Ergodic Theory Of Dynamical Systems Research Articles

LOSIK CLASSES FOR CODIMENSION-ONE FOLIATIONS

AbstractFollowing Losik’s approach to Gelfand’s formal geometry, certain characteristic classes for codimension-one foliations coming from the Gelfand-Fuchs cohomology are considered. Sufficient conditions for nontriviality in terms of dynamical properties of generators of the holonomy groups are found. The nontriviality for the Reeb foliations is shown; this is in contrast with some classical theorems on the Godbillon-Vey class; for example, the Mizutani-Morita-Tsuboi theorem about triviality of the Godbillon-Vey class of foliations almost without holonomy is not true for the classes under consideration. It is shown that the considered classes are trivial for a large class of foliations without holonomy. The question of triviality is related to ergodic theory of dynamical systems on the circle and to the problem of smooth conjugacy of local diffeomorphisms. Certain classes are obstructions for the existence of transverse affine and projective connections.

Search strategy in a complex and dynamic environment: the MH370 case

Search and detection of objects on the ocean surface is a challenging task due to the complexity of the drift dynamics and lack of known optimal solutions for the path of the search agents. This challenge was highlighted by the unsuccessful search for Malaysian Flight 370 (MH370) which disappeared on March 8, 2014. In this paper, we propose an improvement of a search algorithm rooted in the ergodic theory of dynamical systems which can accommodate complex geometries and uncertainties of the drifting search areas on the ocean surface. We illustrate the effectiveness of this algorithm in a computational replication of the conducted search for MH370. We compare the algorithms using many realizations with random initial positions, and analyze the influence of the stochastic drift on the search success. In comparison to conventional search methods, the proposed algorithm leads to an order of magnitude improvement in success rate over the time period of the actual search operation. Simulations of the proposed search control also indicate that the initial success rate of finding debris increases in the event of delayed search commencement. This is due to the existence of convergence zones in the search area which leads to local aggregation of debris in those zones and hence reduction of the effective size of the area to be searched.

Complexity and Stability in Biological Systems

The hypothesis that a positive correlation exists between the complexity of a biological system, as described by its connectance, and its stability, as measured by its ability to recover from disturbance, derives from the investigations of the physiologists, Bernard and Cannon, and the ecologist Elton. Studies based on the ergodic theory of dynamical systems and the theory of large deviations have furnished an analytic support for this hypothesis. Complexity in this context is described by the mathematical object evolutionary entropy, stability is characterized by the rate at which the system returns to its stable conditions (steady state or periodic attractor) after a random perturbation of its robustness. This article reviews the analytical basis of the entropy — robustness theorem — and invokes studies of genetic regulatory networks to provide empirical support for the correlation between complexity and stability. Earlier investigations based on numerical studies of random matrix models and the notion of local stability have led to the claim that complex ecosystems tend to be more dynamically fragile. This article elucidates the basis for this claim which is largely inconsistent with the empirical observations of Bernard, Cannon and Elton. Our analysis thus resolves a long standing controversy regarding the relation between complex biological systems and their capacity to recover from perturbations. The entropy-robustness principle is a mathematical proposition with implications for understanding the basis for the large variations in stability observed in biological systems having evolved under different environmental conditions.

DYNAMICAL SYSTEMS WITH SPECIFICATION

Abstract. In this paper we prove that C 1 -generically, if a difieo-morphism f on a closed C 1 manifold M satisfles weak speciflcationon a locally maximal set ⁄ ‰ M then ⁄ is hyperbolic for f . As acorollary we obtain that C 1 -generically, every difieomorphism withweak speciflcation is Anosov. 1. IntroductionThe notion of the speciflcation was introduced by Bowen [3] to con-struct the equilibrium state and make a Markov partition of Axiom A difieomorphisms. After that the notion of weak speciflcation was intro-duced by Ruelle [10] to study variational principle for Z „ -action. Thenotions have turned out to be important in the study of ergodic theoryof dynamical systems (for more details, see [2, 3, 5, 7, 8, 10, 12, 13]).The deflnition of speciflcation (or weak speciflcation) seems to becomplicated and strong, but it is satisfled by many examples. Forexample, Lind [8] showed that hyperbolic toral automorphisms satisfyspeciflcation and ergodic central spin toral automorphisms satisfy weakspeciflcation. Moreover those properties for solenoidal automorphismsare well discussed by Aoki

Shannon entropy: a rigorous notion at the crossroads between probability, information theory, dynamical systems and statistical physics

Statistical entropy was introduced by Shannon as a basic concept in information theory measuring the average missing information in a random source. Extended into an entropy rate, it gives bounds in coding and compression theorems. In this paper, I describe how statistical entropy and entropy rate relate to other notions of entropy that are relevant to probability theory (entropy of a discrete probability distribution measuring its unevenness), computer sciences (algorithmic complexity), the ergodic theory of dynamical systems (Kolmogorov–Sinai or metric entropy) and statistical physics (Boltzmann entropy). Their mathematical foundations and correlates (the entropy concentration, Sanov, Shannon–McMillan–Breiman, Lempel–Ziv and Pesin theorems) clarify their interpretation and offer a rigorous basis for maximum entropy principles. Although often ignored, these mathematical perspectives give a central position to entropy and relative entropy in statistical laws describing generic collective behaviours, and provide insights into the notions of randomness, typicality and disorder. The relevance of entropy beyond the realm of physics, in particular for living systems and ecosystems, is yet to be demonstrated.

Intrinsic mode functions locate implicit turbulent attractors in time in frontal lobe MEG recordings

In seeking evidence for the presence and characteristic range of coupled time scale(s) of putative implicit turbulent attractors of dorsal frontal lobe magnetic fields, the recorded nonstationary, nonlinear MEG signals were non-orthogonally decomposed using Huang’s Empirical Mode Decomposition, EMD, (Huang and Attoh-Okine, 2005) into 16 Intrinsic Mode Functions, EMD→IMFi, i=1…16. Measures known to be invariant in non-uniformly hyperbolic (turbulent) dynamical systems, topological entropy, hT, metric entropy, hM, non-uniform entropy, hU and power spectral scaling exponent, α, were imposed on each of the IMFi which evidenced most clearly an invariant temporal scale zone of IMFi, i=6…11, for hT, which we have found to be the most robust of invariant measures of MEG’s magnetic field turbulent attractors (Mandell et al., 2011a,b; Mandell, 2013). The ergodic theory of dynamical systems (Walters, 1982; Pollicott and Yuri, 1998) allows the inference that an implicit attractor with consistently hT>0 will also evidence at least one positive Lyapounov exponent indicating the presence of a turbulent attractor with exponential separation of nearby initial conditions, exponential convergence of distant points and disordering, mixing, of orbital sequences. It appears that this approach permits the inference of the presence of chaotic, turbulent attractor and its characteristic time scales without the invocation of arbitrary n-dimensional embedding, phase space reconstructions or (inappropriate) orthogonal decompositions.

Can a metaphor of physics contribute to MEG neuroscience research? Intermittent turbulent eddies in brain magnetic fields

A common manifestation of nonlinear mathematical and experimental neurobiological dynamical systems in transition, intermittence, is currently being attended by concepts from physics such as turbulent eddy and the avalanche of critical systems. Do these concepts constitute an enticing poetry of dynamical universality or do these metaphors from physics generate more specific novel and relevant concepts and experiments in the neurosciences? Using six graphics and ten measures derived from the ergodic theory of dynamical systems, we study the magnetoencephalic, MEG, records of taskless, “resting” human subjects to find consistent evidence for turbulent (chaotic) dynamics marked by intermittent turbulent eddies. This brings up an apparent discrepancy via the juxtaposition of the superposition characteristics of magnetic fields and the non-superposition properties of turbulent flow. Treating this apparent inconsistency as an existent duality, we propose a physical model for how that might be the case. This leaves open the question: has the physical metaphor, turbulent eddy, contributed to a scientific understanding of the human resting MEG?

Correlation Dimension of Complex Networks

We propose a new measure to characterize the dimension of complex networks based on the ergodic theory of dynamical systems. This measure is derived from the correlation sum of a trajectory generated by a random walker navigating the network, and extends the classical Grassberger-Procaccia algorithm to the context of complex networks. The method is validated with reliable results for both synthetic networks and real-world networks such as the world air-transportation network or urban networks, and provides a computationally fast way for estimating the dimensionality of networks which only relies on the local information provided by the walkers.

Correlation dimension estimates of human postural sway

Human postural sway during quiet standing demonstrates a complex structured dynamics, which has been studied by applying numerous methods, such as linear system identification methods, stochastic analysis, and nonlinear system dynamics tools. Although each of the methods applied revealed some particular features of the sway data none of them have succeeded to present a global picture of the quiet stance dynamics, which probably has both stochastic and deterministic properties. In this study we have started applying ergodic theory of dynamical systems to explore statistical characteristic of the sway dynamics observed in successive trials of a subject, different subjects in an age group, and finally different age groups constituted by children, adults, and elderly subjects. Five successive 180-s long trials were performed by each of 28 subjects in four age groups at quiet stance with eyes open. Stationary and ergodic signal characteristics of five successive center of pressure time series collected from a subject in antero-posterior direction (CoPx) were examined. 97% of the trials were found to be stationary by applying Run Test while children and elderly groups demonstrated significant nonstationary behavior. On the other hand 13 out of 24 subjects were found to be nonergodic. We expected to observe differences in complexity of CoPx dynamics due to aging (Farmer, Ott, & Yorke, 1983). However linear metrics such as standard deviation and Fourier spectra of CoPx signals did not show differences due to the age groups. Correlation dimension (Dk) estimates of stationary CoPx signals being an invariant measure of nonlinear system dynamics were computed by using the average displacement method (Eckmann & Ruelle, 1985). Postural dynamics was expanded in m-dimensional space through CoPx signal by introducing optimum time delays, τcritical. 112 out of 136 stationary CoPx signals for 24 stationary subjects converged to Dk estimates. Average of Dk estimates for children and adult groups was 3.67±0.28, whereas mean of Dk estimates for elderly subjects was 4.12±0.59. Nonlinear metrics of postural sway (τcritical, msaturated, and Dk estimates) showed significant differences with respect to the age groups. Dk estimates computed from ergodic subjects’ CoPx sway trajectories revealed that human quiet standing demonstrates multiple degree of freedom dynamics having a fractal structure with a considerable level of noise embedded in the signal whose characteristics is determined individually for each subject. Furthermore by using ergodic theory of complex systems, we have been able to show that the ability to independently control multiple degrees of freedom has been affected by aging.

Predicting the future from the past: An old problem from a modern perspective

The idea of predicting the future from the knowledge of the past is quite natural, even when dealing with systems whose equations of motion are not known. This long-standing issue is revisited in the light of modern ergodic theory of dynamical systems and becomes particularly interesting from a pedagogical perspective due to its close link with Poincaré’s recurrence. Using such a connection, a very general result of ergodic theory—Kac’s lemma—can be used to establish the intrinsic limitations to the possibility of predicting the future from the past. In spite of a naive expectation, predictability is hindered more by the effective number of degrees of freedom of a system than by the presence of chaos. If the effective number of degrees of freedom becomes large enough, whether the system is chaotic or not, predictions turn out to be practically impossible. The discussion of these issues is illustrated with the help of the numerical study of simple models.

Stabilization of nonlinear systems over packet-drop links: Scalar case

In this paper, we study the stabilization problem of a scalar nonlinear system over a packet-drop channel. We consider moment stability notions adopted from the ergodic theory of dynamical systems. The main result of this paper proves that q-moment exponential stability requires a minimal quality of service from the communications link. The necessary conditions presented in the paper relate the positive Lyapunov exponent of the open loop system and the largest probability of erasure. For the first time, the dependence on the Lyapunov exponent highlights the important role played by the global non-equilibrium dynamics in nonlinear stabilization over networks.

Capturing deviation from ergodicity at different scales

We address here the issue of quantifying the extent to which a given dynamical system falls short of being ergodic and introduce a new multiscale technique which we call the “ergodicity defect”. Our approach is aimed at capturing both deviation from ergodicity and its dependence on scale. The method uses ergodic theory of dynamical systems and applies harmonic analysis, in particular the scaling analysis is motivated by wavelet theory. We base the definition of the ergodicity defect on the Birkhoff characterization. We systematically exploit the role of the observation function by using characteristic functions arising from a dyadic equipartition of the phase space. This allows us to view the dependence of the defect on scale. In order to build intuition, we consider the defect for specific examples with known dynamic properties and we are able to explicitly compute the defect for some of these simple examples. We focus on three distinctive cases of the dependence of the defect on scale: (1) a defect value that increases as the scale becomes finer, (2) a defect value decreasing with scale and (3) a defect value independent of scale, which occurs for instance when a map is ergodic. We explain the information contained in these three scenarios. We see more complicated behavior with an example which has invariant subsets at various scales.

Stability criteria for the contextual emergence of macrostates in neural networks

More than thirty years ago, Amari and colleagues proposed a statistical framework for identifying structurally stable macrostates of neural networks from observations of their microstates. We compare their stochastic stability criterion with a deterministic stability criterion based on the ergodic theory of dynamical systems, recently proposed for the scheme of contextual emergence and applied to particular inter-level relations in neuroscience. Stochastic and deterministic stability criteria for macrostates rely on macro-level contexts, which make them sensitive to differences between different macro-levels.

Nonlinear Dynamics and Predictability in the Atmospheric Sciences

Contributions by American scientists to the nonlinear dynamics of intraseasonal, interannual and Quaternary climate changes are summarized for the period 1987–90. Applications of the successive bifurcation approach and of the ergodic theory of dynamical systems to our understanding and prediction of these changes are emphasized.

Application of entropy measures derived from the ergodic theory of dynamical systems to rat locomotor behavior.

Measures of complexity derived from ergodic theory of dynamical systems were developed and applied to an exemplary data set describing locomotor movements of rats in a bounded space. A symbolic dynamical system was obtained by partitioning the event space into equally probable partition elements, using a k-dimensional tree. The measures calculated from the symbolic sequences included the topological entropy (ht)--i.e., the rate of increase of all possible sequences with increasing sequence length--and the metric entropy (hm)--i.e., the rate of increase of all likely sequences with increasing sequence length. These measures were used to assess changes in rat locomotor behavior as recorded in the behavioral pattern monitor (BPM) that were induced by amphetamine (0.25, 0.50, 1.0, or 2.0 mg/kg) and 3,4-methylenedioxymethamphetamine (MDMA; 1.25, 2.5, 5.0, or 10.0 mg/kg). Amphetamine increased the mean activity, ht, and hm. MDMA resulted in a monotonic dose-response curve for activity but exhibited a biphasic dose response in ht and hm. In particular, some animals in the higher dose groups showed a ht in the range of the saline controls, whereas other animals exhibited a significantly reduced ht and a greater decrease in hm, suggesting that two different behavioral reactions coexist within the same higher dose range of MDMA. An important implication of our method is that, in applied ergodic measure-theoretic approaches, the partition that determines the elements of the symbolic dynamical system should not be specified a priori on abstract mathematical grounds but should be chosen relative to its significance with respect to the data set in question. Here, the animal constructs its own spatiotemporal partition in behavioral phase space.

Remarks on quasicrystallic symmetries

Quasicrystallic symmetries are encountered not only in experimental crystallography, but also in some mathematical theories related to quite different physical situations — in the study of singularities of optical ray symmetries, in the Sinai construction of Markov partitions of the ergodic theory of dynamical systems, in computer experiments with iterations of mappings of the plane and in the statistical study of the topology of the level lines of quasiperiodic Hamiltonian functions. In this article, I shall explain some of these relations between such seemingly unrelated topics.

Ergodic theory of chaos and strange attractors

Physical and numerical experiments show that deterministic noise, or chaos, is ubiquitous. While a good understanding of the onset of chaos has been achieved, using as a mathematical tool the geometric theory of differentiable dynamical systems, moderately excited chaotic systems require new tools, which are provided by the ergodic theory of dynamical systems. This theory has reached a stage where fruitful contact and exchange with physical experiments has become widespread. The present review is an account of the main mathematical ideas and their concrete implementation in analyzing experiments. The main subjects are the theory of dimensions (number of excited degrees of freedom), entropy (production of information), and characteristic exponents (describing sensitivity to initial conditions). The relations between these quantities, as well as their experimental determination, are discussed. The systematic investigation of these quantities provides us for the first time with a reasonable understanding of dynamical systems, excited well beyond the quasiperiodic regimes. This is another step towards understanding highly turbulent fluids.

The significance of the ergodic decomposition of stationary measures for the interpretation of probability

De Finetti's representation theorem is a special case of the ergodic decomposition of stationary probability measures. The problems of the interpretation of probabilities centred around de Finetti's theorem are extended to this more general situation. The ergodic decomposition theorem has a physical background in the ergodic theory of dynamical systems. Thereby the interpretations of probabilities in the cases of de Finetti's theorem and its generalization and in ergodic theory are systematically connected to each other.

Total reflection of a plane wave by a semi-infinite random medium

An exact result in the theory of wave propagation in random media is presented. Using the ergodic theory of dynamical systems, it is shown that a semi-infinite, one-dimensional random medium is totally reflecting. A direct numerical study shows that the mean reflection coefficient converges exponentially to one.