Showing papers by "Steven H. Strogatz published in 2008"

Solvable model for chimera states of coupled oscillators

Abstract: Networks of identical, symmetrically coupled oscillators can spontaneously split into synchronized and desynchronized subpopulations. Such chimera states were discovered in 2002, but are not well understood theoretically. Here we obtain the first exact results about the stability, dynamics, and bifurcations of chimera states by analyzing a minimal model consisting of two interacting populations of oscillators. Along with a completely synchronous state, the system displays stable chimeras, breathing chimeras, and saddle-node, Hopf, and homoclinic bifurcations of chimeras.

Distributed synchronization in wireless networks

Abstract: This article has explored history, recent advances, and challenges in distributed synchronization for distributed wireless systems. It is focused on synchronization schemes based on exchange of signals at the physical layer and corresponding baseband processing, wherein analysis and design can be performed using known tools from signal processing. Emphasis has also been given on the synergy between distributed synchronization and distributed estimation/detection problems. Finally, we have touched upon synchronization of nonperiodic (chaotic) signals. Overall, we hope to have conveyed the relevance of the subject and to have provided insight on the open issues and available analytical tools that could inspire further research within the signal processing community.

Stability diagram for the forced Kuramoto model.

Abstract: We analyze the periodically forced Kuramoto model. This system consists of an infinite population of phase oscillators with random intrinsic frequencies, global sinusoidal coupling, and external sinusoidal forcing. It represents an idealization of many phenomena in physics, chemistry, and biology in which mutual synchronization competes with forced synchronization. In other words, the oscillators in the population try to synchronize with one another while also trying to lock onto an external drive. Previous work on the forced Kuramoto model uncovered two main types of attractors, called forced entrainment and mutual entrainment, but the details of the bifurcations between them were unclear. Here we present a complete bifurcation analysis of the model for a special case in which the infinite-dimensional dynamics collapse to a two-dimensional system. Exact results are obtained for the locations of Hopf, saddle-node, and Takens–Bogdanov bifurcations. The resulting stability diagram bears a striking resemblance to that for the weakly nonlinear forced van der Pol oscillator.

Stability diagram for the forced Kuramoto model

Abstract: We analyze the periodically forced Kuramoto model. This system consists of an infinite population of phase oscillators with random intrinsic frequencies, global sinusoidal coupling, and external sinusoidal forcing. It represents an idealization of many phenomena in physics, chemistry and biology in which mutual synchronization competes with forced synchronization. In other words, the oscillators in the population try to synchronize with one another while also trying to lock onto an external drive. Previous work on the forced Kuramoto model uncovered two main types of attractors, called forced entrainment and mutual entrainment, but the details of the bifurcations between them were unclear. Here we present a complete bifurcation analysis of the model for a special case in which the infinite-dimensional dynamics collapse to a two-dimensional system. Exact results are obtained for the locations of Hopf, saddle-node, and Takens-Bogdanov bifurcations. The resulting stability diagram bears a striking resemblance to that for the weakly nonlinear forced van der Pol oscillator.

A Monte Carlo Approach to Joe DiMaggio and Streaks in Baseball

Abstract: We examine Joe DiMaggio's 56-game hitting streak and look at its likelihood, using a number of simple models. And it turns out that, contrary to many people's expectations, an extreme streak, while unlikely in any given year, is not unlikely to have occurred about once within the history of baseball. Surprisingly, however, such a record should have occurred far earlier in baseball history: back in the late 1800's or early 1900's. But not in 1941, when it actually happened.

Complex dynamics of human activity: language, cities, collaboration, and baseball

Abstract: A few areas of human activity are examined here, using a number of different types of mathematical and computational models. First, we examine networks of five languages of the world, with their connectivity derived from the sounds of the words in these languages. We explore the graph-theoretic properties of these networks, finding that these phonological language networks have common properties, and are in turn topologically distinct from other types of complex networks observed in the literature. In addition, we discuss what these common properties imply for how we process language and why natural language is structured the way it is. In addition, by examining the networks of English and Spanish, we explain a surprising difference in processing that was uncovered in some recent experiments, and discuss some more general implications of competition or facilitation between different modes of cognition. We next explore a more macro-scale area of human activity: cities. Superlinear scaling in cities, which appears in sociological quantities such as economic productivity and creative output relative to urban population size, has been observed but not been given a satisfactory theoretical explanation. We provide a model for the superlinear relationship between population size and innovation found in cities, with a reasonable range for the exponent. Next, we examine collaboration and innovation in the scientific world. We attempt to understand how variations in ‘scientific distance’ among collaborators affect the degree to which that collaboration is a productive one. Using both mathematical models and empirical data, we explore the relationship between the scientific or social distance of collaborators and the fruitfulness of their output. Last, we examine Joe DiMaggio's 56-game hitting streak and look at its probability, using a number of simple models. And it turns out that, contrary to many people's expectations, an extreme streak, while unlikely, is not unlikely to have occurred about once within the history of baseball. Surprisingly, however, such a record should have occurred far earlier in baseball history: back in the late 1800's or early 1900's. But not in 1941, when it actually happened.