Manuel A. Matias

Dynamics of localized and patterned structures in the Lugiato-Lefever equation determine the stability and shape of optical frequency combs

This research was supported by the Research Foundation-Flanders (FWO), by the Spanish MINECO, and FEDER under Grants FISICOS (Grant No. FIS2007-60327) and INTENSE@COSYP (Grant No. FIS2012-30634), by Comunitat Autonoma de les Illes Balears, by the Research Council of the Vrije Universiteit Brussel (VUB), and by the Belgian Science Policy Office (BelSPO) under Grant No. IAP 7-35. S. Coen also acknowledges the support of the Marsden Fund of the Royal Society of New Zealand.

TRANSITION TO HIGH-DIMENSIONAL CHAOS THROUGH QUASIPERIODIC MOTION

In this contribution we report on a transition to high-dimensional chaos through three-frequency quasiperiodic behavior. The resulting chaotic attractor has a one positive and two null Lyapunov exponents. The transition occurs at the point at which two symmetry related three-dimensional tori merge in a crisis-like bifurcation. The route can be summarized as: 2D torus → 3D torus → high-dimensional chaotic attractor.

Interaction of chaotic rotating waves in coupled rings of chaotic cells

Abstract The interaction of two chaotic rotating waves of the type recently reported by Matias et al. [Europhys. Lett. 37 (1997) 379] is studied experimentally with arrays of non-linear electronic circuits arranged in ring geometries. Unidirectional coupling is assumed for the cell-to-cell coupling within the same ring, but between rings, cells are coupled diffusively. Depending on the relative sense of driving, competition between a rotating chaotic wave and a global synchronized state has been observed. The results are rationalized by means of a linear stability analysis around the uniform synchronized behavior, where the circulant symmetry of the system allows to express the problem as a superposition of a series of Fourier modes.

Desynchronization Transitions in Rings of Coupled Chaotic Oscillators

Rings of chaotic oscillators coupled unidirectionally through driving are studied. While synchronization is observed for small sizes of the ring, beyond a certain critical size a desynchronizing transition occurs. In the two examples studied here the system exhibits a transition to periodic rotating waves for rings of Lorenz systems, while one finds a sort of chaotic rotating waves when Chuas circuit is used.

Chaotic synchronization in small assemblies of driven Chua's circuits

Chaotic synchronization is studied in experiments performed on dynamic arrays of Chuas circuits that are connected by using a recently introduced driving method especially suited for the design of such arrays. Namely, the driven circuit has the same number of energy storage elements as the driving circuit. The experimental results, which are supported by theoretical analysis, are different depending on the geometric arrangement of the array. In the case of linear arrays, the first circuit always imposes its behavior on the rest of the chain at a finite velocity. Instead, in the case of ring geometries, the chaotic synchronized state is only stable up to a certain size of the ring. Beyond this critical size a desynchronizing bifurcation occurs, leading to a chaotic rotating wave that travels through the array. This instability is explained by performing an analysis in terms of modes.

Formation of localized structures in bistable systems through nonlocal spatial coupling. I. General framework

The present work studies the influence of nonlocal spatial coupling on the existence of localized structures in one-dimensional extended systems. We consider systems described by a real field with a nonlocal coupling that has a linear dependence on the field. Leveraging spatial dynamics we provide a general framework to understand the effect of the nonlocality on the shape of the fronts connecting two stable states. In particular we show that nonlocal terms can induce spatial oscillations in the front tails, allowing for the creation of localized structures, that emerge from pinning between two fronts. In parameter space the region where fronts are oscillatory is limited by three transitions: the modulational instability of the homogeneous state, the Belyakov-Devaney transition in which monotonic fronts acquire spatial oscillations with infinite wavelength, and a crossover in which monotonically decaying fronts develop spatial oscillations with a finite wavelength. We show how these transitions are organized by codimension 2 and 3 points and illustrate how by changing the parameters of the nonlocal coupling it is possible to bring the system into the region where localized structures can be formed.

Formation of localized structures in bistable systems through nonlocal spatial coupling II: The nonlocal Ginzburg Landau Equation

We study the influence of a linear nonlocal spatial coupling on the interaction of fronts connecting two equivalent stable states in the prototypical 1-dimensional real Ginzburg-Landau equation. While for local coupling the fronts are always monotonic and therefore the dynamical behavior leads to coarsening and the annihilation of pairs of fronts, nonlocal terms can induce spatial oscillations in the front, allowing for the creation of localized structures, emerging from pinning between two fronts. We show this for three different nonlocal influence kernels. The first two, mod-exponential and Gaussian, are positive definite and decay exponentially or faster, while the third one, a Mexican-hat kernel, is not positive definite.

Effect of a variable delay in delayed dynamical systems

We report numerical evidence of the eects of a periodic modulation in the delay time of a delayed dynamical system. By referring to a Mackey{Glass equation and by adding a modulation in the delay time, we describe how the solution of the system passes from being chaotic to shadow periodic states. We analyze this transition for both sinusoidal and sawtooth wave modulations, and we give, in the latter case, the relationship between the period of the shadowed orbit and the amplitude of the modulation. Future goals and open questions are highlighted.

Effects of a localized beam on the dynamics of excitable cavity solitons

(Dated: August 19, 2008)We study the dynamical behavior of dissipative solitons in an optical cavity filled with a Kerrmedium when a localized beam is applied on top of the homogeneous pumping. In particular, wereport on the excitability regime that cavity solitons exhibits which is emergent property since thesystem is not locally excitable. The resulting scenario differs in an important way from the caseof a purely homogeneous pump and now two different excitable regimes, both Class I, are shown.The whole scenario is presented and discussed, showing that it is organized by three codimension-2points. Moreover, the localized beam can be used to control important features, such as the excitablethreshold, improving the possibilities for the experimental observation of this phenomenon.

An experimental setup for studying the effect of noise on Chua's circuit

The analog simulation of nonlinear dynamical systems is advantageous in some cases, i.e., when compared with the study using digital computers and, in particular, when one wishes to investigate the role of noise in these systems. In the present work we introduce two different methods of introducing a noise component in the most widely used chaotic circuit, namely, Chuas circuit, and apply these methods to study the effect of noise on identically driven chaotic circuits.