Showing papers by "Steven H. Strogatz published in 2009"

Exact results for the Kuramoto model with a bimodal frequency distribution.

Abstract: We analyze a large system of globally coupled phase oscillators whose natural frequencies are bimodally distributed. The dynamics of this system has been the subject of long-standing interest. In 1984 Kuramoto proposed several conjectures about its behavior; ten years later, Crawford obtained the first analytical results by means of a local center manifold calculation. Nevertheless, many questions have remained open, especially about the possibility of global bifurcations. Here we derive the system's stability diagram for the special case where the bimodal distribution consists of two equally weighted Lorentzians. Using an ansatz recently discovered by Ott and Antonsen, we show that in this case the infinite-dimensional problem reduces exactly to a flow in four dimensions. Depending on the parameters and initial conditions, the long-term dynamics evolves to one of three states: incoherence, where all the oscillators are desynchronized; partial synchrony, where a macroscopic group of phase-locked oscillators coexists with a sea of desynchronized ones; and a standing wave state, where two counter-rotating groups of phase-locked oscillators emerge. Analytical results are presented for the bifurcation boundaries between these states. Similar results are also obtained for the case in which the bimodal distribution is given by the sum of two Gaussians.

Identical phase oscillators with global sinusoidal coupling evolve by Mobius group action.

Abstract: Systems of N identical phase oscillators with global sinusoidal coupling are known to display low-dimensional dynamics. Although this phenomenon was first observed about 20 years ago, its underlying cause has remained a puzzle. Here we expose the structure working behind the scenes of these systems by proving that the governing equations are generated by the action of the Mobius group, a three-parameter subgroup of fractional linear transformations that map the unit disk to itself. When there are no auxiliary state variables, the group action partitions the N-dimensional state space into three-dimensional invariant manifolds (the group orbits). The N−3 constants of motion associated with this foliation are the N−3 functionally independent cross ratios of the oscillator phases. No further reduction is possible, in general; numerical experiments on models of Josephson junction arrays suggest that the invariant manifolds often contain three-dimensional regions of neutrally stable chaos.

Energy landscape of social balance.

Abstract: We model a close-knit community of friends and enemies as a fully connected network with positive and negative signs on its edges. Theories from social psychology suggest that certain sign patterns are more stable than others. This notion of social "balance" allows us to define an energy landscape for such networks. Its structure is complex: numerical experiments reveal a landscape dimpled with local minima of widely varying energy levels. We derive rigorous bounds on the energies of these local minima and prove that they have a modular structure that can be used to classify them.

Phase oscillators with global sinusoidal coupling evolve by Mobius group action

Abstract: Systems of N identical phase oscillators with global sinusoidal coupling are known to display low-dimensional dynamics. Although this phenomenon was first observed about 20 years ago, its underlying cause has remained a puzzle. Here we expose the structure working behind the scenes of these systems, by proving that the governing equations are generated by the action of the Mobius group, a three-parameter subgroup of fractional linear transformations that map the unit disc to itself. When there are no auxiliary state variables, the group action partitions the N-dimensional state space into three-dimensional invariant manifolds (the group orbits). The N-3 constants of motion associated with this foliation are the N-3 functionally independent cross ratios of the oscillator phases. No further reduction is possible, in general; numerical experiments on models of Josephson junction arrays suggest that the invariant manifolds often contain three-dimensional regions of neutrally stable chaos.

Superlinear scaling for innovation in cities.

Abstract: 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. Here we provide a network model for the superlinear relationship between population size and innovation found in cities, with a reasonable range for the exponent.

The Structure of Phonological Networks Across Multiple Languages

Abstract: The network characteristics based on the phonological similarities in the lexicons of several languages were examined. These languages differed widely in their history and linguistic structure, but commonalities in the network characteristics were observed. These networks were also found to be different from other networks studied in the literature. The properties of these networks suggest explanations for various aspects of linguistic processing and hint at deeper organization within human language.

Invariant submanifold for series arrays of Josephson junctions.

Abstract: We study the nonlinear dynamics of series arrays of Josephson junctions in the large-N limit, where N is the number of junctions in the array. The junctions are assumed to be identical, overdamped, driven by a constant bias current, and globally coupled through a common load. Previous simulations of such arrays revealed that their dynamics are remarkably simple, hinting at the presence of some hidden symmetry or other structure. These observations were later explained by the discovery of N−3 constants of motion, the choice of which confines the resulting flow in phase space to a low-dimensional invariant manifold. Here we show that the dimensionality can be reduced further by restricting attention to a special family of states recently identified by Ott and Antonsen. In geometric terms, the Ott–Antonsen ansatz corresponds to an invariant submanifold of dimension one less than that found earlier. We derive and analyze the flow on this submanifold for two special cases: an array with purely resistive loadin...

The Calculus of Friendship: What a Teacher and a Student Learned about Life while Corresponding about Math

Abstract: Prologue ix Continuity (1974-75) 1 Pursuit (1976) 8 Relativity (1977) 13 Irrationality (1978-79) 23 Shifts (1980-89) 34 Proof on a Place Mat (March 1989) 42 The Monk and the Mountain (1989-90) 71 Randomness (1990-91) 84 Infinity and Limits (1991) 94 Chaos (1992-95) 107 Celebration (1996-99) 115 The Path of Quickest Descent (2000-2003) 118 Bifurcation (2004) 128 Hero's Formula (2005-Present) 140 Acknowledgments 155 Further Reading 157 Bibliography 161 Photography Credits 163 Index of Math Problems 165

Sinusoidally coupled phase oscillators evolve by Mobius group action

Abstract: Systems of N identical phase oscillators with global sinusoidal coupling are known to display low-dimensional dynamics. Although this phenomenon was first observed about 20 years ago, its underlying cause has remained a puzzle. Here we expose the structure working behind the scenes of these systems, by proving that the governing equations are generated by the action of the Mobius group, a three-parameter subgroup of fractional linear transformations that map the unit disc to itself. When there are no auxiliary state variables, the group action partitions the N-dimensional state space into three-dimensional invariant manifolds (the group orbits). The N-3 constants of motion associated with this foliation are the N-3 functionally independent cross ratios of the oscillator phases. No further reduction is possible, in general; numerical experiments on models of Josephson junction arrays suggest that the invariant manifolds often contain three-dimensional regions of neutrally stable chaos.

The Calculus of Friendship

Abstract: For the past thirty years I've been corresponding with my high school calculus teacher, Mr. Don Joffray. During that time, he went from the prime of his career to retirement, competed in whitewater kayak at the international level, and lost a son. I matured from teenage math geek to Ivy League professor, suffered the sudden death of a parent, and blundered into a marriage destined to fail. What's remarkable is not that any of this took place—such ups and downs are to be expected in three decades of life—but rather that so little of it is discussed in the letters. Instead, our correspondence , and our friendship itself, is based almost entirely on a shared love of calculus. A s academics, many of us like to isolate a piece of the world to study: an important social issue, a central philosophical problem, a key moment in history. We know we're oversimplifying but we do it anyway—it's the only way to make progress, and what's more, our little worlds are often more beautiful than the real one. In the following excerpt, you'll meet a teacher and a student at the beginning of what would turn into a 30-year correspondence and friendship based on a mutual love of math problems. It may seem an alien world, but it's just another little haven of thought, a place where all problems have solu-tions—except, as you'll see, when they don't. Steven Strogatz is the Jacob Gould Schurman Professor of Applied Mathematics at Cornell University. He works in the areas of chaos and complexity theory, often on topics inspired by the curiosities of everyday life. His latest book, The Calculus of Friendship: What a Teacher and a Student Learned about