the parameter space where either antiphase or in-phase synchronization is stable or where both are stable Although this sort of perturbative analysis is standard in other parts of nonlinear science, surprisingly it has rarely been applied in the context of Huygens’s clocks Unusual features of our approach include its treatment of the escapement mechanism, a small-angle approximation up to cubic order, and both a two- and three-timescale asymptotic analysis

Asymptotic absorption-time distributions in extinction-prone Markov processes

Abstract: We characterize absorption-time distributions for birth-death Markov chains with an absorbing boundary state. Based on generic features of the transition rates, the asymptotic distribution for such "extinction-prone" chains is either Gaussian, Gumbel, or a convolution of Gumbel distributions. We show that several birth-death models of evolutionary dynamics, epidemiology, and chemical reactions fall into these classes, establishing new results for the absorption-time distribution in each case. Possible connections to African sleeping sickness are discussed.

Surprises in a classic boundary-layer problem

Abstract: We revisit a textbook example of a singularly perturbed nonlinear boundary-value problem. Unexpectedly, it shows a wealth of phenomena that seem to have been overlooked previously, including a pitchfork bifurcation in the number of solutions as one varies the small parameter, and transcendentally small terms in the initial conditions that can be calculated by elementary means. Based on our own classroom experience, we believe this problem could provide an enjoyable workout for students in courses on perturbation methods, applied dynamical systems, or numerical analysis.

Basins with tentacles

Abstract: To explore basin geometry in high-dimensional dynamical systems, we consider a ring of identical Kuramoto oscillators. Many attractors are known to coexist in this system; each is a twisted periodic orbit characterized by a winding number $q$, with basin size proportional to $e^{-kq^2}.$ We uncover the geometry behind this size distribution and find the basins are octopus-like, with nearly all their volume in the tentacles, not the head of the octopus (the ball-like region close to the attractor). We suggest that similar basins with tentacles should be generic in high-dimensional systems.

Designing temporal networks that synchronize under resource constraints

Abstract: Being fundamentally a non-equilibrium process, synchronization comes with unavoidable energy costs and has to be maintained under the constraint of limited resources. Such resource constraints are often reflected as a finite coupling budget available in a network to facilitate interaction and communication. Here, we show that introducing temporal variation in the network structure can lead to efficient synchronization even when stable synchrony is impossible in any static network under the given budget, thereby demonstrating a fundamental advantage of temporal networks. The temporal networks generated by our open-loop design are versatile in the sense of promoting synchronization for systems with vastly different dynamics, including periodic and chaotic dynamics in both discrete-time and continuous-time models. Furthermore, we link the dynamic stabilization effect of the changing topology to the curvature of the master stability function, which provides analytical insights into synchronization on temporal networks in general. In particular, our results shed light on the effect of network switching rate and explain why certain temporal networks synchronize only for intermediate switching rate.

Modeling the interplay between seasonal flu outcomes and individual vaccination decisions

Abstract: Seasonal influenza presents an ongoing challenge to public health. The rapid evolution of the flu virus necessitates annual vaccination campaigns, but the decision to get vaccinated or not in a given year is largely voluntary, at least in the United States, and many people decide against it. In early attempts to model these yearly flu vaccine decisions, it was often assumed that individuals behave rationally, and do so with perfect information -- assumptions that allowed the techniques of classical economics and game theory to be applied. However, the usual assumptions are contradicted by the emerging empirical evidence about human decision-making behavior in this context. We develop a simple model of coupled disease spread and vaccination dynamics that instead incorporates experimental observations from social psychology to model annual vaccine decision-making more realistically. We investigate population-level effects of these new decision-making assumptions, with the goal of understanding whether the population can self-organize into a state of herd immunity, and if so, under what conditions. Our model agrees with established results while also revealing more subtle population-level behavior, including biennial oscillations about the herd immunity threshold.