Non-stationary Dynamical Systems Research Articles

NONSTATIONARY MIXING AND THE UNIQUE ERGODICITY OF ADIC TRANSFORMATIONS

We define topological and measure-theoretic mixing for nonstationary dynamical systems and prove that for a nonstationary subshift of finite type, topological mixing implies the minimality of any adic transformation defined on the edge space, while if the Parry measure sequence is mixing, the adic transformation is uniquely ergodic. We also show this measure theoretic mixing is equivalent to weak ergodicity of the edge matrices in the sense of inhomogeneous Markov chain theory.

Investigation of Condition Monitoring of a Flap System

This paper presents an initiative research work that applies cointegration testing method to the condition monitoring and fault diagnosis of nonstationary dynamic engineering systems. The cointegration testing method seeks a linear combination of a set of nonstationary system variables which describes the dynamic equilibrium relations among them. The fact that the cointegrational model could be violated on the occurrence of a fault is utilised to detect abnormal conditions. The model residuals are processed and analyzed to extract the fault features. In this paper it is demonstrated that the cointegration testing method is a power full means for condition monitoring for nonstationary engineering systems.

Inferential framework for non-stationary dynamics: theory and applications

An extended Bayesian inference framework is presented, aiming to infer time-varyingparameters in non-stationary nonlinear stochastic dynamical systems. The convergence ofthe method is discussed. The performance of the technique is studied using, as an example,signal reconstruction for a system of neurons modeled by FitzHugh–Nagumooscillators: it is applied to reconstruction of the model parameters and elements of themeasurement matrix, as well as to inference of the time-varying parameters ofthe non-stationary system. It is shown that the proposed approach is able toreconstruct unmeasured (hidden) variables of the system, to determine the modelparameters, to detect stepwise changes of control parameters for each oscillatorand to track the continuous evolution of the control parameters in the adiabaticlimit.

Using the Kalman Filtering for the Fault Detection and Isolation (FDI) in the Nonlinear Dynamic Processes

This paper presents a Fault Detection and Isolation (FDI) method for stochastic nonlinear dynamic systems. First, the developed fault detection method is based on statistical information generated by the extended Kalman filter (EKF) and is intended to reveal any drift from the normal behaviour of the process. A fault of a chemical origin in a perfectly stirred batch chemical reactor, occurring at an unknown instant, is simulated. The purpose is to detect the presence of this abrupt change, and pinpoint the moment it occurred. It is also shown that the convergence of the EKF is accomplished more or less rapidly according to the nature of the noise generated by the measurement sensors. The state estimate is observed and discussed, as well as the time delay in detection according to the decision threshold. Then, this study shows another method of tackling the problem of the physical origin diagnosis of faults by combining the technique based on the standardized innovations and the technique using the multiple extended Kalman filters for a strongly non-stationary nonlinear dynamic system. The usefulness of this combination is the implementation of all the fault dynamics models if the decision threshold on the standardized innovation exceeds a determined threshold. In the other case, one EKF is enough to estimate all the process state. An algorithm is described and applied to a perfectly stirred chemical reactor operating in a semi-batch mode. The chemical reaction used is an exothermic second order one.

Optimal pulse observation of one type of systems with delay

Consideration is given to a linear problem of optimal pulse observation of a non-stationary dynamic system with delay in an equation of its mathematical model. To compute estimates of an unknown vector parameter of the initial state of the system, fast direct and dual methods are proposed. The main role belongs to quasi-reduction of the fundamental matrix of solutions to systems with delay. As is shown, to perform iterations of the method, integration of auxiliary systems of ordinary differential equations on small time intervals is sufficient. An algorithm of the operation of an optimal estimator--device for computing estimates of current states--is described. The results are illustrated by the problem of optimal observation of the fourth-order system with delay.

Characterization of nonstationary chaotic systems

Nonstationary dynamical systems arise in applications, but little has been done in terms of the characterization of such systems, as most standard notions in nonlinear dynamics such as the Lyapunov exponents and fractal dimensions are developed for stationary dynamical systems. We propose a framework to characterize nonstationary dynamical systems. A natural way is to generate and examine ensemble snapshots using a large number of trajectories, which are capable of revealing the underlying fractal properties of the system. By defining the Lyapunov exponents and the fractal dimension based on a proper probability measure from the ensemble snapshots, we show that the Kaplan-Yorke formula, which is fundamental in nonlinear dynamics, remains valid most of the time even for nonstationary dynamical systems.

PARAMETER SELECTION FOR PERMUTATION ENTROPY MEASUREMENTS

We investigate the applicability of the permutation entropy H and a synchronization index γ that is based on the changing tendency of temporal permutation entropies to analyze noisy time series from nonstationary dynamical systems with poorly understood properties. Using model systems, we first study the interdependencies of parameters involved in the calculation of both measures. Having identified appropriate parameter settings we then analyze long-lasting EEG time series recorded from an epilepsy patient. Our findings indicate that γ could be of interest for studies on the predictability of epileptic seizures.

Characterization of Synchrony with Applications to Epileptic Brain Signals

Measurement of synchrony in networks of complex or high-dimensional, nonstationary, and noisy systems such as the mammalian brain is technically difficult. We present a general method to analyze synchrony from multichannel time series. The idea is to calculate the phase-synchronization times and to construct a matrix. We develop a random-matrix-based criterion for proper choosing of the diagonal matrix elements. Monitoring of the eigenvalues and the determinant provides an effective way to assess changes in synchrony. The method is tested using a prototype nonstationary dynamical system, electroencephalogram (scalp) data from absence seizures for which enhanced synchrony is presumed, and electrocorticogram (intracranial) data from subjects having partial seizures with secondary generalization.

Effect of resonant-frequency mismatch on attractors

Resonant perturbations are effective for harnessing nonlinear oscillators for various applications such as controlling chaos and inducing chaos. Of physical interest is the effect of small frequency mismatch on the attractors of the underlying dynamical systems. By utilizing a prototype of nonlinear oscillators, the periodically forced Duffing oscillator and its variant, we find a phenomenon: resonant-frequency mismatch can result in attractors that are nonchaotic but are apparently strange in the sense that they possess a negative Lyapunov exponent but its information dimension measured using finite numerics assumes a fractional value. We call such attractors pseudo-strange. The transition to pesudo-strange attractors as a system parameter changes can be understood analytically by regarding the system as nonstationary and using the Melnikov function. Our results imply that pseudo-strange attractors are common in nonstationary dynamical systems.

Detecting and characterizing phase synchronization in nonstationary dynamical systems

We propose a general framework for detecting and characterizing phase synchronization from noisy, nonstationary time series. For detection, we propose to use the average phase-synchronization time and show that it is extremely sensitive to parameter changes near the onset of phase synchronization. To characterize the degree of temporal phase synchronization, we suggest to monitor the evolution of phase diffusion from a moving time window and argue that this measure is practically useful as it can be enhanced by increasing the size of the window. While desynchronization events can be caused by either a lack of sufficient deterministic coupling or noise, we demonstrate that the time scales associated with the two mechanisms are quite different. In particular, noise-induced desynchronization events tend to occur on much shorter time scales. This allows for the effect of noise on phase synchronization to be corrected in a practically doable manner. We perform a control study to substantiate these findings by constructing and investigating a prototype model of nonstationary dynamical system that consists of coupled chaotic oscillators with time-varying coupling parameter.

Artificial life approach for continuous optimisation of non-stationary dynamical systems

In this paper, we develop an intelligent system to approach dynamical optimisation problems emerging in control of complex systems. In particular our proposal is to exploit the adaptivity of an artificial life (alife) environment in order to achieve

Testing for autocorrelation in non-stationary dynamic systems of equations

Using Monte Carlo methods, the properties of systemwise generalizations of the Breusch-Godfrey test for autocorrelated errors are studied in integrated cointegrated systems of equations. Our analysis, regarding the size of the test, reveals that the corrected LR tests have been shown to perform satisfactorily even in cases when the exogenous variables follow a unit root process, whilst the commonly used TR2 test behaves badly even in single equations. All tests perform badly, however, when the number of equations increases and the exogenous variables are highly autocorrelated.

A Scaling Property in the Non-stationary Dissipative Dynamical Systems**The project supported by the Advanced Research Project PRA M97-05 No. 3, co-organized by the Ministry of Science and Technology of China, the Association Franco-Chinoise pour la Recherche Scientifique et Technique (AFCRST) of France, and National Natural Science Foundation of China under Grant No. 19904004

A scaling property about the parameter shifts for the critical points of the period-doubling bifurcation is exploited for the non-stationary dynamical systems.

Nonstationary processes: Decreasing forcing frequency

Nonstationary ( NS) processes or NS dynamical systems ( DS) are characterized by the appearance of the process ( p) on the Control Parameters ( CP) in the governing operators. That is, CP(NS) or Cp(p) where p is arbitrary processes: deterministic or random, and by the evolution paths (ϕ) traced by CPs within the bifurcation regions. Kryolov and Bogolyubov, Russian mathematicians, made first step in introducing NS processes. They also introduced asymptotic method for solution of nonlinear equations. Next, Mitropolskii [1], head of the Ukrainian School of mathematics and dynamics, expanded the asymptotic method. Evan-Iwanowski [2], Syracuse NY and his school, expanded the concept of nonstationarity and the asymptotic method to include the multiple-resonance (combination resonance) systems. Extensive experimental work showed excellent agreement with the analysis. Next, the authors dealt with the NS topics related to the precursors to chaos and NS period doubling applications to structural mechanics [3–8]. The new method of NS bifurcation maps allowed determining the dynamical contents of the response within time ranges or cycles for extended time-flow sample [9]. In that, the method is more effective than the well known Poincaré maps by eliminating overlapping responses. This paper presents the study of the effects of decreasing forcing frequency Ω NS = Ω 0(1 + α νt) in the Duffing nonlinear oscillator, where α ν is a negative number. The new dynamic responses appear which are not encountered for the increasing forcing frequency. The authors are convinced that the new physical, chemical and biodynamic nonstationary interpretation eventually will show up based on this study.

Identification of nonstationary dynamics in physiological recordings.

We present a novel framework for the analysis of time series from dynamical systems that alternate between different operating modes. The method simultaneously segments and identifies the dynamical modes by using predictive models. In extension to previous approaches, it allows an identification of smooth transition between successive modes. The method can be used for analysis, diagnosis, prediction, and control. In an application to EEG and respiratory data recorded from humans during afternoon naps, the obtained segmentations of the data agree with the sleep stage segmentation of a medical expert to a large extent. However, in contrast to the manual segmentation, our method does not require a priori knowledge about physiology. Moreover, it has a high temporal resolution and reveals previously unclassified details of the transitions. In particular, a parameter is found that is potentially helpful for vigilance monitoring. We expect that the method will generally be useful for the analysis of nonstationary dynamical systems, which are abundant in medicine, chemistry, biology and engineering.

A design algorithm for self-improving nonstationary dynamic systems using adaptive coordinate-parametric control

The article considers the class of self-improving dynamic systems that, functioning under conditions of nonstationary with failures and element aging, are capable of assessing the external operating conditions and the degree of functional readiness, while at the same time raising the degree of readiness to the maximum attainable on the current time interval. Concurrently, the system estimates the reachability of the goals that can be accomplished at the current instant and activates the control strategy to achieve the highest-priority goal among the reachable goals. The principle of adaptive coordinate-parametric control is applied to design self-improving dynamic systems.

Sample-Path Stability of Non-Stationary Dynamic Economic Systems

The goal of this paper is to introduce and illustrate a new approach to the stability analysis of sample-paths of nonlinear stochastic economic models with non-stationary components. We place our study within the mathematical theory of random dynamical systems and apply the concept of a random fixed point which is tailor-made for the study of the long-term behavior of sample-paths in stochastic systems. The main tool for the application of this approach is a Banach-type fixed point theorem for non-stationary random dynamical systems which is proved here. The concept and the theorem are thoroughly explained and illustrated by two examples from stochastic growth theory.

Modeling of nonlinear nonstationary dynamic systems with a novel class of artificial neural networks

This paper introduces a novel neural-network architecture that can be used to model time-varying Volterra systems from input-output data. The Volterra systems constitute a very broad class of stable nonlinear dynamic systems that can be extended to cover nonstationary (time-varying) cases. This novel architecture is composed of parallel subnets of three-layer perceptrons with polynomial activation functions, with the output of each subnet modulated by an appropriate time function that gives the summative output its time-varying characteristics. The paper shows the equivalence between this network architecture and the class of time-varying Volterra systems, and demonstrates the range of applicability of this approach with computer-simulated examples and real data. Although certain types of nonstationarities may not be amenable to this approach, it is hoped that this methodology will provide the practical tools for modeling some broad classes of nonlinear, nonstationary systems from input-output data, thus advancing the state of the art in a problem area that is widely viewed as a daunting challenge.

Robust Algorithms of Ellipsoidal State Estimation for Multidimensional Non-stationary Continuous Dynamic Systems

Robust Algorithms of Ellipsoidal State Estimation for Multidimensional Non-stationary Continuous Dynamic Systems

Ellipsoidal Observer of State for Continuous Dynamic Systems with Non-Controlled Disturbances

Algorithm of the state observer for linear multidimensional non-stationary dynamic systems operating under the influence of restricted non-controlled disturbances with statistically indefinite properties is developed. Efficiency of the proposed algorithm is demonstrated by computer modeling for adaptive control system of spacecraft orientation.