Estimation Of Lyapunov Exponent Research Articles

Letter to the Editor: “Sensitivity of the Wolf’s and Rosenstein’s Algorithms to Evaluate Local Dynamic Stability from Small Gait Data Sets”

algorithms to evaluatedynamic stability of gait. Properly assessing stability ofgait may be of great benefit but is fraught with diffi-culties, especially when using short datasets, renderingsuch a comparison important. Nonetheless we wouldlike to express some concerns about validity of meth-ods and interpretation of results in that study.Cignetti et al. estimated Lyapunov exponents forthe seminal Lorenz system and subsequently comparedLyapunov exponents of young and old subjects whilewalking. For the latter the duration of experimentaltrials was fixed to 3 min. By fixing trial duration,however, the number of stride cycles may have differedbetween groups because younger subjects walked fas-ter. The fact that the speed difference between groupswas not significant does not exclude speed as confo-under, as the authors erroneously state. Moreover,stride times have not been reported but might havebeen shorter for young subjects. Shortening stridetimes could have resulted in larger Lyapunov expo-nents, since more stride cycles were included in theanalysis of the younger subjects.

Nonlinear analysis using Lyapunov exponents in breast thermograms to identify abnormal lesions

Breast diseases are one of the major issues in women’s health today. Early detection of breast cancer plays a significant role in reducing the mortality rate. Breast thermography is a potential early detection method which is non-invasive, non-radiating, passive, fast, painless, low cost, risk free with no contact with the body. By identifying and removing malignant tumors in early stages before they metastasize and spread to neighboring regions, cancer threats can be minimized. Cancer is often characterized as a chaotic, poorly regulated growth. Cancerous cells, tumors, and vasculature defy have irregular shapes which have potential to be described by a nonlinear dynamical system. Chaotic time series can provide the tools necessary to generate the procedures to evaluate the nonlinear system. Computing Lyapunov exponents is thus a powerful means of quantifying the degree of the chaos.In this paper, we present a novel approach using nonlinear chaotic dynamical system theory for estimating Lyapunov exponents in establishing possible difference between malignant and benign patterns. In order to develop the algorithm, the first hottest regions of breast thermal images are identified first, and then one dimensional scalar time series is obtained in terms of the distance between each subsequent boundary contour points and the center of the mass of the first hottest region. In the next step, the embedding dimension is estimated, and by time delay embedding method, the phase space is reconstructed. In the last step, the Lyapunov exponents are computed to analyze normality or abnormality of the lesions. Positive Lyapunov exponents indicates abnormality while negative Lyapunov exponents represent normality. The normalized errors show the algorithm is satisfactorily, and provide a measure of chaos. It is shown that nonlinear analysis of breast thermograms using Lyapunov exponents may potentially capable of improving reliability of thermography in breast tumor detection as well as the possibility of differentiating between different classes of breast lesions.

Estimation of Lyapunov exponents from a time series for n-dimensional state space using nonlinear mapping

It has been demonstrated that when estimating Lyapunov exponents using a time series, nonlinear mapping used for characterizing the evolution of the neighbors leads to more accurate negative exponents and is more robust to noise in the times series. However, the number of unknown elements of the matrices associated with nonlinear mapping increases significantly with the embedding dimensions of the state space where the dynamics is reconstructed. Such unknown coefficients are solved from a set of linear algebraic equations based on the least square-root fit method. Derivation of such linear equations and computer programming are tedious and error prone especially for the systems with high embedding dimensions. In this work, we develop a general form of the linear algebraic equations and the corresponding computer program in terms of arbitrary embedding dimensions. A stable robotic system with all negative Lyapunov exponents and the Lorenz system with positive, zero, and negative exponents are used to demonstrate the efficacy of the proposed method. The work can contribute significantly to estimating Lyapunov exponents for systems with large embedding dimensions.

A NEURO-FUZZY BASED MODEL FOR ACCURATE ESTIMATION OF THE LYAPUNOV EXPONENTS OF AN UNKNOWN DYNAMICAL SYSTEM

A neuro-fuzzy based model is proposed in this paper for estimating the Lyapunov exponents of an unknown dynamical system according solely to a set of observations. Several approaches have been presented in recent years; most of them using the approximation of both the function of the trajectory of observations and the partial derivatives, to yield the Jacobian matrix of the function. The Jacobian matrix is then employed in the Jacobian-based methods that extract the Lyapunov exponents by QR-decomposition. While the accurate estimation of Lyapunov exponents has been sought, most of the related papers mainly focus on the accuracy of the trajectory approximation. In this paper, an Adaptive Neuro-Fuzzy Inference System is presented and stated to be an efficient tool for such a purpose. Structural parameters of the proposed model as the embedding dimension and the delay time are calculated by the Takens theorem and autocorrelation function, respectively. Model validation is performed by cross approximate entropy. Then, the promising performance of the proposed model as an accurate estimation of the Lyapunov exponents and its robustness to the measurement noise are finally evaluated.

Modified permutation-entropy analysis of heartbeat dynamics

Heart rate variability (HRV) contains important information about the modulation of the cardiovascular system. Various methods of nonlinear dynamics (e.g., estimating Lyapunov exponents) and complexity measures (e.g., correlation dimension or entropies) have been applied to HRV analysis. Permutation entropy, which was proposed recently, has been widely used in many fields due to its conceptual and computational simplicity. It maps a time series onto a symbolic sequence of permutation ranks. The original permutation entropy assumes the time series under study has a continuous distribution, thus equal values are rare and can be ignored by ranking them according to their order of emergence, or broken by adding small random perturbations to ensure every symbol in a sequence is different. However, when the observed time series is digitized with lower resolution leading to a greater number of equal values, or the equalities represent certain characteristic sequential patterns of the system, it may not be rational to simply ignore or break them. In the present paper, a modified permutation entropy is proposed that, by mapping the equal value onto the same symbol (rank), allows for a more accurate characterization of system states. The application of the modified permutation entropy to the analysis of HRV is investigated using clinically collected data. Results show that modified permutation entropy can greatly improve the ability to distinguish the HRV signals under different physiological and pathological conditions. It can characterize the complexity of HRV more effectively than the original permutation entropy.

Gottwald Melborune (0–1) test for chaos in a plasma

Abstract. Plasma is a highly complex system exhibiting a rich variety of nonlinear dynamical phenomena. In the last two decades or so there has been a spurt of growth in exploring unconventional nonlinear dynamical methods of analysis, like chaos theory, multi fractal analysis, self organized criticality etc. of experimental data from different plasma systems. Investigation of fluctuating plasma parameters is very important since they are correlated with transport of particles, and energy. In time series analysis, it is considered of key importance to determine whether the data measured from the system is regular, deterministically chaotic, or random. The two important parameters that are in general estimated are the correlation dimension and the Lyapunov exponent. Though correlation dimension helps in determining the complexity of a system, Lyapunov exponent reveals if the system is chaotic or not and also helps in prediction to some extent. In spite of its extensive usage, estimation of Lyapunov exponent can be quite tedious and sometimes suffers from some disadvantages like reliability in the presence of noise, requirement of phase space reconstruction etc., and hence it is necessary to explore other possibilities of estimating the chaoticity of a data. In this paper we have analysed for chaoticity, the nonlinear floating potential fluctuations from a glow discharge plasma system by the 0–1 test and compared it with the results obtained from Lyapunov exponent.

Quantification of surface acoustic wave induced chaotic mixing-flows in microfluidic wells

We report the use of 20 MHz surface acoustic waves (SAWs) on lithium niobate (LiNbO 3) chips to generate fast mixing flows in microfluidic wells. Whilst the chaotic nature of these SAW-driven flows have been speculated in the past, we provide quantitative evidence of the existence of such chaotic advection in these systems over a range of viscosities and input powers through the estimation of the finite time Lyapunov exponent (FTLE), which is a measure of the strength of the chaotic flow. The strongest mixing flows were most evident at higher SAW excitation amplitudes, as expected, since the increasing amounts of inertia in the system served to amplify disturbances in the system that promote the random stretching and folding of the fluid elements over a cascade of length scales in a way that introduces criss-crossing of the streamlines. What is less expected, however, is the effect of fluid viscosity. In contrast to classical acoustic streaming theories where the increase in viscous dissipation is offset by the intensification of the streaming due to increased acoustic energy absorption in systems with larger viscosities, we observe that increases in viscosity essentially suppresses the chaotic advection and hence the mixing effect, which is more akin to most other flow systems. This can be attributed to nonlinear effects due to convective acceleration that cannot be neglected in the fast streaming flows induced at the high MHz frequencies associated with the SAWs. The evidence of the chaotic advection in these SAW-driven flows is further verified through a pixel intensity analysis, in which mixing times, quantified through a normalised mixing index, were observed to be inversely proportional to the Lyapunov exponent, characteristic of processes in which transport is dominated by chaotic advection. Practically, these results, which show mixing events taking place in just seconds, demonstrate the utility of SAWs for the design of effective microfluidic mixers.

On stability analysis via Lyapunov exponents calculated based on radial basis function networks

The concept of Lyapunov exponents is a powerful tool for analysing the stability of nonlinear dynamic systems, especially when the mathematical models of the systems are available. For real world systems, such models are often unknown. Estimating Lyapunov exponents using a time series has the advantage in that no mathematical model is required. However the time-series-based methods are believed to be reliable only for estimating positive exponents. Furthermore, when nonlinear mapping is applied for deriving the neighbourhood-to-neighbourhood matrices, the loads of mathematical deduction and programming increase significantly, which makes it unfeasible to nonlinear systems with high dimensions. In contrary, the model-based methods are constructive and reliable for calculating both positive and non-positive exponents. The use of the system Jacobians is the key to the advantage of the model-based methods. In this article, a novel approach is proposed, where the system Jacobians are derived based on system approximation using the radial basis function network. The proposed method inherits the advantage of the model-based methods, yet no mathematical model is required. Two case studies are presented to demonstrate the efficacy of the proposed method. We believe that the work can contribute to the stability analysis of nonlinear systems of which the dynamics are either difficult to model due to complexities or unknown.

Quantifying chaos: A tale of two maps

In many applications, there is a desire to determine if the dynamics of interest are chaotic or not. Since positive Lyapunov exponents are a signature for chaos, they are often used to determine this. Reliable estimates of Lyapunov exponents should demonstrate evidence of convergence; but literature abounds in which this evidence lacks. This Letter presents two maps through which it highlights the importance of providing evidence of convergence of Lyapunov exponent estimates. The results suggest cautious conclusions when confronted with real data. Moreover, the maps are interesting in their own right.

A single bout of resistance exercise does not affect nonlinear dynamics of lower extremity kinematics during treadmill walking

Peripheral fatigue has been known to result in altered force output and muscle recruitment patterns by the CNS. These changes may affect lower extremity movement during gait, and such behavior may present implications for the interpretation of nonlinear analysis of gait in situations where a subject might become fatigued. The purpose of this study was to examine the effects of a single bout of resistance training on lower extremity movement during treadmill walking in healthy subjects. Fifteen recreationally active subjects performed two 10 min trials of treadmill walking at their preferred speed while knee and ankle kinematics of their right limb were recorded via optical motion capture. Between walking trials, subjects performed a series of lower extremity resistance exercises designed to induce moderate muscular fatigue. Detrended fluctuation analysis of stride length and stride time revealed that statistical persistence was unaffected by moderate muscle fatigue. Estimates of finite-time maximal Lyapunov exponents for ankle angle, knee angle, and vertical ankle movement over the short (0–1 stride) and long (4–10 strides) term were also unaffected by a single bout of resistance training. These results suggest that control of locomotion in healthy individuals, as measured by the nonlinear dynamics of lower extremity movement used here, is relatively robust to moderate muscle fatigue. Additional work with greater levels of fatigue will be necessary to fully characterize the effects of muscular fatigue on gait.

Estimation of the largest Lyapunov exponent from the perturbation vector and its derivative dot product

Nowadays, the largest Lyapunov exponent (LLE) is employed in many different areas of the scientific research. Thus, there is still need to elaborate fast and simple methods of LLE calculation. The new method of the LLE estimation is presented in this paper. The method (LLEDP) applies the perturbation vector and its derivative dot product to calculate Lyapunov exponent in the direction of disturbance. Value of this exponent is the LLE. The theoretical improvement was introduced. Results of the numerical simulations were shown and compared with the Stefanski method (Stefanski in Chaos Solitons Fractals 11(15):2443–2451, 2000; Stefanski and Kapitaniak in Chaos Solitons Fractals 15:233–244, 2003; Stefanski et al. in Chaos Solitons Fractals 23:1651–1659, 2005; Stefanski in J. Theor. Appl. Mech. 46:665–678, 2008). Investigations of the Duffing oscillators with external excitation and three Duffing oscillators coupled in ring scheme without external excitation were made with use of the presented method. Fast time LLE estimation results convergence was shown.

Behaviour of stock markets’ memories

In this article, we show that the memory of markets has nonchaotic behaviour. Its time trend is neutral and nonlinearity tests such as Brock, Dechert, Sheinkman (BDS) rejects nonlinearity in stock markets’ memories. The estimation of fractional differencing parameters is carried out by various methods such as Maximum Likelihood Estimation (MLE), Nonlinear Least Squares (NLS), Hurst exponents, Gewek, Porter- Hudak (GPH), wavelet transformation, and Whittle. Also Lyapunov exponents are estimated by two methods of Rosenstein and Jacobian. Results of Lyapunov exponent estimation shows memory of markets are not chaotic. Furthermore, there are no any Autoregressive Conditional Heteroscedasticity (ARCH) effects in memory of markets. ARCH test is more specific than the BDS test and it may powerful test for detecting possible ARCH effects. All of tests show memory of markets has random behaviour.

A robust method on estimation of Lyapunov exponents from a noisy time series

Lyapunov exponents can indicate the asymptotic behaviors of nonlinear systems, and thus can be used for stability analysis. However, it is notoriously difficult to estimate these exponents reliably from experimental data due to the measurement error (noise). In this paper, a novel method for estimating Lyapunov exponents from a time series in the presence of additive noise corruption is presented. The method combines the ideas of averaging the noisy data to form new neighbors and of nonlinear mapping to determine neighborhood mapping matrices. Two case studies of balancing control of a bipedal robot and the Lorenz systems are presented to demonstrate the efficacy of the proposed method. The bipedal robot system has two negative Lyapunov exponents while the Lorenz system has one positive, zero, and negative exponents, respectively. It is shown that, as compared with the existing methods, our proposed one is more robust to the ratio of signal to noise, and is particularly effective in estimating negative Lyapunov exponents. We believe that the work can contribute significantly to the stability analysis of nonlinear systems using a noisy time series.

On new estimates for Lyapunov exponents of discrete time varying linear systems

In this paper, we propose certain new bounds for the Lyapunov exponents of discrete time varying linear systems. The bounds are expressed in terms of spectral radii of matrix coefficients and therefore may be used to establish the exponential stability of time varying system on the basis of eigenvalues of individual coefficient. This approach is known in the literature as frozen time method.

10.1007/s11444-008-1004-3

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 Poincaré 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 Poincaré maps.

Analysis of startup oscillations in natural circulation boiling systems

The aim of the present work is to get an insight of the phenomena behind the low-pressure low-power transients that occur during startup of a natural circulation boiling system. A RELAP5 model developed for a test facility and its prototype is used to record additional system parameters that were not included in the data obtained from experiments. The flow oscillations observed during experimental and numerical studies are analyzed and classified. It is inferred that the low amplitude oscillations are not condensation induced geysering instabilities, but a density wave instability supported by flashing. The similarity between the nature of startup transients observed in the test facility and the prototype is also examined. The effect of flashing is more pronounced in the prototype due to the strong variation of saturation temperature as the length scale is 4 times that of the model. The time series data obtained from experimental observations and numerical simulations are analyzed to identify the structural nature of flow oscillations. The power spectral density estimated using fast Fourier transform (FFT) algorithm illustrates the chaotic nature of the signals. The nonlinear time series analysis (TISEAN) package has been used for the estimation of Lyapunov exponent and the Poincaré section. The Poincaré section and the Lyapunov exponent confirm the chaotic nature of the flow oscillations.

Analyzing the variation of Lyapunov exponents of solar and geomagnetic activity indices during coronal mass ejections

Hostile space weather is one of the principal threats to modern human technology because of the dependency of our lives on satellites. Solar coronal mass ejections, solar flares, and high‐speed solar wind streams often cause some sequences of damaging disturbances within the Earth's magnetosphere, in the atmosphere, and even on the Earth's surface. The time‐varying Sun, as the main source of space weather causes geomagnetic storms and substorms in many ways with coronal mass ejections (CME) being possibly the most harmful among them. CMEs cause transient magnetized plasma flows as interplanetary consequences, which drive the geomagnetic storms and substorms in the Earth's magnetosphere. It is shown that the cyclic solar activity has chaotic characteristics. It seems that the chaotic modeling of solar and geomagnetic activity indices is an important approach for analysis and prediction of the solar extreme events. This is particularly useful for engineers and designers interested in space weather prediction and its applications. One of the most important tools for eliciting the chaotic trends is the “Lyapunov exponents,” which is a useful measure of the stability of a dynamic system. This paper analyzes the variation of Lyapunov exponents (LE) for three solar and geomagnetic activity indices during CMEs: the sunspot number, disturbance storm time, and proton temperature. The analysis has been carried out using the technique of adaptive LE estimation adopted from previous works; the suitable window for which has been determined via trial and error. It is shown that the LE of these solar and geomagnetic activity indices varies rapidly during CMEs. The variation in LEs creates a pattern as a precursor for the forthcoming CME. This precursor, which is an oscillation in the values of Les, begins several steps sooner than the CME's occurrence. Then, during the CME, the LEs decrease to a small positive or a negative value, which demonstrates that during an anomaly such as a CME the chaotic characteristics of solar and geomagnetic activity indices decrease and solar and geomagnetic activity indices follow more regular dynamics.

Nonlinear time series analysis of knee and ankle kinematics during side by side treadmill walking

Nonlinear time series analysis was used to estimate maximal Lyapunov exponents of select ankle and knee kinematics during three different conditions of treadmill walking: independent, side by side, and side by side with forced synchronization of stepping. Stride to stride variability was significantly increased for the condition in which individuals walked side by side and synchronized unintentionally when compared to the conditions of forced synchronization and independent walking. In addition, standard deviations of three kinematic variables of lower extremity movement were significantly increased during the condition in which unintentional synchronization occurred. No relationship was found between standard deviation and estimates of maximal Lyapunov exponents. An increase in kinematic variability during side by side walking for nonimpaired individuals who are not at risk of falling suggests that variability in certain aspects of performance might be indicative of a healthy system. Modeling this variability for an impaired individual to imitate may have beneficial effects on locomotor function. These results may therefore have implications for the rehabilitation of gait in humans by suggesting that a different functional outcome might be achieved by practicing side by side walking as opposed to more commonly used strategies involving independent walking.

Diagnosis of spatiotemporal chaos in wave envelopes of a nematic electroconvection pattern

In this paper we report and analyze complex spatiotemporal dynamics recorded in electroconvection in the nematic liquid crystal I52, driven by an ac voltage slightly above the onset value. The instability mechanism creating the pattern is an oscillatory (Hopf) instability, giving rise to two pairs of counterpropagating rolls traveling in oblique directions relative to the unperturbed director axis. If a system of nonlinear partial differential equations shows the same set of unstable modes, the pattern above the onset is represented in a weakly nonlinear analysis as a superposition of the traveling rolls in terms of wave envelopes varying slowly in space and time. Motivated by this procedure, we extract slowly varying envelopes from the space-time data of the pattern, using a four-wave demodulation based on Fourier analysis. In order to characterize the spatiotemporal dynamics, we apply a variety of diagnostic methods to the envelopes, including the calculation of mean intensities and correlation lengths, global and local Karhunen-Loève decompositions in Fourier space and physical space, the location of holes, the identification of coherent vertical structures, and estimates of Lyapunov exponents. The results of this analysis provide strong evidence that our pattern exhibits extensive spatiotemporal chaos. One of its main characteristics is the presence of coherent structures of low and high intensities extended in the vertical (parallel to the director) direction.

Estimation of Lyapunov exponents for a system with sensitive friction model

Mathematical modelling and numerical analysis of a vibrating system with dry friction is presented. Three qualitatively different friction characteristics are considered. One of them is an example of so-called sensitive friction characteristic. Their influence on the dynamics on the attractor of the friction oscillator is investigated through bifurcational analysis. This analysis is supported by Lyapunov exponents estimated using approach for the systems with discontinuities. Theoretical background for such a synchronisation-based method of determining the largest Lyapunov exponent is explained. The results obtained through the proposed approach approximate the LLE with a good precision.