Estimation Of Lyapunov Exponent Research Articles

Accounting for observation noise in equation‐free forecasting: The hidden‐Markov S‐map

Abstract Ecosystems contain numerous species interacting with each other and the environment. However, data on all relevant state variables is rarely available, hampering inference and prediction. Empirical dynamic modelling (EDM) is a valuable tool for prediction, inference and control in such partially observed systems. However, EDM typically assumes that the available time series are observed without error. Failing to account for observation noise strongly biases estimates of Lyapunov exponents and reduces forecast accuracy. To address this limitation, we propose incorporating EDM into a hidden Markov framework and using an iterative scheme based on the expectation maximization (EM) algorithm to obtain filtered state and parameter estimates. We evaluate the performance of this approach on several simulated dynamical systems with a range of additive noise levels, as well as on insect population time series. Accounting for observation noise improved accuracy of population forecasts and estimates of Lyapunov exponents (LE) over a wide range of noise levels relevant to ecological time series.

Functional data learning using convolutional neural networks

In this paper, we show how convolutional neural networks (CNNs) can be used in regression and classification learning problems for noisy and non-noisy functional data (FD). The main idea is to transform the FD into a 28 by 28 image. We use a specific but typical architecture of a CNN to perform all the regression exercises of parameter estimation and functional form classification. First, we use some functional case studies of FD with and without random noise to showcase the strength of the new method. In particular, we use it to estimate exponential growth and decay rates, the bandwidths of sine and cosine functions, and the magnitudes and widths of curve peaks. We also use it to classify the monotonicity and curvatures of FD, the algebraic versus exponential growth, and the number of peaks of FD. Second, we apply the same CNNs to Lyapunov exponent estimation in noisy and non-noisy chaotic data, in estimating rates of disease transmission from epidemic curves, and in detecting the similarity of drug dissolution profiles. Finally, we apply the method to real-life data to detect Parkinson’s disease patients in a classification problem. We performed ablation analysis and compared the new method with other commonly used neural networks for FD and showed that it outperforms them in all applications. Although simple, the method shows high accuracy and is promising for future use in engineering and medical applications.

Quantification of chaos in a time series generated from a traffic flow model using the extended entropic chaos degree

Several Traffic flow models have been proposed to address the underlying causes of traffic congestion and develop effective methods for its suppression. Typically, these models are represented using differential or difference equations. Traffic flow models can exhibit chaotic behaviors. Traffic situations become unstable if chaos prevails in the traffic flow. Therefore, the conditions under which chaos occurs in the traffic flow and the intensity of chaos at that time must be investigated. The Lyapunov exponent is used for quantifying the chaos in the dynamical systems. However, computing the Lyapunov exponent over time series without a dynamical map is challenging. Recently, the extended entropic chaos degree has been introduced as an extension of the entropic chaos degree. The extended entropic chaos degree can be computed directly for any time series. Analytically, the extended entropic chaos degree equals the sum of all the Lyapunov exponents for multidimensional non-periodic maps. Moreover, the extended entropic chaos degree is mathematically shown to equal the sum of one positive and one negative Lyapunov exponents for two-dimensional typical chaotic maps, such as a generalized Baker’s map, and a standard map.In this study, we utilize the extended entropic chaos degree to quantify chaos in a time series with less computing complexity compared with conventional methods. First, we demonstrate that chaos can be quantified in a time series generated from a traffic flow model using an extended entropic chaos degree. Lyapunov exponent for a time series of time traveling between two adjacent traffic lights in the traffic flow model is not directly computable because its dynamics are not given as an explicit differential map. Second, we demonstrate that the computational complexity of the extended entropic chaos degree only corresponds to O(M). However, the computational complexities of the estimated Lyapunov exponents using the Wolf method are O(M2), where M is the number of points on a one-dimensional time series. Thus, we show that the extended chaos degree can quantify the chaos in the traffic flow model, such as one vehicle moving through a sequence of traffic lights in one dimension, with less complexity, where the chaos occurs depending on the timing of signal switching.

Noise-robust estimation of the maximal Lyapunov exponent based on state space reconstruction with principal components

The maximal Lyapunov exponent (MLE) is a widely used indicator of the dependence on initial conditions of various systems in many fields. However, both the accuracy and the consistency of the conventional method for estimating MLE heavily depend on the noise level. We developed a method for estimating MLE with higher accuracy and noise-robustness compared with the conventional method. Considering that state space reconstruction using principal components can enhance noise-robustness when appropriate parameter values are used, we devised a set of algorithms for finding proper window length and the number of principal components required for state space reconstruction and MLE estimation. Numerical simulations of multiple dynamical systems with various data lengths and noise levels verified that the devised algorithm yields more accurate, consistent, and noise-robust estimation of MLE than the conventional method.

Quantitative Aeroelastic Stability Prediction of Wings Exhibiting Nonlinear Restoring Forces

In engineering practice, eigen-solution is used to assess the stability of linear dynamical systems. However, the linearity assumption in dynamical systems sometimes implies simplifications, particularly when strong nonlinearities exist. In this case, eigen-analysis requires linerisation of the problem and hence fails to provide a direct stability estimation. For this reason, a more reliable tool should be implemented to predict nonlinear phenomena such as chaos or limit cycle oscillations. One method to overcome this difficulty is the Lyapunov Characteristic Exponents (LCEs), which provide quantitative indications of the stability characteristics of dynamical systems governed by nonlinear time-dependent differential equations. Stability prediction using Lyapunov Characteristic Exponents is compatible with the eigen-solution when the problem is linear. Moreover, LCE estimations do not need a steady or equilibrium solution and they can be calculated as the system response evolves in time. Hence, they provide a generalization of traditional stability analysis using eigenvalues. These properties of Lyapunov Exponents are very useful in aeroelastic problems possessing nonlinear characteristics, which may significantly alter the aeroelastic characteristics, and result in chaotic and limit cycle behaviour. A very common nonlinearity in flexible systems is the nonlinear restoring force such as cubic stiffness, which would substantially benefit from using LCEs in stability assessment. This work presents the quantitative evaluation of aeroelastic stability indicators in the presence of nonlinear restoring force. The method is demonstrated on a two-dimensional aeroelastic problem by comparing the system behaviour and estimated Lyapunov Exponents.

Supervised machine learning to estimate instabilities in chaotic systems: Estimation of local Lyapunov exponents

AbstractIn chaotic dynamical systems such as the weather, prediction errors grow faster in some situations than in others. Real‐time knowledge about the error growth could enable strategies to adjust the modelling and forecasting infrastructure on the fly to increase accuracy and/or reduce computation time. For example, one could change the ensemble size, the distribution and type of target observations, and so forth. Local Lyapunov exponents are known indicators of the rate at which very small prediction errors grow over a finite time interval. However, their computation is very expensive: it requires maintaining and evolving a tangent linear model, orthogonalisation algorithms and storing large matrices. In this feasibility study, we investigate the accuracy of supervised machine learning in estimating the current local Lyapunov exponents, from input of current and recent time steps of the system trajectory, as an alternative to the classical method. Thus machine learning is not used here to emulate a physical model or some of its components, but “nonintrusively” as a complementary tool. We test four popular supervised learning algorithms: regression trees, multilayer perceptrons, convolutional neural networks, and long short‐term memory networks. Experiments are conducted on two low‐dimensional chaotic systems of ordinary differential equations, the Rössler and Lorenz 63 models. We find that on average the machine learning algorithms predict the stable local Lyapunov exponent accurately, the unstable exponent reasonably accurately, and the neutral exponent only somewhat accurately. We show that greater prediction accuracy is associated with local homogeneity of the local Lyapunov exponents on the system attractor. Importantly, the situations in which (forecast) errors grow fastest are not necessarily the same as those in which it is more difficult to predict local Lyapunov exponents with machine learning.

Subcritical behaviour of short cylindrical journal bearings under periodic excitation

Rotating machinery supported on journal bearings is affected by forces due to rotating unbalance and pressure gradients in the oil film. The interaction of these forces can evoke nonlinear behaviour, including asynchronous motion and even chaos. This work attempts to characterise the sub-synchronous motion of the rigid rotor supported on cylindrical journal bearings due to the abovementioned interaction. The analysis focuses on the rotor behaviour at the rotor speeds lower than the threshold speed for oil whirl, associated with sub-synchronous vibration of magnitude equaling the bearing clearance. It is shown that the sub-synchronous vibration can occur well before reaching the threshold speed and that the underlying period-doubling bifurcation depends on the amount of the rotating unbalance. The rotor response and stability are analysed using a numerical continuation method employing the infinitely short journal bearing model. Continuation results are further validated by time simulations which utilise the finite difference method to compute the hydrodynamic forces. The validation process employs bifurcation diagrams, Poincaré sections and numerical estimates of the largest Lyapunov exponents.

Detecting the dynamical instability of complex time series via partitioned entropy.

A method is proposed to detect the dynamical instability of complex time series. We focus on how the partitioned entropy of an initially localized region of the attractor evolves in time and show that its growth rate corresponds to the first Lyapunov exponent. To avoid spurious detection of the dynamical instability, a criterion is further introduced to distinguish chaos from limit cycles or tori. Numerical experiments using prototypical models of chaotic systems demonstrate that the growth rate of the partitioned entropy indeed provides a good estimate of the first Lyapunov exponent. The method is also shown to be robust against observational noise and dynamical noise. Analysis of experimental data measured from a physical model of the vocal folds highlights the practical applicability of the present method to real-world data. Advantages of the present method over conventional methods are also discussed.

Changes in Dynamic Stability of Cardiovascular Time Series by Local Largest Lyapunov Exponent in the Time-Frequency Domain in Vasovagal Syncope

Orthostatic stress via a head-up tilt test (HUTT) has been clinically used to rule out vasovagal syncope (VVS), which is generated by an inappropriate autonomic nervous system (ANS) regulation. VVS has been evaluated mainly by hemodynamic variables and linear analysis of heart rate variability. In this study, the estimation of the local largest Lyapunov exponent (LLE[Formula: see text] was used to assess the predictability of the beat-to-beat interval (BBI), systolic (SBP), and diastolic blood pressure (DBP) times series to analyze in a nonlinear and time-varying fashion the impact of HUTT on ANS regulation in young females, healthy and patients with VVS. The estimation of LLEL of the time series was achieved in both the original domain (full-band) and narrow-band signals by the complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN). Results pointed out that during the orthostatic phase (OPH), the VVS patients showed: (1) significantly higher predictability (less complexity) of the BBI time series, i.e., significant altered cardiac vagal modulation, (2) less predictability of SBP variability suggesting altered sympathetic modulation to the vasculature, and (3) LLEL in conjunction with CEEMDAN evidenced an impairment of cardiovascular autonomic control.

Accurate Bounds on Lyapunov Exponents for Expanding Maps of the Interval

In this short note we describe a simple but remarkably effective method for rigorously estimating Lyapunov exponents for expanding maps of the interval. We illustrate the applicability of this method with some standard examples.

Quasi-Stability of Coexisting Attractors of a Neurodynamic Model with Delay

The problem of the coexistence of quasi-stable attractors of a neurodynamic model with delay is considered. The model is a system of two specially connected differential-difference equations. An algorithm for estimating Lyapunov exponents for the system is developed.

Estimating Lyapunov exponents on a noisy environment by global and local Jacobian indirect algorithms

Most of the existing methods and techniques for the detection of chaotic behaviour from empirical time series try to quantify the well-known sensitivity to initial conditions through the estimation of the so-called Lyapunov exponents corresponding to the data generating system, even if this system is unknown. Some of these methods are designed to operate in noise-free environments, such as those methods that directly quantify the separation rate of two initially close trajectories. As an alternative, this paper provides two nonlinear indirect regression methods for estimating the Lyapunov exponents on a noisy environment. We extend the global Jacobian method, by using local polynomial kernel regressions and local neural net kernel models. We apply such methods to several noise-contaminated time series coming from different data generating processes. The results show that in general, the Jacobian indirect methods provide better results than the traditional direct methods for both clean and noisy time series. Moreover, the local Jacobian indirect methods provide more robust and accurate fit than the global ones, with the methods using local networks obtaining more accurate results than those using local polynomials.

Assessment of relative dispersion in the Gulf of Tonkin using numerical modeling and HF radar observations of surface currents

Particle pair statistics from synthetic drifter trajectories reconstructed from realistic, high-resolution numerical simulations (SYMPHONIE model) and HF radar velocity measurements are used to investigate the dispersion properties in the Gulf of Tonkin (GoT). This study takes an approach based on two-particles statistics providing the relative dispersion, relative diffusivity and Finite Size Lyapunov Exponent (FSLE) estimates. In the GoT, the relative dispersion follows the predictions from the theory of two-dimensional turbulence with two inertial subranges identified in the kinetic energy spectrum with the spectral slopes −5/3 and −3. The time evolution of dispersion shows an exponential growth during 5–8 days, followed by a power law regime during the next 6–20 days. Fixed-length indicators from the relative diffusivity and the FSLE reveal a local dispersion at large and intermediate scales (above Rossby radius of deformation) and non-local dispersion in sub-mesoscale range (below Rossby radius of deformation). The effect of river runoff on the local hydrodynamics and dispersion processes is assessed using the numerical model simulations without river discharge. The results show that in the model, the river plume, when present, highly impacts the Lagrangian statistics. High gradients of buoyancy reinforce the sub-mesoscale circulation in a large region along the Vietnamese coast and modify the scales and intensity of turbulent dispersion. However, a clear change of dispersion regime in the sub-mesoscale range is not identified, suggesting that the mesoscale circulation in the GoT largely governs particle spreading even at small scale.

An empirical dynamic modeling framework for missing or irregular samples

Empirical dynamic modeling (EDM) is a powerful method for forecasting and analyzing nonlinear dynamics. However, typical applications of EDM assume that samples are evenly spaced over time. This presents problems in ecology, in which data are often missing or sampled irregularly. Standard methods for handling irregularity in EDM suffer under conditions that are common in ecology, such as short time series and large dynamic fluctuations, so there is a need to adapt the framework to cope with these challenges more effectively. Here we consider a variable step-size extension of EDM, which incorporates the temporal spacing between samples into EDM delay-coordinate vectors and circumvents the challenges faced by other approaches. We evaluated the forecast accuracy of the variable step-size method along with that of two other methods: (1) exclusion of delay-coordinate vectors with missing data and (2) linear interpolation along with ordinary EDM. We tested these methods using simulated data from three chaotic ecological models with various amounts and patterns of missing data. We also evaluated them using two empirical datasets: laboratory rotifer dynamics and aphid dynamics from the field. Results showed that while exclusion and linear interpolation can produce accurate forecasts in some scenarios, the variable step-size method consistently gives accurate forecasts in a wide range of scenarios. Our analysis demonstrates that variable step-size EDM is an effective method for coping with missing or irregular samples and expands the number of datasets to which EDM can be applied. Furthermore, EDM can be extended to estimate Lyapunov exponents from irregularly sampled time series and approximate continuous dynamics from discrete-time data.

Computational Approaches and Tools as Applied to the Study of Rhythms and Chaos in Biology.

The temporal dynamics in biological systems displays a wide range of behaviors, from periodic oscillations, as in rhythms, bursts, long-range (fractal) correlations, chaotic dynamics up to brown and white noise. Herein, we propose a comprehensive analytical strategy for identifying, representing, and analyzing biological time series, focusing on two strongly linked dynamics: periodic (oscillatory) rhythms and chaos. Understanding the underlying temporal dynamics of a system is of fundamental importance; however, it presents methodological challenges due to intrinsic characteristics, among them the presence of noise or trends, and distinct dynamics at different time scales given by molecular, dcellular, organ, and organism levels of organization. For example, in locomotion circadian and ultradian rhythms coexist with fractal dynamics at faster time scales. We propose and describe the use of a combined approach employing different analytical methodologies to synergize their strengths and mitigate their weaknesses. Specifically, we describe advantages and caveats to consider for applying probability distribution, autocorrelation analysis, phase space reconstruction, Lyapunov exponent estimation as well as different analyses such as harmonic, namely, power spectrum; continuous wavelet transforms; synchrosqueezing transform; and wavelet coherence. Computational harmonic analysis is proposed as an analytical framework for using different types of wavelet analyses. We show that when the correct wavelet analysis is applied, the complexity in the statistical properties, including temporal scales, present in time series of signals, can be unveiled and modeled. Our chapter showcase two specific examples where an in-depth analysis of rhythms and chaos is performed: (1) locomotor and food intake rhythms over a 42-day period of mice subjected to different feeding regimes; and (2) chaotic calcium dynamics in a computational model of mitochondrial function.

Detection of Chaos Using Reservoir Computing Approach

Detection of chaos in time series is of utmost importance in many scientific fields. Indeed, the presence of chaos and its significance, especially in multidimensional systems, plays an essential role in the control and analysis of such systems, and in their practical use in a variety of applications. In this paper, we demonstrate a new methodology for the detection of chaos in time series using a reservoir computing (RC) paradigm called conceptor-driven network (ConDN). Case studies on the known chaotic attractors (i.e. Lorenz, Rossler, Chua) of integer (conventional) and non-integer (fractional-order) orders, as well as a physically simulated and designed spintronic device (NCVO) are used in this study to validate the proposed chaos detection approach. The proposed chaos detection approach is tested on clean and noisy time series of the mentioned attractors. It outperforms the 0-1 chaos detection test and the largest Lyapunov exponent (LLE) estimation approach especially in the high noise-level conditions. In addition, the proposed approach is capable of differentiating the time series generated by the systems whose dynamics is at the edge of chaos. The simplicity of use of the proposed chaos detection approach can be counted, as well, as one of its main advantages over traditional chaos detection methods.

Recalculation of error growth models' parameters for the ECMWF forecast system

Abstract. This article provides a new estimate of error growth models' parameters approximating predictability curves and their differentials, calculated from data of the ECMWF forecast system over the 1986 to 2011 period. Estimates of the largest Lyapunov exponent are also provided, along with model error and the limit value of the predictability curve. The proposed correction is based on the ability of the Lorenz (2005) system to simulate the predictability curve of the ECMWF forecasting system and on comparing the parameters estimated for both these systems, as well as on comparison with the largest Lyapunov exponent (λ=0.35 d−1) and limit value of the predictability curve (E∞=8.2) of the Lorenz system. Parameters are calculated from the quadratic model with and without model error, as well as by the logarithmic, general, and hyperbolic tangent models. The average value of the largest Lyapunov exponent is estimated to be in the < 0.32; 0.41 > d−1 range for the ECMWF forecasting system; limit values of the predictability curves are estimated with lower theoretically derived values, and a new approach for the calculation of model error based on comparison of models is presented.

Alternative Methods of the Largest Lyapunov Exponent Estimation with Applications to the Stability Analyses Based on the Dynamical Maps-Introduction to the Method.

Controlling stability of dynamical systems is one of the most important challenges in science and engineering. Hence, there appears to be continuous need to study and develop numerical algorithms of control methods. One of the most frequently applied invariants characterizing systems’ stability are Lyapunov exponents (LE). When information about the stability of a system is demanded, it can be determined based on the value of the largest Lyapunov exponent (LLE). Recently, we have shown that LLE can be estimated from the vector field properties by means of the most basic mathematical operations. The present article introduces new methods of LLE estimation for continuous systems and maps. We have shown that application of our approaches will introduce significant improvement of the efficiency. We have also proved that our approach is simpler and more efficient than commonly applied algorithms. Moreover, as our approach works in the case of dynamical maps, it also enables an easy application of this method in noncontinuous systems. We show comparisons of efficiencies of algorithms based our approach. In the last paragraph, we discuss a possibility of the estimation of LLE from maps and for noncontinuous systems and present results of our initial investigations.

Space-Quasiperiodic and Time-Chaotic Parametric Patterns in a Magnonic Quasicrystal Active Ring Resonator

We demonstrate the dissipative spatiotemporal patterns that are generated by an active ring resonator based on a magnonic quasicrystal (MQC) with Fibonacci-type structure. The dissipative patterns are formed through the magnetostatic surface spin-wave (MSSW) parametric decay into exchange spin waves (SWs), temporal dispersion of the ring resonator and amplification. In the spatial domain, the MSSW and SW parametric patterns measured with the help of Brillouin light spectroscopy have the quasiperiodic spatial localization in crests and grooves of the MQC, respectively. In contrast to the optical quasiperiodic parametric spatial conservative solitons, the amplitude profiles of the magnonic quasiperiodic parametric spatial dissipative patterns correspond to the profiles of the crests and grooves of the MQC. In the time domain, the packets of the chaotic parametric pulses, which are analogs of temporal solitons, are generated at each point of the spatial patterns. The chaotic nature of such pulses is confirmed by an estimate of a highest Lyapunov exponent from the experimental time series. The pulse packets are formed through the time-filtering technique using the external microwave (MW) pulses to control a ring gain. It is shown that at a certain duty factor of the external MW pulses, the parametric MSSW pulses have the amplitude and phase profiles corresponding to the profiles of a conservative bright envelope soliton.

Effective estimates of Lyapunov exponents for random products of positive matrices

In this note we describe estimates on the error when calculating the Lyaponov exponent for random products of positive matrices using dynamical determinants. This extends the results in (Jurga N and Morris I 2019 Nonlinearity 32 4117–46; Pollicott M 2010 Invent. Math. 181 209–26) by drawing upon a new approach introduced in (Jenkinson O and Pollicott M 2018 Adv. Math. 325 87–115; Jenkinson O, Pollicott M and Vytnova P 2018 J. Stat. Phys. 170 221–53).