Estimation Of Lyapunov Exponent Research Articles

Razumikhin-type theorems on stability of stochastic retarded systems

The Razumikhin-type theorems have been well developed and are very effective for the stability of deterministic retarded dynamic systems. Chang (1984) studied the Razumikhin-type stability conditions for stochastic retarded functional differential equations (SRFDEs). But the stability is not in a global sense. Recently, by utilising the Razumikhin-type argument, Mao (1997a) provided some useful criteria for exponential stability of stochastic retarded functional differential equations. However, the important general asymptotic stability has not been considered. The aim of this article is to establish Razumikhin-type theorems on asymptotic stability of stochastic retarded systems, which is a generalisation of the result obtained by Mao. Moreover, generalised exponential stability is considered for better estimate of Lyapunov exponent of time-variant systems. In the conclusion, this article is to show the underlying principle of the Razumikhin-type theorems on stability of stochastic retarded functional differential equations. With this idea, it is not difficult to apply Razumikhin-type techniques to stochastic retarded systems.

Hamiltonian Chaos in a Model Alveolus

In the pulmonary acinus, the airflow Reynolds number is usually much less than unity and hence the flow might be expected to be reversible. However, this does not appear to be the case as a significant portion of the fine particles that reach the acinus remains there after exhalation. We believe that this irreversibility is at large a result of chaotic mixing in the alveoli of the acinar airways. To test this hypothesis, we solved numerically the equations for incompressible, pulsatile, flow in a rigid alveolated duct and tracked numerous fluid particles over many breathing cycles. The resulting Poincare sections exhibit chains of islands on which particles travel. In the region between these chains of islands, some particles move chaotically. The presence of chaos is supported by the results of an estimate of the maximal Lyapunov exponent. It is shown that the streamfunction equation for this flow may be written in the form of a Hamiltonian system and that an expansion of this equation captures all the essential features of the Poincare sections. Elements of Kolmogorov-Arnol'd-Moser theory, the Poincare-Birkhoff fixed-point theorem, and associated Hamiltonian dynamics theory are then employed to confirm the existence of chaos in the flow in a rigid alveolated duct.

Evidence of low-dimensional chaos in magnetized plasma turbulence

We analyze probe data obtained from a toroidal magnetized plasma configuration suitable for studies of low-frequency gradient-driven instabilities. These instabilities give rise to field-aligned convection rolls analogous to Rayleigh-Benard cells in neutral fluids, and may theoretically develop similar routes to chaos. When using mean-field dimension analysis, we observe low dimensionality, but this could originate from either low-dimensional chaos, periodicity or quasi-periodicity. Therefore, we apply recurrence plot analysis as well as estimation of the largest Lyapunov exponent. These analyses provide evidence of low-dimensional chaos, in agreement with theoretical predictions. Our results can be applied to other magnetized plasma configurations, where gradient-driven instabilities dominate the dynamics of the system.

Lyapunov exponents estimation for hysteretic systems

This article discusses the Lyapunov exponent estimation of non-linear hysteretic systems by adapting the classical algorithm by Wolf and co-workers [Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A., 1985. Determining Lyapunov exponents from a times series. Physica D 16, 285–317.]. This algorithm evaluates the divergence of nearby orbits by monitoring a reference trajectory, evaluated from the equations of motion of the original hysteretic system, and a perturbed trajectory resulting from the integration of the linearized equations of motion. The main issue of using this algorithm for non-linear, rate-independent, hysteretic systems is related to the procedure of linearization of the equations of motion. The present work establishes a procedure of linearization performing a state space split and assuming an equivalent viscous damping in order to represent hysteretic dissipation in the linearized system. The dynamical response of a single-degree of freedom pseudoelastic shape memory alloy (SMA) oscillator is discussed as an application of the proposed algorithm. The restitution force of the oscillator is provided by an SMA element described by a rate-independent, hysteretic, thermomechanical constitutive model. Two different modeling cases are considered for isothermal and non-isothermal heat transfer conditions, and numerical simulations are performed for both cases. The evaluation of the Lyapunov exponents shows that the proposed procedure is capable of quantifying chaos capturing the non-linear dissipation of hysteretic systems.

Adiabatic chaos in the Prometheus-Pandora system

The chaotic orbital motion of Prometheus and Pandora, the 16th and 17th satellites of Saturn, is studied. Chaos in their orbital motion, as found by Goldreich & Rappaport and Renner & Sicardy, is due to interaction of resonances in the resonance multiplet corresponding to the 121:118 commensurability of the mean motions of the satellites. It is shown rigorously that the system moves in adiabatic regime. The Lyapunov time (the ‘time horizon of predictability’ of the motion) is calculated analytically and compared to the available numerical–experimental estimates. For this purpose, a method of analytical estimation of the maximum Lyapunov exponent in the perturbed pendulum model of non-linear resonance is applied. The method is based on the separatrix map theory. An analytical estimate of the width of the chaotic layer is made as well, based on the same theory. The ranges of chaotic diffusion in the mean motion are shown to be almost twice as big compared to previous estimates for both satellites.

Lyapunov exponent estimates of a class of higher-order stochastic Anderson models

In this article, we propose a class of high-order stochastic partial differential equations (SPDEs) for spatial dimensions d ≤ 5 d\leq 5 which might be called high-order stochastic Anderson models. This class of the equations is perturbed by a space-time white noise when d ≤ 3 d\leq 3 and by a space-correlated Gaussian noise when d = 4 , 5 d=4,5 . The objectives of this article are to get some estimates on the Lyapunov exponent of the solutions and to study the convergence rates of the chaos expansions of the solutions for the models.

Mapped Clock Oscillators as Ring Devices and Their Application to Neuronal Electrical Rhythms

The mapped clock oscillator (MCO) is a second-order, Winfree-type oscillator generating two instantaneous clock variables (amplitude and phase) that are mapped to an observable output variable (voltage) via a static nonlinearity. Two fundamental classes of ring devices are presented. Their respective dynamics give rise to two oscillator forms--the labile clock and the clock--which can be coupled together in various configurations to create higher-order systems with sufficient complexity to capture the dynamics of neuronal assemblies. To demonstrate the applicability of MCOs in modelling neuronal rhythms, a hippocampal network model of four coupled oscillators was constructed and shown to exhibit rhythmic activity of varying complexity, depending on model parameters. The dynamics of the network were quantified through estimation of the maximum lyapunov exponent and the correlation dimension. Synthesis of complex neuronal rhythms may have therapeutic implications. The modular and efficient design of the MCO should facilitate the process of implementing coupled MCO networks in electronic hardware as potential neural prostheses for treating dynamic diseases such as epilepsy.

Realization of synchronization in time-delayed systems with stochastic perturbation

Since noise is ubiquitous in both nature and artificial systems, the stochastic perturbation influence on the dynamics of the unidirectionally coupled Ikeda models is investigated in this paper. On the one hand, sufficient conditions on the complete synchronization between these noise-perturbed and chaotic models are mathematically established, and an estimation of the sample transverse Lyapunov exponent is rigorously derived. On the other hand, specific examples and their numerical simulations are provided to illustrate the feasibility of our theoretical results. Moreover, the results on the Ikeda models are further generalized to a wide class of coupled nonlinear systems with multiple time delays and a common additive noise. It is believed that the idea and approach developed in this paper could be further generalized to investigate some other problems on chaos synchronization and chaos control with stochastic perturbation.

A comparison of various linear and non-linear signal processing techniques to separate uterine EMG records of term and pre-term delivery groups

Various linear and non-linear signal-processing techniques were applied to three-channel uterine EMG records to separate term and pre-term deliveries. The linear techniques were root mean square value, peak and median frequency of the signal power spectrum and autocorrelation zero crossing; while the selected non-linear techniques were estimation of the maximal Lyapunov exponent, correlation dimension and calculating sample entropy. In total, 300 records were grouped into four groups according to the time of recording (before or after the 26th week of gestation) and according to the total length of gestation (term delivery records--pregnancy duration >or=37 weeks and pre-term delivery records--pregnancy duration <37 weeks). The following preprocessing band-pass Butterworth filters were tested: 0.08-4, 0.3-4, and 0.3-3 Hz. With the 0.3-3 Hz filter, the median frequency indicated a statistical difference between those term and pre-term delivery records recorded before the 26th week (p = 0.03), and between all term and all pre-term delivery records (p = 0.012). With the same filter, the sample entropy indicated statistical differences between those term and pre-term delivery records recorded before the 26th week (p = 0.035), and between all term and all pre-term delivery records (p = 0.011). Both techniques also showed noticeable differences between term delivery records recorded before and after the 26th week (p <or= 0.001).

Turbulent vertical diffusivity in the sub-tropical stratosphere

Abstract. Vertical (cross-isentropic) mixing is produced by small-scale turbulent processes which are still poorly understood and paramaterized in numerical models. In this work we provide estimates of local equivalent diffusion in the lower stratosphere by comparing balloon borne high-resolution measurements of chemical tracers with reconstructed mixing ratio from large ensembles of random Lagrangian backward trajectories using European Centre for Medium-range Weather Forecasts analysed winds and a chemistry-transport model (REPROBUS). We focus on a case study in subtropical latitudes using data from HIBISCUS campaign. An upper bound on the vertical diffusivity is found in this case study to be of the order of 0.5 m2 s−1 in the subtropical region, which is larger than the estimates at higher latitudes. The relation between diffusion and dispersion is studied by estimating Lyapunov exponents and studying their variation according to the presence of active dynamical structures.

Quantifying the synchronizability of externally driven oscillators

This paper is focused on the problem of complete synchronization in arrays of externally driven identical or slightly different oscillators. These oscillators are coupled by common driving which makes an occurrence of generalized synchronization between a driving signal and response oscillators possible. Therefore, the phenomenon of generalized synchronization is also analyzed here. The research is concentrated on the cases of an irregular (chaotic or stochastic) driving signal acting on continuous-time (Duffing systems) and discrete-time (Henon maps) response oscillators. As a tool for quantifying the robustness of the synchronized state, response (conditional) Lyapunov exponents are applied. The most significant result presented in this paper is a novel method of estimation of the largest response Lyapunov exponent. This approach is based on the complete synchronization of two twin response subsystems via additional master-slave coupling between them. Examples of the method application and its comparison with the classical algorithm for calculation of Lyapunov exponents are widely demonstrated. Finally, the idea of effective response Lyapunov exponents, which allows us to quantify the synchronizability in case of slightly different response oscillators, is introduced.

Analysis of stellar trajectories in an open cluster model

A two-point model of an unisolated star cluster moving in a circular orbit in the Galactic plane is analyzed. The equations of stellar motion are linearized in the neighborhood of the singular point at the zero-velocity surface (ZVS), and also in the neighborhood of a point below the critical ZVS on a trajectory with less than the critical stellar ‘energy.’ We find the eigenvalues and eigenvectors of these equations and point out the instability of the two singular points on the critical ZVS; the separatrix connecting these points is determined numerically. For trajectories located below the critical ZVS, the absolute values of the eigenvalues of the linearized equations of motion increase with decreasing energy of the star and decreasing maximum distance between the trajectory and the cluster center of mass. This results in an increase of the numerical estimates of the maximum characteristic Lyapunov exponents for trajectories located closer to the center of mass of the cluster. We use Poincare sections and the maximum characteristic Lyapunov exponents to analyze the properties of the stellar trajectories. A number of periodic orbits for different stellar energies are found, and the properties of the trajectories in the vicinity of these periodic orbits analyzed. Almost all the stellar trajectories considered are stochastic, with the degree of stochasticity increasing with decreasing stellar energy. Domains with different degrees of stochasticity are identified in the Poincare maps.

Epileptic seizure characterization by Lyapunov exponent of EEG signal

PurposeThe purpose of this paper is to develop the new method of estimation of the short‐term largest Lyapunov exponent of electroencephalogram (EEG) waveforms for the detection and prediction of the epileptic seizure.Design/methodology/approachThe paper proposed the modifications concerned with the way of selection of the segments of EEG waveforms taking part in estimation of Lyapunov exponent, as well as determination of the distances between two time series. The proposed method is based on Kolmogorov‐Smirnov test of similarity of two vectors. Through the application of this test more accurate and less parameterized approach to the estimation of the short‐term largest Lyapunov exponent of EEG waveforms has been obtained.FindingsThe results of performed experiments have shown that in most cases our modified method has outperformed the classical procedure, leading to more stable results, closer to the neurologist indications. The analysis of the data has proved that the change of the largest Lyapunov exponent provides a lot of information regarding the epileptic seizure. The minimum value of Lyapunov exponent indicates fairly well the seizure moment. The Tindex applied for few different electrode sites can provide good advanced prediction of the incoming epileptic seizure.Practical implicationsAfter additional experiments this method may find practical application for supporting the medical diagnosis of the epilepsy.Originality/valueThe proposed modification of the estimation of the short‐term largest Lyapunov exponent of the EEG waveforms eliminates some arbitrarily chosen parameters tuned by the user and leads to more accurate estimate. Such estimation results are better suited for the characterization of the epileptic activity.

Caos, statistica e metodi di ricampionamento

Chaos theory offers to the statistician new perspectives for time series analysis as well as concepts and ideas that have a through contribution to statistics. On the other hand, statistical methodology has shown to play a crucial role for the comprehension of nonlinear and chaotic phenomena. From this standpoint we present some essential notions for the analysis of chaotic time series. Particular attention is given to the problem of estimating Lyapunov exponents, together with the derivation of confidence intervals for estimates. For this latter problem we propose a solution based on resampling by means of spline interpolation. We show from simulations that the distribution of the maximal Lyapunov exponent obtained by way of resampling a single series with our method, agrees with the true distribution obtained from series with different initial conditions.

SCALE-INVARIANCE OF NORMALIZED YEARLY MEAN GRAIN YIELD ANOMALY SERIES

In crop science, tools of non-linear dynamics, fractals, chaos, intermittency and self-organized criticality may be employed and applied to the analysis of spatial variability and temporal behavior of agro-meteorological variables, soil properties, plant attributes, commercial yields, and prices of the agricultural products in order to gain knowledge about underlying complex processes. A search on the occurrence of particular scaling laws in Mexico's normalized yearly mean grain yield anomaly series of maize (Zea mays L.), beans (Phaseolus vulgaris L.), wheat (Triticum aestivum L.) and rice (Oriza sativa L.), using a variography approach is reported in this work. Additionally, power spectrum determination, time-frequency analysis, and estimation of Lyapunov exponent were performed for each profile in order to obtain useful information on the frequency contents and signs at which important frequencies occur as well as to determine their sensitivity to initial conditions. Fractal analysis gives us the order maize &lt; wheat &lt; rice &lt; beans of sensitivity to external forcings, which was the same as that obtained through the Lyapunov exponent values. Results confirm that the final outputs (crop yields) in agricultural systems are affected by the magnetic and sunspot cycles of the Sun, the El Niño Southern Oscillation (ENSO), and the quasi-biannual oscillation, and possibly by the so-called heliospheric mid-term-quasi-periodicities, which act on different time (or spatial and spatial-time) scales. In particular, the maize normalized year anomaly series is clearly correlated with March, April, May and June series of the ENSO index.

Data-Driven Dynamic Modeling and Control of a Surface Aeration System

In this study we have developed a support vector regression (SVR) based data-driven model for predicting two important design parameters of surface aerators, namely, the volumetric mass transfer coefficient (kLa) and fractional gas hold-up (eG). The dynamical state of the surface aerator system was captured by acquiring pressure fluctuation signals (PFSs) at various design and operating conditions. The most informative features from PFS were extracted using the chaos analysis technique, which includes estimation of Lyapunov exponent, correlation dimensions, and Kolmogorov entropy. At similar conditions the values of kLa and eG were also measured. Two different SVR models for predicting the volumetric mass transfer coefficient (kLa) and overall gas hold-up (eG) as a function of chaotic invariants, design parameters, and operating parameters were developed showing test accuracies of 98.8% and 97.1%, respectively. Such SVM based models for the surface aerator can be potentially useful on a commercial scale f...

Effect of noise on fractal structure

In this paper, I investigate the effect of dynamical noise on the estimation of the Hurst exponent and the fractal dimension of time series. Recently, Serletis et al. [Serletis, Apostolos, Asghar Shahmoradi, Demitre Serletis. Effect of noise on estimation of Lyapunov exponents from a time series. Chaos, Solitons & Fractals, forthcoming] have shown that dynamical noise can make the detection of chaotic dynamics very difficult, and Serletis et al. [Serletis, Apostolos, Asghar Shahmoradi, Demitre Serletis. Effect of noise on the bifurcation behavior of dynamical systems. Chaos, Solitons & Fractals, forthcoming] have shown that dynamical noise can also shift bifurcation points and produce noise-induced transitions, making the determination of bifurcation boundaries difficult. Here I apply the detrending moving average (DMA) method, recently developed by Alessio et al. [Alessio E, Carbone A, Castelli G, Frappietro V. Second-order moving average and scaling of stochastic time series. The Eur Phys J B 2002;27:197–200] and Carbone et al. [Carbone A, Castelli G, Stanley HE. Time-dependent Hurst exponent in financial time series. Physica A 2004;344:267–71; Carbone A, Castelli G, Stanley HE. Analysis of clusters formed by the moving average of a long-range correlated time series. Phys Rev E 2004;69:026105], to estimate the Hurst exponent of a Brownian walk with a Hurst exponent of 0.5, coupled with low and high intensity noise, and show that dynamical noise has no effect on fractal structure.

Phase Resetting Curves and Oscillatory Stability in Interneurons of Rat Somatosensory Cortex

Synchronous oscillations in neural activity are found over wide areas of the cortex. Specific populations of interneurons are believed to play a significant role in generating these synchronized oscillations through mutual synaptic and gap-junctional interactions. Little is known, though, about the mechanism of how oscillations are maintained stably by particular types of interneurons and by their local networks. To obtain more insight into this, we measured membrane-potential responses to small current-pulse perturbations during regular firing, to construct phase resetting curves (PRCs) for three types of interneurons: nonpyramidal regular-spiking (NPRS), low-threshold spiking (LTS), and fast-spiking (FS) cells. Within each cell type, both monophasic and biphasic PRCs were observed, but the proportions and sensitivities to perturbation amplitude were clearly correlated to cell type. We then analyzed the experimentally measured PRCs to predict oscillation stability, or firing reliability, of cells for a complex stochastic input, as occurs in vivo. To do this, we used a method from random dynamical system theory to estimate Lyapunov exponents of the simplified phase model on the circle. The results indicated that LTS and NPRS cells have greater oscillatory stability (are more reliably entrained) in small noisy inputs than FS cells, which is consistent with their distinct types of threshold dynamics.

Quantifying noise-induced stability of a cortical fast-spiking cell model with Kv3-channel-like current

Population oscillations in neural activity in the gamma (>30 Hz) and higher frequency ranges are found over wide areas of the mammalian cortex. Recently, in the somatosensory cortex, the details of neural connections formed by several types of GABAergic interneurons have become apparent, and they are believed to play a significant role in generating these oscillations through synaptic and gap-junctional interactions. However, little is known about the mechanism of how such oscillations are maintained stably by particular interneurons and by their local networks, in a noisy environment with abundant synaptic inputs. To obtain more insight into this, we studied a fast-spiking (FS)-cell model including Kv3-channel-like current, which is a distinctive feature of these cells, from the viewpoint of nonlinear dynamical systems. To examine the specific role of the Kv3-channel in determining oscillation properties, we analyzed basic properties of the FS-cell model, such as the bifurcation structure and phase resetting curves (PRCs). Furthermore, to quantitatively characterize the oscillation stability under noisy fluctuations mimicking small fast synaptic inputs, we applied a recently developed method from random dynamical system theory to estimate Lyapunov exponents, both for the original four-dimensional dynamics and for a reduced one-dimensional phase-equation on the circle. The results indicated that the presence of the Kv3-channel-like current helps to regulate the stability of noisy neural oscillations and a transient-period length to stochastic attractors.

On estimates of Lyapunov exponents of synchronized coupled systems

Lyapunov exponents of a synchronized coupled system consist of those of the underlying individual systems and the transverse systems, based on a mode decomposition along the synchronization manifold. Estimates of bounds on the Lyapunov exponents (including transverse Lyapunov exponents) are derived. Several examples are used to validate the theoretical estimates.