Asymptotics of large bound states of localized structures

We analyze stationary fronts connecting uniform and periodic states emerging from a pattern-forming instability. The size of the resulting periodic domains cannot be predicted with weakly nonlinear methods. We show that what determine this size are exponentially small (but exponentially growing in space) terms. These can only be computed by going beyond all orders of the usual multiple-scale expansion. We apply the method to the Swift-Hohenberg equation and derive analytically a snaking bifurcation curve. At each fold of this bifurcation curve, a new pair of peaks is added to the periodic domain, which can thus be seen as a bound state of localized structures. Such scenarios have been reported with optical localized structures in nonlinear cavities and localized buckling

Phase Diagram for the Winfree Model of Coupled Nonlinear Oscillators

In 1967 Winfree proposed a mean-field model for the spontaneous synchronization of chorusing crickets, flashing fireflies, circadian pacemaker cells, or other large populations of biological oscillators. Here we give the first bifurcation analysis of the model, for a tractable special case. The system displays rich collective dynamics as a function of the coupling strength and the spread of natural frequencies. Besides incoherence, frequency locking, and oscillator death, there exist novel hybrid solutions that combine two or more of these states. We present the phase diagram and derive several of the stability boundaries analytically.Comment: 4 pages, 4 figure

Dynamics of delayed-coupled chaotic logistic maps: influence of network topology, connectivity and delay times

We review our recent work on the synchronization of a network of delay-coupled maps, focusing on the interplay of the network topology and the delay times that take into account the finite velocity of propagation of interactions. We assume that the elements of the network are identical ($N$ logistic maps in the regime where the individual maps, without coupling, evolve in a chaotic orbit) and that the coupling strengths are uniform throughout the network. We show that if the delay times are sufficiently heterogeneous, for adequate coupling strength the network synchronizes in a spatially homogeneous steady-state, which is unstable for the individual maps without coupling. This synchronization behavior is referred to as ``suppression of chaos by random delays'' and is in contrast with the synchronization when all the interaction delay times are homogeneous, because with homogeneous delays the network synchronizes in a state where the elements display in-phase time-periodic or chaotic oscillations. We analyze the influence of the network topology considering four different types of networks: two regular (a ring-type and a ring-type with a central node) and two random (free-scale Barabasi-Albert and small-world Newman-Watts). We find that when the delay times are sufficiently heterogeneous the synchronization behavior is largely independent of the network topology but depends on the networks connectivity, i.e., on the average number of neighbors per node.Comment: 5 pages, 7 figures. Also submitted to Pramana: the journal of the Indian Academy of Sciences. To appear in the Proceedings of "Perspectives on Nonlinear Dynamics 2007

Antiphase dynamics and self-pulsing due to a low-frequency spatial population grating in a multimode laser

We study analytically equations that extend the Tang-Statz-deMars rate equations for a multimode Fabry-Perot laser by including the low-spatial-frequency population grating and the inhomogeneous pumping rate along the cavity axis [Quant. Semiclassic Opt. 5, L17 (1997)]. First, we prove the theorem that is the foundation of the antiphase dynamics: The total intensity transients are characterized by only one frequency, the single-mode relaxation oscillation. Second, we study the three-mode laser operation. In this context, we derive analytic expressions for the steady-state intensities, their linear stability, and the bifurcation points. We prove that strictly multimode solutions display a Hopf bifurcation leading to passive Q-switched solutions. Numerically, we have found that these time-periodic regimes may bifurcate to quasiperiodic and chaotic states and that there are many domains of bistability.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Bragg localized structures in a passive cavity with transverse modulation of the refractive index and the pump

We consider a passive optical cavity containing a photonic crystal and a purely absorptive two-level medium. The cavity is driven by a superposition of two coherent beams forming a periodically modulated pump. Using a coupled mode reduction and direct numerical modeling of the full system we demonstrate the existence of bistability between uniformly periodic states, modulational instabilities and localized structures of light. All are found to exist within the conduction band of the photonic material. Moreover, contrary to similar previously found intra-band structures, we show that these localized structures can be truly stationary states.Journal Articleinfo:eu-repo/semantics/publishe

Synchronization of weakly stable oscillators and semiconductor laser arrays

info:eu-repo/semantics/publishe

Japanese oysters in Dutch waters

We study a number of aspects of the colonisation of the Eastern Scheldt by the Japanese Oyster. We formulate and analyse some simple models of the spatial spreading, and determine a rough dependence of the spreading behaviour on parameters. We examine the suggestion of reducing salinity by opening freshwater dams, with the aim of reducing oyster fertility, and make predictions of the effect of such measures. Finally, we present an outline of a large-scale simulation taking into account detailed data on the geometry and sea fl oor properties of the Eastern Scheldt

Japanese oysters in Dutch waters

We study a number of aspects of the colonisation of the Eastern Scheldt by the Japanese Oyster. We formulate and analyse some simple models of the spatial spreading, and determine a rough dependence of the spreading behaviour on parameters. We examine the suggestion of reducing salinity by opening freshwater dams, with the aim of reducing oyster fertility, and make predictions of the effect of such measures. Finally, we present an outline of a large-scale simulation taking into account detailed data on the geometry and sea fl oor properties of the Eastern Scheldt