S. Boccaletti

The synchronization of chaotic systems

Abstract Synchronization of chaos refers to a process wherein two (or many) chaotic systems (either equivalent or nonequivalent) adjust a given property of their motion to a common behavior due to a coupling or to a forcing (periodical or noisy). We review major ideas involved in the field of synchronization of chaotic systems, and present in detail several types of synchronization features: complete synchronization, lag synchronization, generalized synchronization, phase and imperfect phase synchronization. We also discuss problems connected with characterizing synchronized states in extended pattern forming systems. Finally, we point out the relevance of chaos synchronization, especially in physiology, nonlinear optics and fluid dynamics, and give a review of relevant experimental applications of these ideas and techniques.

The control of chaos: Theory and applications

Abstract Control of chaos refers to a process wherein a tiny perturbation is applied to a chaotic system, in order to realize a desirable (chaotic, periodic, or stationary) behavior. We review the major ideas involved in the control of chaos, and present in detail two methods: the Ott–Grebogi–Yorke (OGY) method and the adaptive method. We also discuss a series of relevant issues connected with chaos control, such as the targeting problem, i.e., how to bring a trajectory to a small neighborhood of a desired location in the chaotic attractor in both low and high dimensions, and point out applications for controlling fractal basin boundaries. In short, we describe procedures for stabilizing desired chaotic orbits embedded in a chaotic attractor and discuss the issues of communicating with chaos by controlling symbolic sequences and of synchronizing chaotic systems. Finally, we give a review of relevant experimental applications of these ideas and techniques.

Pattern formation and competition in nonlinear optics

Abstract Pattern formation and competition occur in a nonlinear extended medium if dissipation allows for attracting sets, independently of initial and boundary conditions. This intrinsic patterning emerges from a reaction diffusion dynamics (Turing chemical patterns). In optics, the coupling of an electromagnetic field to a polarizable medium and the presence of losses induce a more general (diffraction-diffusion) mechanism of pattern formation. The presence of a coherent phase propagation may lead to a large set of unstable bands and hence to a richer variety with respect to the chemical case. A review of different experimental situations is presented, including a discussion on suitable indicators which characterize the different regimes. Vistas on perspective new phenomena and applications include an extension to atom optics.

Localized versus delocalized patterns in a nonlinear optical interferometer

We compare the morphology of the spatial structures displayed by an optical system consisting of a liquid crystal light valve (LCLV) in a feedback configuration, as the feedback is gradually tuned from purely diffractive to mixed interferential and diffractive. Different kinds of spatially coherent structures (e.g. hexagons, rolls), as well as localized structures and space-time turbulent patterns are observed. The features of the localized structures change with the parameter setting, and certain regions of the parameter space provide stable clusters of isolated spots (`molecules). Numerical simulations based on a Kerr-like model of the LCLV are in agreement with the experimental observations. We analyse the links between the observed behaviours and the results of a linear-stability analysis of the underlying homogeneous stationary states.

TRANSPORT INDUCED PATTERN SELECTION IN A NONLINEAR OPTICAL SYSTEM

The pattern selection process in a two-dimensional optical system, with a Kerr nonlinearity inserted in an optical feedback loop, is strongly affected by nonlocal interactions. Experimentally, nonlocality is introduced by means of a lateral transport of the feedback signal. As the transport length is increased, the system displays a sequence of transition over different classes of patterns. In the case of purely diffractive feedback, patterns corresponding to both focusing and defocusing nonlinearities are reported. When interference instead of diffraction operates in the feedback loop, new selection rules govern both the scale and the spatial symmetry of the observed structures. In all the cases considered, the stability analysis of the model equations yields theoretical predictions in good agreement with the experimental results.

Predicting Phase Synchronization for Homoclinic Chaos in a CO2 Laser

A novel approach is presented for the reconstruction of phase synchronization phenomena in a chaotic CO2 laser from experimental data. We analyze this laser system in a regime of homoclinic chaos, which is able to phase synchronize with a weak sinusoidal forcing. Our technique recovers the synchronization diagram of the experimental system from only few measurement data sets, thus allowing the prediction of the regime of phase synchronization as well as non‐synchronization in a broad parameter space of forcing frequency and amplitude without further experiments.

Information encoding in a chaotic laser

In this work we describe a simple method of encoding in real time information in the inter-spike intervals of a homoclinic chaotic system. This has been experimentally tested by means of an instantaneous synchronization between the laser intensity of a CO2 laser with feedback in the regime of homoclinic chaos and an external pulsed signal of very low power. The information is previously encoded in the temporal intervals between consecutive pulses of the external signal. The value of the inter-pulse intervals is varied each time a new pulse is generated. (Summary only available)