On the statistical distribution of first--return times of balls and cylinders in chaotic systems

We study returns in dynamical systems: when a set of points, initially populating a prescribed region, swarms around phase space according to a deterministic rule of motion, we say that the return of the set occurs at the earliest moment when one of these points comes back to the original region. We describe the statistical distribution of these "first--return times" in various settings: when phase space is composed of sequences of symbols from a finite alphabet (with application for instance to biological problems) and when phase space is a one and a two-dimensional manifold. Specifically, we consider Bernoulli shifts, expanding maps of the interval and linear automorphisms of the two dimensional torus. We derive relations linking these statistics with Renyi entropies and Lyapunov exponents.Comment: submitted to Int. J. Bifurcations and Chao

Existence and Stability of Steady Fronts in Bistable CML

We prove the existence and we study the stability of the kink-like fixed points in a simple Coupled Map Lattice for which the local dynamics has two stable fixed points. The condition for the existence allows us to define a critical value of the coupling parameter where a (multi) generalized saddle-node bifurcation occurs and destroys these solutions. An extension of the results to other CML's in the same class is also displayed. Finally, we emphasize the property of spatial chaos for small coupling.Comment: 18 pages, uuencoded PostScript file, J. Stat. Phys. (In press

Adaptive coupling for achieving stable synchronization of chaos

We consider synchronization of coupled chaotic systems and propose an adaptive strategy that aims at evolving the strength of the coupling to achieve stability of the synchronized evolution. We test this idea in a simple configuration in which two chaotic systems are unidirectionally coupled (a sender and a receiver) and we study conditions for the receiver to adaptively synchronize with the sender. Numerical simulations show that, under certain conditions, our strategy is successful in dynamically evolving the coupling strength until it converges to a value that is compatible with synchronization.Comment: 12 Pages, 9 figures, accepted for publication in Physical Review

Detectability of non-differentiable generalized synchrony

Generalized synchronization of chaos is a type of cooperative behavior in directionally-coupled oscillators that is characterized by existence of stable and persistent functional dependence of response trajectories from the chaotic trajectory of driving oscillator. In many practical cases this function is non-differentiable and has a very complex shape. The generalized synchrony in such cases seems to be undetectable, and only the cases, in which a differentiable synchronization function exists, are considered to make sense in practice. We show that this viewpoint is not always correct and the non-differentiable generalized synchrony can be revealed in many practical cases. Conditions for detection of generalized synchrony are derived analytically, and illustrated numerically with a simple example of non-differentiable generalized synchronization.Comment: 8 pages, 8 figures, submitted to PR

Synchronization of dynamical hypernetworks: dimensionality reduction through simultaneous block-diagonalization of matrices

We present a general framework to study stability of the synchronous solution for a hypernetwork of coupled dynamical systems. We are able to reduce the dimensionality of the problem by using simultaneous block-diagonalization of matrices. We obtain necessary and sufficient conditions for stability of the synchronous solution in terms of a set of lower-dimensional problems and test the predictions of our low-dimensional analysis through numerical simulations. Under certain conditions, this technique may yield a substantial reduction of the dimensionality of the problem. For example, for a class of dynamical hypernetworks analyzed in the paper, we discover that arbitrarily large networks can be reduced to a collection of subsystems of dimensionality no more than 2. We apply our reduction techique to a number of different examples, including a class of undirected unweighted hypermotifs of three nodes.Comment: 9 pages, 6 figures, accepted for publication in Phys. Rev.

The stability of adaptive synchronization of chaotic systems

In past works, various schemes for adaptive synchronization of chaotic systems have been proposed. The stability of such schemes is central to their utilization. As an example addressing this issue, we consider a recently proposed adaptive scheme for maintaining the synchronized state of identical coupled chaotic systems in the presence of a priori unknown slow temporal drift in the couplings. For this illustrative example, we develop an extension of the master stability function technique to study synchronization stability with adaptive coupling. Using this formulation, we examine local stability of synchronization for typical chaotic orbits and for unstable periodic orbits within the synchronized chaotic attractor (bubbling). Numerical experiments illustrating the results are presented. We observe that the stable range of synchronism can be sensitively dependent on the adaption parameters, and we discuss the strong implication of bubbling for practically achievable adaptive synchronization.Comment: 21 pages, 6 figure

Continuation of connecting orbits in 3D-ODEs: (I) Point-to-cycle connections

We propose new methods for the numerical continuation of point-to-cycle connecting orbits in 3-dimensional autonomous ODE's using projection boundary conditions. In our approach, the projection boundary conditions near the cycle are formulated using an eigenfunction of the associated adjoint variational equation, avoiding costly and numerically unstable computations of the monodromy matrix. The equations for the eigenfunction are included in the defining boundary-value problem, allowing a straightforward implementation in AUTO, in which only the standard features of the software are employed. Homotopy methods to find connecting orbits are discussed in general and illustrated with several examples, including the Lorenz equations. Complete AUTO demos, which can be easily adapted to any autonomous 3-dimensional ODE system, are freely available.Comment: 18 pages, 10 figure

Isolated ionospheric disturbances as deduced from global GPS network

International audienceWe investigate an unusual class of medium-scale traveling ionospheric disturbances of the nonwave type, isolated ionospheric disturbances (IIDs) that manifest themselves in total electron content (TEC) variations in the form of single aperiodic negative TEC disturbances of a duration of about 10min (the total electron content spikes, TECS). The data were obtained using the technology of global detection of ionospheric disturbances using measurements of TEC variations from a global network of receivers of the GPS. For the first time, we present the TECS morphology for 170 days in 1998?2001. The total number of TEC series, with a duration of each series of about 2.3h (2h18m), exceeded 850000. It was found that TECS are observed in no more than 1?2% of the total number of TEC series mainly in the nighttime in the spring and autumn periods. The TECS amplitude exceeds the mean value of the "background" TEC variation amplitude by a factor of 5?10 as a minimum. TECS represent a local phenomenon with a typical radius of spatial correlation not larger than 500km. The IID-induced TEC variations are similar in their amplitude, form and duration to the TEC response to shock-acoustic waves (SAW) generated during rocket launchings and earthquakes. However, the IID propagation velocity is less than the SAW velocity (800?1000m/s) and are most likely to correspond to the velocity of background medium-scale acoustic-gravity waves, on the order of 100?200m/s. Key words. Ionosphere (ionospheric irregularities, instruments and techniques) - Radio science (ionospheric propagation

Kneadings, Symbolic Dynamics and Painting Lorenz Chaos. A Tutorial

A new computational technique based on the symbolic description utilizing kneading invariants is proposed and verified for explorations of dynamical and parametric chaos in a few exemplary systems with the Lorenz attractor. The technique allows for uncovering the stunning complexity and universality of bi-parametric structures and detect their organizing centers - codimension-two T-points and separating saddles in the kneading-based scans of the iconic Lorenz equation from hydrodynamics, a normal model from mathematics, and a laser model from nonlinear optics.Comment: Journal of Bifurcations and Chaos, 201