Computation Of Lyapunov Exponents Research Articles

Real-time multi-state classification of short-term voltage stability based on multivariate time series machine learning

The analysis of the significant amount of data collected by PMU devices in wide-area monitoring, protection and control (WAMPAC) applications has been a great challenge for assessing power system fast phenomena. This paper presents a novel methodology for real-time assessment of short-term voltage stability (STVS) under large disturbances with an approach based on data mining and machine learning. This methodology classifies off-line the power system stability in multiple operating states through the calculation of the maximal Lyapunov exponent and dynamic voltage indices. This allows identifying not only fast voltage collapses (FVC) but also fault-induced delayed voltage recovery (FIDVR) events. The multivariate time series data of the power system dynamic response are processed and transformed with a symbolic representation technique. This together with the operating state classification are used to train an intelligent machine based on Random Forest proposed for applying in real time to classify the post-disturbance operating state. This methodology is tested in the New England 39-bus system. The performance of the methodology to classify the STVS in real time was verified, obtaining a classification error less than 2% using a post-contingency data window of 0.58 s and less than one-third of all the bus voltage measurements. Results show the ability of the methodology to predict voltage stability problems, having enough time to carry out automatic control actions to prevent or mitigate problems.

A robust method to estimate the largest Lyapunov exponent of noisy signals: A revision to the Rosenstein’s algorithm

This study proposed a revision to the Rosenstein’s method of numerical calculation of the largest Lyapunov exponent (LyE) to make it more robust to noise. To this aim, the effect of increasing number of initial neighboring points on the LyE value was investigated and compared to values obtained by filtering the time series. Both simulated (Lorenz and passive dynamic walker) and experimental (human walking) time series were used to calculate the LyE. The number of initial neighbors used to calculate LyE for all time series was 1 (the original Rosenstein’s method), 2, 3, 4, 5, 10, 15, 20, 25, and 30 data points. The results demonstrated that the LyE graph reached a plateau at the 15-point neighboring condition implying that the LyE values calculated using at least 15 neighboring points were consistent. The proposed method could be used to calculate more consistent LyE values in experimental time series acquired from biological systems where noise is omnipresent.

Cluster Synchronization in Multilayer Networks: A Fully Analog Experiment with LC Oscillators with Physically Dissimilar Coupling.

We investigate cluster synchronization in experiments with a multilayer network of electronic Colpitts oscillators, specifically a network with two interaction layers. We observe and analytically characterize the appearance of several cluster states as we change coupling in the layers. In this study, we innovatively combine bifurcation analysis and the computation of transverse Lyapunov exponents. We observe four kinds of synchronized states, from fully synchronous to a clustered quasiperiodic state-the first experimental observation of the latter state. Our work is the first to study fundamentally dissimilar kinds of coupling within an experimental multilayer network.

BIFURCATIONS AND CHAOS CONTROL IN A DISCRETE-TIME PREDATOR-PREY SYSTEM OF LESLIE TYPE

We investigate the dynamics of a discrete-time predator-prey system of Leslie type. We show algebraically that the system passes through a flip bifurcation and a Neimark-Sacker bifurcation in the interior of $ \mathbb{R}^{2}_+ $ using center manifold theorem and bifurcation theory. Numerical simulations are implimented not only to validate theoretical analysis but also exhibits chaotic behaviors, including phase portraits, period-11 orbits, invariant closed circle, and attracting chaotic sets. Furthermore, we compute Lyapunov exponents and fractal dimension numerically to justify the chaotic behaviors of the system. Finally, a state feedback control method is applied to stabilize the chaotic orbits at an unstable fixed point.

Minimal Realizations of Autonomous Chaotic Oscillators Based on Trans-Immittance Filters

This review paper describes a design process toward fully analog realizations of chaotic dynamics that can be considered canonical (minimum number of the circuit elements), robust (exhibit structurally stable strange attractors), and novel. Each autonomous chaotic lumped circuit proposed in this paper can be understood as a looped system, where linear trans-immittance frequency filter interacts with an active nonlinear two-port. The existence of chaos is demonstrated via well-established numerical algorithms that represent the current standard in the field of nonlinear dynamics, i.e., by calculation of the largest Lyapunov exponent and high-resolution 1-D bifurcation diagrams. The achieved numerical results are put into the context of experimental measurement; observed state trajectories prove a one-to-one correspondence between theoretical expectations and practical outputs, i.e., prescribed strange attractors do not represent the chaotic transients. Finally, short term unpredictability of the chaotic flow is demonstrated via calculation of Kaplan–Yorke dimension that is high, i.e., generated waveforms can find interesting applications in the fields of chaotic masking, modulation, or chaos-based cryptography.

Stochastic bifurcations in a nonlinear tri-stable energy harvester under colored noise

In this paper, the stochastic bifurcations and the performance analysis of a strongly nonlinear tri-stable energy harvesting system with colored noise are investigated. Using the stochastic averaging method, the averaged Fokker–Plank–Kolmogorov equation and the stationary probability density (SPD) of the amplitude are obtained, respectively. Meanwhile, the Monte Carlo simulations are performed to verify the effectiveness of the theoretical results. D-bifurcation is studied through the largest Lyapunov exponent calculations, which implies the system undergoes D-bifurcation twice with increasing the nonlinear stiffness coefficients. The effects of the nonlinear stiffness coefficients, noise intensity and correlation time on P-bifurcation are discussed by the qualitative changes of the SPD. Moreover, the relationship between D- and P-bifurcation is explored. If the strength of stochastic jump has obvious gap with respect to the two statuses before and after the occurrence of P-bifurcation, the D-bifurcation will happen, too. Finally, the performance and the capability of harvesting energy from ambient random excitation are analyzed.

Identifying the linear region based on machine learning to calculate the largest Lyapunov exponent from chaotic time series.

To reduce the error caused by the human factor, this paper proposes a modification of a well-known small data method to obtain the largest Lyapunov exponent more accurately, which is based on machine learning for better identification of linear region. Firstly, we use the k-d tree neighborhood search algorithm to improve the computational efficiency of the average divergence index data. Secondly, the unsaturated data are obtained by the density peak based clustering algorithm from the average divergence index data. Thirdly, we use the density peak based clustering algorithm to identify the linear region from the first-order difference curve of the retained data. Finally, the largest Lyapunov exponent is obtained by using the least squares method to fit the linear region. Our method is applied to simulate five famous theoretical chaotic systems, the results show that the proposed method can automatically identify the linear region, which is more accurate than the small data method for the largest Lyapunov exponent calculation and the effectiveness of our method is verified through the simulation of two real-world time series.

Analyzing Complex Dynamic Modes in Different Modifications of Relativistic Generators on a Virtual Cathode

Features of chaotic vibrations generated in one-beam and multibeam vircators are studied. The transition to the chaotic generation is found to pass in accordance with a complex quasi-periodical dynamics, which is confirmed by the calculated Lyapunov exponents.

Computational algorithms of time series for stick-slip dynamics and time-delayed feedback control of chaos for a class of discontinuous friction systems

Abstract A modification of the Henon method is proposed for the straightforward and simultaneous computation of time series with “exact” stick-slip transitions and fixed step time series related to nonlinear dynamics of Filippov-type discontinuous mechanical friction systems. Indeed, unaccessible original trajectories are obtained with the proposed new method. Also, the “exact” numerical simulations of those systems under time delay effects are now possible. So, time-delayed feedback control is used, thanks to an additional method based on that novel approach, to stabilize unstable periodic orbits embedded in chaotic attractors of particular discontinuous systems. Valuable information on the control and for the Poincare map and Lyapunov exponent computations are obtained. The proposed methods seem even to be the key to address non-conventional bifurcations occurring in autonomous delayed systems with dry friction. The modification technique provided is applicable to event-driven schemes in general.

Chaos of Wilson loop from string motion near black hole horizon

To find the origin of chaos near black hole horizon in string-theoretic AdS/CFT correspondence, we perform a chaos analysis of a suspended string in AdS black hole backgrounds. It has a definite CFT interpretation: chaos of Wilson loops, or in other words, sensitive time-evolution of a quark antiquark force in thermal gauge theories. Our nonlinear numerical simulation of the suspended Nambu-Goto string shows chaos, which would be absent in pure AdS background. The calculated Lyapunov exponent $\lambda$ satisfies the universal bound $\lambda \leq 2\pi T_{\rm H}$ for the Hawking temperature $T_{\rm H}$. We also analyze a toy model of a rectangular string probing the horizon and show that it contains a universal saddle characterized by the surface gravity $2\pi T_{\rm H}$. Our work demonstrates that the black hole horizon is the origin of the chaos, and suggests a close interplay between chaos and quark deconfinement.

Forecast of Chaotic Series in a Horizon Superior to the Inverse of the Maximum Lyapunov Exponent

In this article, two models of the forecast of time series obtained from the chaotic dynamic systems are presented: the Lorenz system, the manufacture system, and the volume of the Great Salt Lake of Utah. The theory of the nonlinear dynamic systems indicates the capacity of making good-quality predictions of series coming from dynamic systems with chaotic behavior up to a temporal horizon determined by the inverse of the major Lyapunov exponent. The analysis of the Fourier power spectrum and the calculation of the maximum Lyapunov exponent allow confirming the origin of the series from a chaotic dynamic system. The delay time and the global dimension are employed as parameters in the models of forecast of artificial neuronal networks (ANN) and support vector machine (SVM). This research demonstrates how forecast models built with ANN and SVM have the capacity of making forecasts of good quality, in a superior temporal horizon at the determined interval by the inverse of the maximum Lyapunov exponent or theoretical forecast frontier before deteriorating exponentially.

Chaotic behavior and fractals discovered in the time evolution of discharge current at atmospheric pressure

In this study, nonlinear behaviors in the time evolution of discharge current at atmospheric pressure are investigated from the viewpoints of chaos and fractals. When the electric field between two electrodes placed in atmospheric pressure increases and exceeds a certain value, the insulation breakdown of air and corona discharge occur. With the further increase in the electric field, the corona discharge gradually transforms into an arc discharge along with the increasing discharge current. It is found from the chaotic analysis results of the calculation of the largest Lyapunov exponents that the system sustains more chaos in the corona than the arc discharge, i.e., the system is stabilized with the system changing from a corona to arc discharge. The correlation dimension of the time evolution of the discharge current saturates at a non-integer value at every discharge current, and 1f-type spectra are observed in the power spectrum, i.e., the system exhibits fractal properties.

Highly efficient computation of Finite-Time Lyapunov Exponents (FTLE) on GPUs based on three-dimensional SPH datasets

“Smoothed particle hydrodynamics” (SPH) is a particle method that becomes increasingly popular in different fields of science and engineering. Reason for the popularity are the different advantages in comparison to conventional grid-based computational fluid dynamics (CFD). One example is the much cheaper identification of ”Lagrangian coherent structures” (LCS) in fluid flows by means of the “finite-time Lyapunov exponent” (FTLE). Schemes for the evaluation of FTLE fields based on SPH datasets already exist. Despite the smaller computational effort required in case of SPH data, their evaluation is still costly. This may be the reason that no investigations have been published presently which address the application of existing schemes to SPH-data in 3-D. Therefore in the current paper a new and highly efficient GPU implementation of an existing scheme for the evaluation of FTLE fields is proposed that enables the interactive analysis of large SPH datasets. The suitability of the scheme in case of 3-D datasets and the computational efficiency of the novel GPU implementation are demonstrated. Furthemore, the so called particle birthtime is presented as a cheap alternative to FTLE fields, even though it has a variety of limitations compared to FTLE fields.

Multistability in a three-dimensional oscillator: tori, resonant cycles and chaos

The emergence of multistability in a simple three-dimensional autonomous oscillator is investigated using numerical simulations, calculations of Lyapunov exponents and bifurcation analysis over a broad area of two-dimensional plane of control parameters. Using Neimark–Sacker bifurcation of 1:1 limit cycle as the starting regime, many parameter islands with the coexisting attractors were detected in the phase diagram, including the coexistence of torus, resonant limit cycles and chaos; and transitions between the regimes were considered in detail. The overlapping between resonant limit cycles of different winding numbers, torus and chaos forms the multistability.

On the contact interaction of a two-layer beam structure with clearance described by kinematic models of the first, second and third order approximation

Abstract In this article, the contact interaction of two geometrically non-linear beams with a small clearance is studied by employing various approximations of the kinematic beam models. Our investigation concerns two-case studies, i.e. when the upper beam is governed by the second order Timoshenko model (i) or by third order Pelekh-Sheremetev-Reddy-Levinson model (ii), whereas in both cases the lower beam is described by the kinematic (Bernoulli-Euler) model of the first approximation. The upper beam is subjected to the transversal, uniformly distributed harmonic load, whereas the beam interaction follows the classical Kantor’s model. The problem is highly non-linear due to occurrence of the geometric von Karman non-linearity and the contact interaction between beams (structural non-linearity). The governing PDEs are reduced to ODEs by the method of finite differences (FDM) of the second order. The obtained ODEs are solved by a few Runge-Kutta type methods of different orders. Results of convergence versus the number of partition points along the spatial co-ordinate and time steps are investigated. New non-linear phenomena of the studied structural package are detected, illustrated and discussed with emphasis on the “true” chaotic vibrations. In particular, three qualitatively different algorithms for computation of largest Lyapunov exponents are employed, and the differences between geometrically linear versus non-linear problems are reported.

Calculation of Lyapunov exponents in impacted beam on distributed contact

This paper presents the calculation of Lyapunov exponents in a beam subjected to distributed impact. In this case, discontinuity sporadically occurs over the contacted area. The impact area is spatially discretized and the transition condition at impact instance is applied to all impact nodes in the finite element manner. The solution of the continuous vibration in the impacted beam is expressed in the modal expansion form. Then, the spectra of Lyapunov exponents in the periodic and chaotic motion are estimated for the truncated number of modes. Numerical results show that the calculated largest Lyapunov exponent agrees with the bifurcation plot.

Region of interest encryption for color images based on a hyperchaotic system with three positive Lyapunov exponets

We propose a hyperchaotic encryption algorithm for the region of interest (ROI) in a color image. The ROI for the human face is first identified based on the Open Source Computer Vision library by using a Gaussian mixture model. It is then encrypted using a novel hyperchaotic system, which has three positive Lyapunov exponents. The system can be in a hyperchaotic state with the parameter changing over a wide region, which is proved by Lyapunov exponent spectrum calculations. Furthermore, hierarchical keys are adopted to ensure the security of the encrypted sequences generated. In the encryption process, crossed image row–column scrambling and pixel value diffusion are experienced. The hyperchaotic sequences required for the encryption relate not only to the keys but also to the plaintext images. Consequently, the algorithm is effective against plaintext and ciphertext attacks. The security performance is also verified in detail with respect to histograms, correlations, information entropy, key space, key sensitivity analysis, number of pixel change rate, unified average changing intensity, and known and chosen attacks. The results indicate that the proposed algorithm has a large key space and a low correlation between adjacent pixels, hence, it can also be usefully employed to resist brute-force and statistical attacks.

Computation of the largest positive Lyapunov exponent using rounding mode and recursive least square algorithm

It has been shown that natural interval extensions (NIE) can be used to calculate the largest positive Lyapunov exponent (LLE). However, the elaboration of NIE are not always possible for some dynamical systems, such as those modelled by simple equations or by Simulink-type blocks. In this paper, we use rounding mode of floating-point numbers to compute the LLE. We have exhibited how to produce two pseudo-orbits by means of different rounding modes; these pseudo-orbits are used to calculate the Lower Bound Error (LBE). The LLE is the slope of the line gotten from the logarithm of the LBE, which is estimated by means of a recursive least square algorithm (RLS). The main contribution of this paper is to develop a procedure to compute the LLE based on the LBE without using the NIE. Additionally, with the aid of RLS the number of required points has been decreased. Eight numerical examples are given to show the effectiveness of the proposed technique.

Application of chaos theory to the particle dynamics of asymmetry-induced transport

The techniques of chaos theory are employed in an effort to better understand the complex single-particle dynamics of asymmetry-induced transport in non-neutral plasmas. The dynamical equations are re-conceptualized as describing time-independent trajectories in a four-dimensional space consisting of the radius r, rotating frame angle ψ, axial position z, and axial velocity v. Results include the identification of an integral of the motion, fixed-point analysis of the dynamical equations, the construction and interpretation of Poincaré sections to visualize the dynamics, and, for the case of chaotic motion, numerical calculation of the largest Lyapunov exponent. Chaotic cases are shown to be associated with the overlap of resonance islands formed by the applied asymmetry.

Dynamical behaviors in a discrete fractional-order predator-prey system

This paper is related to the dynamical behaviors of a discrete-time fractional-order predatorprey model. We have investigated existence of positive fixed points and parametric conditions for local asymptotic stability of positive fixed points of this model. Moreover, it is also proved that the system undergoes Flip bifurcation and Neimark-Sacker bifurcation for positive fixed point. Various chaos control strategies are implemented for controlling the chaos due to Flip and Neimark-Sacker bifurcations. Finally, numerical simulations are provided to verify theoretical results. These results of numerical simulations demonstrate chaotic behaviors over a broad range of parameters. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behaviors in the model.