arXiv subjects

Steven H. Strogatz

Publications and source records attributed to Steven H. Strogatz.

At least 37 records · Page 2Linked to original sources

arXiv2018

Volcano transition in a solvable model of oscillator glass

In 1992 a puzzling transition was discovered in simulations of randomly coupled limit-cycle oscillators. This so-called volcano transition has resisted analysis ever since. It was originally conjectured to mark the emergence of an oscillator glass, but here we show it need not. We introduce and solve a simpler model with a qualitatively identical volcano transition and find, unexpectedly, that its supercritical state is not glassy. We discuss the implications for the original model and suggest experimental systems in which a volcano transition and oscillator glass may appear.

math.DS↗

arXiv2017

Oscillators that sync and swarm

Synchronization occurs in many natural and technological systems, from cardiac pacemaker cells to coupled lasers. In the synchronized state, the individual cells or lasers coordinate the timing of their oscillations, but they do not move through space. A complementary form of self-organization occurs among swarming insects, flocking birds, or schooling fish; now the individuals move through space, but without conspicuously altering their internal states. Here we explore systems in which both synchronization and swarming occur together. Specifically, we consider oscillators whose phase dynamics and spatial dynamics are coupled. We call them swarmalators, to highlight their dual character. A case study of a generalized Kuramoto model predicts five collective states as possible long-term modes of organization. These states may be observable in groups of sperm, Japanese tree frogs, colloidal suspensions of magnetic particles, and other biological and physical systems in which self-assembly and synchronization interact.

nlin.AO↗

arXiv2017

Evolutionary dynamics of incubation periods

The incubation period of a disease is the time between an initiating pathologic event and the onset of symptoms. For typhoid fever, polio, measles, leukemia and many other diseases, the incubation period is highly variable. Some affected people take much longer than average to show symptoms, leading to a distribution of incubation periods that is right skewed and often approximately lognormal. Although this statistical pattern was discovered more than sixty years ago, it remains an open question to explain its ubiquity. Here we propose an explanation based on evolutionary dynamics on graphs. For simple models of a mutant or pathogen invading a network-structured population of healthy cells, we show that skewed distributions of incubation periods emerge for a wide range of assumptions about invader fitness, competition dynamics, and network structure. The skewness stems from stochastic mechanisms associated with two classic problems in probability theory: the coupon collector and the random walk. Unlike previous explanations that rely crucially on heterogeneity, our results hold even for homogeneous populations. Thus, we predict that two equally healthy individuals subjected to equal doses of equally pathogenic agents may, by chance alone, show remarkably different time courses of disease.

q-bio.PE↗

arXiv2017

Takeover times for a simple model of network infection

We study a stochastic model of infection spreading on a network. At each time step a node is chosen at random, along with one of its neighbors. If the node is infected and the neighbor is susceptible, the neighbor becomes infected. How many time steps $T$ does it take to completely infect a network of $N$ nodes, starting from a single infected node? An analogy to the classic "coupon collector" problem of probability theory reveals that the takeover time $T$ is dominated by extremal behavior, either when there are only a few infected nodes near the start of the process or a few susceptible nodes near the end. We show that for $N \gg 1$, the takeover time $T$ is distributed as a Gumbel for the star graph; as the sum of two Gumbels for a complete graph and an Erdős-Rényi random graph; as a normal for a one-dimensional ring and a two-dimensional lattice; and as a family of intermediate skewed distributions for $d$-dimensional lattices with $d \ge 3$ (these distributions approach the sum of two Gumbels as $d$ approaches infinity). Connections to evolutionary dynamics, cancer, incubation periods of infectious diseases, first-passage percolation, and other spreading phenomena in biology and physics are discussed.

q-bio.PE↗

arXiv2017

Dynamics on expanding spaces: modeling the emergence of novelties

Novelties are part of our daily lives. We constantly adopt new technologies, conceive new ideas, meet new people, experiment with new situations. Occasionally, we as individuals, in a complicated cognitive and sometimes fortuitous process, come up with something that is not only new to us, but to our entire society so that what is a personal novelty can turn into an innovation at a global level. Innovations occur throughout social, biological and technological systems and, though we perceive them as a very natural ingredient of our human experience, little is known about the processes determining their emergence. Still the statistical occurrence of innovations shows striking regularities that represent a starting point to get a deeper insight in the whole phenomenology. This paper represents a small step in that direction, focusing on reviewing the scientific attempts to effectively model the emergence of the new and its regularities, with an emphasis on more recent contributions: from the plain Simon's model tracing back to the 1950s, to the newest model of Polya's urn with triggering of one novelty by another. What seems to be key in the successful modelling schemes proposed so far is the idea of looking at evolution as a path in a complex space, physical, conceptual, biological, technological, whose structure and topology get continuously reshaped and expanded by the occurrence of the new. Mathematically it is very interesting to look at the consequences of the interplay between the "actual" and the "possible" and this is the aim of this short review.

physics.soc-ph↗

arXiv2016

Correlated disorder in the Kuramoto model: Effects on phase coherence, finite-size scaling, and dynamic fluctuations

We consider a mean-field model of coupled phase oscillators with quenched disorder in the natural frequencies and coupling strengths. A fraction $p$ of oscillators are positively coupled, attracting all others, while the remaining fraction $1-p$ are negatively coupled, repelling all others. The frequencies and couplings are deterministically chosen in a manner which correlates them, thereby correlating the two types of disorder in the model. We first explore the effect of this correlation on the system's phase coherence. We find that there is a a critical width $γ_c$ in the frequency distribution below which the system spontaneously synchronizes. Moreover, this $γ_c$ is independent of $p$. Hence, our model and the traditional Kuramoto model (recovered when $p=1$) have the same critical width $γ_c$. We next explore the critical behavior of the system by examining the finite-size scaling and the dynamic fluctuation of the traditional order parameter. We find that the model belongs to the same universality class as the Kuramoto model with deterministically (not randomly) chosen natural frequencies for the case of $p<1$.

cond-mat.stat-mech↗

arXiv2016

Dynamics of a population of oscillatory and excitable elements

We analyze a variant of a model proposed by Kuramoto, Shinomoto, and Sakaguchi for a large population of coupled oscillatory and excitable elements. Using the Ott-Antonsen ansatz, we reduce the behavior of the population to a two-dimensional dynamical system with three parameters. We present the stability diagram and calculate several of its bifurcation curves analytically, for both excitatory and inhibitory coupling. Our main result is that when the coupling function is broad, the system can display bistability between steady states of constant high and low activity, whereas when the coupling function is narrow and inhibitory, one of the states in the bistable regime can show persistent pulsations in activity.

nlin.CD↗

arXiv2016

Phase coherence induced by correlated disorder

We consider a mean-field model of coupled phase oscillators with quenched disorder in the coupling strengths and natural frequencies. When these two kinds of disorder are uncorrelated (and when the positive and negative couplings are equal in number and strength), it is known that phase coherence cannot occur and synchronization is absent. Here we explore the effects of correlating the disorder. Specifically, we assume that any given oscillator either attracts or repels all the others, and that the sign of the interaction is deterministically correlated with the given oscillator's natural frequency. For symmetrically correlated disorder with zero mean, we find that the system spontaneously synchronizes, once the width of the frequency distribution falls below a critical value. For asymmetrically correlated disorder, the model displays coherent traveling waves: the complex order parameter becomes nonzero and rotates with constant frequency different from the system's mean natural frequency. Thus, in both cases, correlated disorder can trigger phase coherence.

cond-mat.stat-mech↗

arXiv2015

Evolutionary game dynamics of controlled and automatic decision-making

We integrate dual-process theories of human cognition with evolutionary game theory to study the evolution of automatic and controlled decision-making processes. We introduce a model where agents who make decisions using either automatic or controlled processing compete with each other for survival. Agents using automatic processing act quickly and so are more likely to acquire resources, but agents using controlled processing are better planners and so make more effective use of the resources they have. Using the replicator equation, we characterize the conditions under which automatic or controlled agents dominate, when coexistence is possible, and when bistability occurs. We then extend the replicator equation to consider feedback between the state of the population and the environment. Under conditions where having a greater proportion of controlled agents either enriches the environment or enhances the competitive advantage of automatic agents, we find that limit cycles can occur, leading to persistent oscillations in the population dynamics. Critically, however, these limit cycles only emerge when feedback occurs on a sufficiently long time scale. Our results shed light on the connection between evolution and human cognition, and demonstrate necessary conditions for the rise and fall of rationality.

math.DS↗

arXiv2015

Nonlinear Dynamics of the Rock-Paper-Scissors Game with Mutations

We analyze the replicator-mutator equations for the Rock-Paper-Scissors game. Various graph-theoretic patterns of mutation are considered, ranging from a single unidirectional mutation pathway between two of the species, to global bidirectional mutation among all the species. Our main result is that the coexistence state, in which all three species exist in equilibrium, can be destabilized by arbitrarily small mutation rates. After it loses stability, the coexistence state gives birth to a stable limit cycle solution created in a supercritical Hopf bifurcation. This attracting periodic solution exists for all the mutation patterns considered, and persists arbitrarily close to the limit of zero mutation rate and a zero-sum game.

math.DS↗

arXiv2015

Limit Cycles Sparked by Mutation in the Repeated Prisoner's Dilemma

We explore a replicator-mutator model of the repeated Prisoner's Dilemma involving three strategies: always cooperate (ALLC), always defect (ALLD), and tit-for-tat (TFT). The dynamics resulting from single unidirectional mutations are considered, with detailed results presented for the mutations TFT $\rightarrow$ ALLC and ALLD $\rightarrow$ ALLC. For certain combinations of parameters, given by the mutation rate $μ$ and the complexity cost $c$ of playing tit-for-tat, we find that the population settles into limit cycle oscillations, with the relative abundance of ALLC, ALLD, and TFT cycling periodically. Surprisingly, these oscillations can occur for unidirectional mutations between any two strategies. In each case, the limit cycles are created and destroyed by supercritical Hopf and homoclinic bifurcations, organized by a Bogdanov-Takens bifurcation. Our results suggest that stable oscillations are a robust aspect of a world of ALLC, ALLD, and costly TFT; the existence of cycles does not depend on the details of assumptions of how mutation is implemented.

math.DS↗

arXiv2015

Simple model for the Darwinian transition in early evolution

It has been hypothesized that in the era just before the last universal common ancestor emerged, life on earth was fundamentally collective. Ancient life forms shared their genetic material freely through massive horizontal gene transfer (HGT). At a certain point, however, life made a transition to the modern era of individuality and vertical descent. Here we present a minimal model for this hypothesized "Darwinian transition." The model suggests that HGT-dominated dynamics may have been intermittently interrupted by selection-driven processes during which genotypes became fitter and decreased their inclination toward HGT. Stochastic switching in the population dynamics with three-point (hypernetwork) interactions may have destabilized the HGT-dominated collective state and led to the emergence of vertical descent and the first well-defined species in early evolution. A nonlinear analysis of a stochastic model dynamics covering key features of evolutionary processes (such as selection, mutation, drift and HGT) supports this view. Our findings thus suggest a viable route from early collective evolution to the start of individuality and vertical Darwinian evolution, enabling the emergence of the first species.

q-bio.PE↗

arXiv2015

Synchronization as Aggregation: Cluster Kinetics of Pulse-Coupled Oscillators

We consider models of identical pulse-coupled oscillators with global interactions. Previous work showed that under certain conditions such systems always end up in sync, but did not quantify how small clusters of synchronized oscillators progressively coalesce into larger ones. Using tools from the study of aggregation phenomena, we obtain exact results for the time-dependent distribution of cluster sizes as the system evolves from disorder to synchrony.

nlin.AO↗

arXiv2014

Self-organization in soliton modelocked parametric frequency combs

We show that self-organization occurs in the phase dynamics of soliton modelocking in paramet- ric frequency combs. Reduction of the Lugiato-Lefever equation (LLE) to a simpler set of phase equations reveals that this self-organization arises via mechanisms akin to those in the Kuramoto model for synchronization of coupled oscillators. In addition, our simulations show that the phase equations evolve to a broadband phase-locked state, analogous to the soliton formation process in the LLE. Our simplified equations intuitively explain the origin of the pump phase offset in soliton- modelocked parametric frequency combs. They also predict that the phase of the intracavity field undergoes an anti-symmetrization that precedes phase synchronization, and they clarify the role of chaotic states in soliton formation in parametric combs.

nlin.PS↗

arXiv2013

Phase diagram for the Kuramoto model with van Hemmen interactions

We consider a Kuramoto model of coupled oscillators that includes quenched random interactions of the type used by van Hemmen in his model of spin glasses. The phase diagram is obtained analytically for the case of zero noise and a Lorentzian distribution of the oscillators' natural frequencies. Depending on the size of the attractive and random coupling terms, the system displays four states: complete incoherence, partial synchronization, partial antiphase synchronization, and a mix of antiphase and ordinary synchronization.

nlin.CD↗

arXiv2012

Encouraging moderation: Clues from a simple model of ideological conflict

Some of the most pivotal moments in intellectual history occur when a new ideology sweeps through a society, supplanting an established system of beliefs in a rapid revolution of thought. Yet in many cases the new ideology is as extreme as the old. Why is it then that moderate positions so rarely prevail? Here, in the context of a simple model of opinion spreading, we test seven plausible strategies for deradicalizing a society and find that only one of them significantly expands the moderate subpopulation without risking its extinction in the process.

nlin.AO↗

arXiv2011

Conformists and contrarians in a Kuramoto model with identical natural frequencies

We consider a variant of the Kuramoto model, in which all the oscillators are now assumed to have the same natural frequency, but some of them are negatively coupled to the mean field. These "contrarian" oscillators tend to align in antiphase with the mean field, whereas the positively coupled "conformist" oscillators favor an in-phase relationship. The interplay between these effects can lead to rich dynamics. In addition to a splitting of the population into two diametrically opposed factions, the system can also display traveling waves, complete incoherence, and a blurred version of the two-faction state. Exact solutions for these states and their bifurcations are obtained by means of the Watanabe-Strogatz transformation and the Ott-Antonsen ansatz. Curiously, this system of oscillators with identical frequencies turns out to exhibit more complicated dynamics than its counterpart with heterogeneous natural frequencies.

nlin.CD↗

arXiv2010

Analysis of a continuous-time model of structural balance

It is not uncommon for certain social networks to divide into two opposing camps in response to stress. This happens, for example, in networks of political parties during winner-takes-all elections, in networks of companies competing to establish technical standards, and in networks of nations faced with mounting threats of war. A simple model for these two-sided separations is the dynamical system dX/dt = X^2 where X is a matrix of the friendliness or unfriendliness between pairs of nodes in the network. Previous simulations suggested that only two types of behavior were possible for this system: either all relationships become friendly, or two hostile factions emerge. Here we prove that for generic initial conditions, these are indeed the only possible outcomes. Our analysis yields a closed-form expression for faction membership as a function of the initial conditions, and implies that the initial amount of friendliness in large social networks (started from random initial conditions) determines whether they will end up in intractable conflict or global harmony.

nlin.AO↗