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Applied Delay Differential Equations

16 Apr 2009-
TL;DR: In this article, Bernoulli's equation and phase equations are used to describe the phase equation of phase equation.Stability.Vibrations.Vectors. Lasers.
Abstract: Stability.- Biology.- Bernoulli's equation.- Chemistry.- Mechanical vibrations.- Lasers.- Phase equations.
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Journal Article•DOI•
TL;DR: This work reports an optoelectronic implementation of reservoir computing based on a recently proposed architecture consisting of a single non linear node and a delay line that is sufficiently fast for real time information processing.
Abstract: Reservoir computing is a recently introduced, highly efficient bio-inspired approach for processing time dependent data. The basic scheme of reservoir computing consists of a non linear recurrent dynamical system coupled to a single input layer and a single output layer. Within these constraints many implementations are possible. Here we report an optoelectronic implementation of reservoir computing based on a recently proposed architecture consisting of a single non linear node and a delay line. Our implementation is sufficiently fast for real time information processing. We illustrate its performance on tasks of practical importance such as nonlinear channel equalization and speech recognition, and obtain results comparable to state of the art digital implementations.

606 citations

Journal Article•DOI•
TL;DR: In this paper, a review of the properties of single and two delay-coupled laser systems is presented, with a particular emphasis on emerging complex behavior, deterministic chaos, synchronization phenomena, and application of these properties that range from encrypted communication and fast random bit sequence generators to bioinspired information processing.
Abstract: Complex phenomena in photonics, in particular, dynamical properties of semiconductor lasers due to delayed coupling, are reviewed. Although considered a nuisance for a long time, these phenomena now open interesting perspectives. Semiconductor laser systems represent excellent test beds for the study of nonlinear delay-coupled systems, which are of fundamental relevance in various areas. At the same time delay-coupled lasers provide opportunities for photonic applications. In this review an introduction into the properties of single and two delay-coupled lasers is followed by an extension to network motifs and small networks. A particular emphasis is put on emerging complex behavior, deterministic chaos, synchronization phenomena, and application of these properties that range from encrypted communication and fast random bit sequence generators to bioinspired information processing.

494 citations

Journal Article•DOI•
TL;DR: A powerful and highly controllable experiment based on an optoelectronic delayed feedback applied to a wavelength tuneable semiconductor laser, with which a wide variety of chimera patterns can be accurately investigated and interpreted, and a cascade of higher-order chimeras as a pattern transition from N to N+1 clusters of chaoticity is uncovered.
Abstract: A chimera state is a rich and fascinating class of self-organized solutions developed in high-dimensional networks. Necessary features of the network for the emergence of such complex but structured motions are non-local and symmetry breaking coupling. An accurate understanding of chimera states is expected to bring important insights on deterministic mechanism occurring in many structurally similar high-dimensional dynamics such as living systems, brain operation principles and even turbulence in hydrodynamics. Here we report on a powerful and highly controllable experiment based on an optoelectronic delayed feedback applied to a wavelength tuneable semiconductor laser, with which a wide variety of chimera patterns can be accurately investigated and interpreted. We uncover a cascade of higher-order chimeras as a pattern transition from N to N+1 clusters of chaoticity. Finally, we follow visually, as the gain increases, how chimera state is gradually destroyed on the way to apparent turbulence-like system behaviour. Chimera states are a class of self-organized solutions of high-dimensional networks with non-local and symmetry breaking coupling. Here the authors study the chimera patterns generated in a non-linear optical setup and uncover the transition between chimera orders as a pattern across clusters of chaoticity.

176 citations

Journal Article•DOI•
TL;DR: This work studies two different frameworks for delay-adaptive prediction-based control design for nonlinear systems with unknown long actuator delay, and proves global asymptotic convergence of the proposed adaptive controller and local regulation.
Abstract: We present a systematic delay-adaptive prediction-based control design for nonlinear systems with unknown long actuator delay. Our approach is based on the representation of the constant actuator delay as a transport Partial Differential Equation (PDE) in which the convective speed is inversely proportional to the unknown delay. We study two different frameworks, assuming first that the actuator state is measured and relaxing afterward. For the full-state feedback case, we prove global asymptotic convergence of the proposed adaptive controller while, in the second case, replacing the actuator state by its adaptive estimate, we prove local regulation. The relevance of the obtained results are illustrated by simulations of a biological activator/repressor system.

143 citations

Journal Article•DOI•
TL;DR: In this article, a reaction diffusion model with logistic type growth, nonlocal delay effect and Dirichlet boundary condition is considered, and combined effect of the time delay and nonlocal spatial dispersal provides a more realistic way of modeling the complex spatiotemporal behavior.

136 citations