Showing papers by "Steven H. Strogatz published in 1999"

Time Delay in the Kuramoto Model of Coupled Oscillators

Abstract: We generalize the Kuramoto model of coupled oscillators to allow time-delayed interactions. New phenomena include bistability between synchronized and incoherent states, and unsteady solutions with time-dependent order parameters. We derive exact formulas for the stability boundaries of the incoherent and synchronized states, as a function of the delay, in the special case where the oscillators are identical. The experimental implications of the model are discussed for populations of chirping crickets, where the finite speed of sound causes communication delays, and for physical systems such as coupled phase-locked loops or lasers.

Small-world networks

Abstract: Everyone is familiar with the small-world phenomenon: soon after meeting a stranger, we are often suprised to discover that we have a mutual friend, or that we are somehow linked by a short chain of friends. In this talk, I’ll present evidence that the small-world phenomenon is more than a curiosity of social networks — it is actually a general property of large, sparse networks whose topology is neither completely regular nor completely random. To check this idea, Duncan Watts and I have analyzed three networks of scientific interest: the neural network of the nematode worm C. elegans, the electrical power grid of the western United States, and the collaboration graph of actors in feature films. All three are small worlds, in the sense that the average number of “handshakes” separating any two members is extremely small (close to the theoretical lower limit set by a random graph). Yet at the same time, all three networks exhibit much more local clustering than a random net, demonstrating that they are not random. I’ll also discuss a class of model networks that interpolate between regular lattices and random graphs. Previous theoretical research on complex systems in a wide range of disciplines has focused almost exclusively on networks that are either regular or random. Real networks often lie somewhere in between. Our mathematical model shows that networks in this middle ground tend to exhibit the small-world phenomenon, thanks to the presence of a few long-range edges that link parts of the graph that would otherwise be far apart. Furthermore, we find that when various dynamical systems are coupled in a small-world fashion, they exhibit much greater propagation speed, computational power, and synchronizability than their locally connected, regular counterparts. We explore the implications of these results for simple models of disease spreading, global computation in cellular automata, and collective locking of biological oscillators.

Pattern Formation in Continuous and Coupled Systems : A Survey Volume

Abstract: Rayleigh-Benard convection with rotation at small Prandtl numbers.- Chaotic intermittency of patterns in symmetric systems.- Heteroclinic cycles and phase turbulence.- Hopf bifurcation in anisotropic systems.- Heteroclinic cycles in symmetrically coupled systems.- Symmetry and pattern formation in coupled cell networks.- Spatial hidden symmetries in pattern formation.- Stability boundaries of the dynamic states in pulsating and cellular flames.- A quantitative description of the relaxation of textured patterns.- Forced symmetry breaking: theory and applications.- Spatiotemporal patterns in electrochemical systems.- Memory effects and complex patterns in a catalytic surface reaction.- Bursting mechanisms for hydrodynamical systems.- Bifurcation from periodic solutions with spatiotemporal symmetry.- Resonant pattern formation in a spatially extended chemical system.- Time-dependent pattern formation for two-layer convection.- Localized structures in pattern-forming systems.- Pattern formation in a surface reaction with global coupling.- Dynamical behavior of patterns with Euclidean symmetry.- Pattern selection in a diffusion-reaction system with global or long-range interaction.- Dynamics of kinks and vortices in Josephson-junction arrays.- Josephson junction arrays: Puzzles and prospects.- List of Participants.