Ergodic projection for quantum dynamical semigroups

This paper deals with problems concerning ergodic projection for semigroups of bounded linear mappings on aW ^*-algebra. A criterion for the normality of this projection is given and the structure of the fixed-point space is analyzed.     

Permutation complexity of interacting dynamical systems

The coupling complexity index is an information measure introduced within the framework of ordinal symbolic dynamics. This index is used to characterize the complexity of the relationship between dynamical system components. In this work, we clarify the meaning of the coupling complexity by discussing in detail some cases leading to extreme values, and present examples using synthetic data to describe its properties. We also generalize the coupling complexity index to the multivariate case and derive a number of important properties by exploiting the structure of the symmetric group. The applicability of this index to the multivariate case is demonstrated with a real-world data example. Finally, we define the coupling complexity rate of random and deterministic time series. Some formal results about the multivariate coupling complexity index have been collected in an Appendix.     

Functional holography analysis: simplifying the complexity of dynamical networks

We present a novel functional holography (FH) analysis devised to study the dynamics of task-performing dynamical networks. The latter term refers to networks composed of dynamical systems or elements, like gene networks or neural networks. The new approach is based on the realization that task-performing networks follow some underlying principles that are reflected in their activity. Therefore, the analysis is designed to decipher the existence of simple causal motives that are expected to be embedded in the observed complex activity of the networks under study. First we evaluate the matrix of similarities (correlations) between the activities of the network's components. We then perform collective normalization of the similarities (or affinity transformation) to construct a matrix of functional correlations. Using dimension reduction algorithms on the affinity matrix, the matrix is projected onto a principal three-dimensional space of the leading eigenvectors computed by the algorithm. To retrieve back information that is lost in the dimension reduction, we connect the nodes by colored lines that represent the level of the similarities to construct a holographic network in the principal space. Next we calculate the activity propagation in the network (temporal ordering) using different methods like temporal center of mass and cross correlations. The causal information is superimposed on the holographic network by coloring the nodes locations according to the temporal ordering of their activities. First, we illustrate the analysis for simple, artificially constructed examples. Then we demonstrate that by applying the FH analysis to modeled and real neural networks as well as recorded brain activity, hidden causal manifolds with simple yet characteristic geometrical and topological features are deciphered in the complex activity. The term "functional holography" is used to indicate that the goal of the analysis is to extract the maximum amount of functional information about the dynamical network as a whole unit.     

Changing dynamical complexity with time delay in coupled fiber laser oscillators

We investigate the complexity of the dynamics of two mutually coupled systems with internal delays and vary the coupling delay over 4 orders of magnitude. Karhunen-Loève decomposition of spatiotemporal representations of fiber laser intensity data is performed to examine the eigenvalue spectrum and significant orthogonal modes. We compute the Shannon information from the eigenvalue spectra to quantify the dynamical complexity. A reduction in complexity occurs for short coupling delays while a logarithmic growth is observed as the coupling delay is increased.     

Detecting dynamical complexity changes in time series using the base-scale entropy

Timely detection of dynamical complexity changes in natural and man-made systems has deep scientific and practical meanings. We introduce a complexity measure for time series： the base-scale entropy. The definition directly applies to arbitrary real-word data. We illustrate our method on a practical speech signal and in a theoretical chaotic system. The results show that the simple and easily calculated measure of base-scale entropy can be effectively used to detect qualitative and quantitative dynamical changes.     

Adaptive complete synchronization of chaotic dynamical network with unknown and mismatched parameters

In this paper, an adaptive controller is designed to synchronize the chaotic dynamical network with unknown and mismatched parameters. Based on the invariance principle of differential equations, some generic sufficient conditions for asymptotic synchronization are obtained. In order to demonstrate the effectiveness of the proposed method, an example is provided and numerical simulations are performed. The numerical results show that our control scheme is very effective and robust against the weak noise.     

Reiterated Ergodic Algebras and Applications

We redefine the homogenization algebras without requiring the separability assumption. We show that this enables one to treat more complicated homogenization problems than those solved by the previous theory. In particular we exhibit an example of algebra which, contrary to the algebra of almost periodic functions, induces no homogenization algebra. We prove some general compactness results which are then applied to the resolution of some homogenization problems related to the generalized Reynolds type equations and to some nonlinear hyperbolic equations.     

Ergodic Jacobi Matrices and Conformal Maps

We study structural properties of the Lyapunov exponent γ and the density of states k for ergodic (or just invariant) Jacobi matrices in a general framework. In this analysis, a central role is played by the function w = − γ + iπ k as a conformal map between certain domains. This idea goes back to Marchenko and Ostrovskii, who used this device in their analysis of the periodic problem.     

Infinite Random Matrices and Ergodic Measures

We introduce and study a 2-parameter family of unitarily invariant probability measures on the space of infinite Hermitian matrices. We show that the decomposition of a measure from this family on ergodic components is described by a determinantal point process on the real line. The correlation kernel for this process is explicitly computed. At certain values of parameters the kernel turns into the well-known sine kernel which describes the local correlation in Circular and Gaussian Unitary Ensembles. Thus, the random point configuration of the sine process is interpreted as the random set of “eigenvalues” of infinite Hermitian matrices distributed according to the corresponding measure. Received: 22 January 2001 / Accepted: 30 May 2001     

Infinite Ergodic Theory and Non-extensive Entropies

We recapitulate results from the infinite ergodic theory that are relevant to the theory of non-extensive entropies. In particular, we recall that the Lyapunov exponent of the corresponding systems is zero and that the deviation between neighboring trajectories does not necessarily grow polynomially. Nonetheless, as we show, no single quantity can describe this subexponential growth, the generalized q-exponential exp _q being, in particular, ruled out. We also revisit a number of dynamical systems preserving nonfinite ergodic measure.     

Invariant Observables and the Individual Ergodic Theorem

The notion of the almost everywhere equality of observables is introduced. The limit of Cesaro means is an invariant observable with respect to this notion.     

Ergodic properties of equilibrium states

Possible ergodic properties of Gibbs states are discussed by constructing a number of examples. In particular existence of Gibbs states which are mixing but not extremal is shown.     

Iterative procedure for analysis of linear dynamical stochastic systems with fluctuating parameters

An iterative procedure for the analysis of linear dynamical stochastic systems with fluctuating parameters is described. We propose an algorithm for obtaining an ensemble-averaged solution of a dynamical equation with arbitrary fluctuation statistics. Institute of Applied Optics, Belarus Academy of Sciences, Mogilev, Belarus. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 40, No. 11, pp. 1448–1452, November, 1997.     

Duality Theorems in Ergodic Transport

We analyze several problems of Optimal Transport Theory in the setting of Ergodic Theory. In a certain class of problems we consider questions in Ergodic Transport which are generalizations of the ones in Ergodic Optimization. Another class of problems is the following: suppose ?? is the shift acting on Bernoulli space X={1,2,??,d}^?, and, consider a fixed continuous cost function c:X×X???. Denote by ?? the set of all Borel probabilities ?? on X×X, such that, both its x and y marginals are ??-invariant probabilities. We are interested in the optimal plan ?? which minimizes ??c? d?? among the probabilities in ??. We show, among other things, the analogous Kantorovich Duality Theorem. We also analyze uniqueness of the optimal plan under generic assumptions on c. We investigate the existence of a dual pair of Lipschitz functions which realizes the present dual Kantorovich problem under the assumption that the cost is Lipschitz continuous. For continuous costs c the corresponding results in the Classical Transport Theory and in Ergodic Transport Theory can be, eventually, different. We also consider the problem of approximating the optimal plan ?? by convex combinations of plans such that the support projects in periodic orbits.     

Ergodic theorems for reaction-diffusion processes

New sufficient conditions are given for the ergodicity of reaction-diffusion processes which improve both Neuhauser's recent result and the present author's previous result. In the main criterion; contrary to the previous ones, the pure birth rate of the reaction plays a critical role. To do this, a new but natural coupling is introduced. It is proved that this coupling is the best one in some sense. One of the main results says that the reaction-diffusion processes are ergodic for all large enough pure birth rates.     

Ergodic stream-lines in steady convection

Stream-lines of steady Rayleigh-Bénard convection with square planform are displayed using Poincaré maps. As the second order mode becomes more important the flow becomes ergodic from the boundaries inward, like a perturbed, integrable hamiltonian system.     

Ergodic Properties of Microcanonical Observables

The problem of the existence of a strong stochasticity threshold in the FPU- model is reconsidered, using suitable microcanonical observables of thermodynamic nature, like the temperature and the specific heat. Explicit expressions for these observables are obtained by exploiting rigorous methods of differential geometry. Measurements of the corresponding temporal autocorrelation functions locate the threshold at a finite value of the energy density, which is independent of the number of degrees of freedom.