Showing papers by "Steven H. Strogatz published in 2016"
TL;DR: In this article, the authors review 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.
Abstract: Novelties are part of our daily lives. We constantly adopt new technologies, conceive new ideas, meet new people, and experiment with new situations. Occasionally, we as individual, 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 modeling schemes proposed so far is the idea of looking at evolution as a path in a complex space, physical, conceptual, biological, and 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.
75 citations
TL;DR: In this article, the authors show that self-organization occurs in the phase dynamics of soliton modelocking in parametric frequency combs and that the phase equations evolve to a broadband phase-locked state, analogous to the soliton formation process in the LLE.
Abstract: 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.
27 citations
TL;DR: A mean-field model of coupled phase oscillators with quenched disorder in the coupling strengths and natural frequencies finds that the system spontaneously synchronizes, once the width of the frequency distribution falls below a critical value.
Abstract: 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.
22 citations
TL;DR: A variant of a model proposed by Kuramoto, Shinomoto, and Sakaguchi for a large population of coupled oscillatory and excitable elements 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.
Abstract: 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.
20 citations
TL;DR: This analysis yields the exact prefactors in the scaling laws, which also match the numerics, in the Kuramoto model of coupled oscillators where the number of oscillators, N, is large but finite, and the oscillators' natural frequencies are evenly spaced on a given interval.
Abstract: We study phase locking in the Kuramoto model of coupled oscillators in the special case where the number of oscillators, $N$, is large but finite, and the oscillators' natural frequencies are evenly spaced on a given interval. In this case, stable phase-locked solutions are known to exist if and only if the frequency interval is narrower than a certain critical width, called the locking threshold. For infinite $N$, the exact value of the locking threshold was calculated 30 years ago; however, the leading corrections to it for finite $N$ have remained unsolved analytically. Here we derive an asymptotic formula for the locking threshold when $N\ensuremath{\gg}1$. The leading correction to the infinite-$N$ result scales like either ${N}^{\ensuremath{-}3/2}$ or ${N}^{\ensuremath{-}1}$, depending on whether the frequencies are evenly spaced according to a midpoint rule or an end-point rule. These scaling laws agree with numerical results obtained by Paz\'o [D. Paz\'o, Phys. Rev. E 72, 046211 (2005)]. Moreover, our analysis yields the exact prefactors in the scaling laws, which also match the numerics.
13 citations
TL;DR: It is shown that whether a ring synchronizes more readily than a chain usually does, but not always, and that the ratio between the locking thresholds of a ring and its matched chain must be derived.
Abstract: We present a case study of how topology can affect synchronization. Specifically, we consider arrays of phase oscillators coupled in a ring or a chain topology. Each ring is perfectly matched to a chain with the same initial conditions and the same random natural frequencies. The only difference is their boundary conditions: periodic for a ring and open for a chain. For both topologies, stable phase-locked states exist if and only if the spread or "width" of the natural frequencies is smaller than a critical value called the locking threshold (which depends on the boundary conditions and the particular realization of the frequencies). The central question is whether a ring synchronizes more readily than a chain. We show that it usually does, but not always. Rigorous bounds are derived for the ratio between the locking thresholds of a ring and its matched chain, for a variant of the Kuramoto model that also includes a wider family of models.
12 citations
TL;DR: In this paper, a mean-field model of coupled phase oscillators with quenched disorder in the natural frequencies and coupling strengths was considered and the effect of this correlation on the system's phase coherence was explored.
Abstract: 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 critical width γc in the frequency distribution below which the system spontaneously synchronizes. Moreover, this γc is independent of p. Hence, our model and the traditional Kuramoto model (recovered when p = 1) have the same critical width γc. 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.
9 citations
TL;DR: This work considers a mean-field model of coupled phase oscillators with quenched disorder in the natural frequencies and coupling strengths and finds that the model belongs to the same universality class as the Kuramoto model with deterministically chosen natural frequencies for the case of p < 1.
Abstract: 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 $\gamma_c$ in the frequency distribution below which the system spontaneously synchronizes. Moreover, this $\gamma_c$ is independent of $p$. Hence, our model and the traditional Kuramoto model (recovered when $p=1$) have the same critical width $\gamma_c$. 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 $p<1$.
9 citations
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TL;DR: In this article, the authors compute the shareability curves for each city, and find that a natural rescaling collapses them onto a single, universal curve, and explain this scaling law theoretically with a simple model that predicts the potential for ride sharing in any city, using a few basic urban quantities and no adjustable parameters.
Abstract: Sharing rides could drastically improve the efficiency of car and taxi transportation. Unleashing such potential, however, requires understanding how urban parameters affect the fraction of individual trips that can be shared, a quantity that we call shareability. Using data on millions of taxi trips in New York City, San Francisco, Singapore, and Vienna, we compute the shareability curves for each city, and find that a natural rescaling collapses them onto a single, universal curve. We explain this scaling law theoretically with a simple model that predicts the potential for ride sharing in any city, using a few basic urban quantities and no adjustable parameters. Accurate extrapolations of this type will help planners, transportation companies, and society at large to shape a sustainable path for urban growth.
1 citations