Lyapunov Exponent Method Research Articles

Cluster-based reduced-order modelling of a mixing layer

AbstractWe propose a novel cluster-based reduced-order modelling (CROM) strategy for unsteady flows. CROM combines the cluster analysis pioneered in Gunzburger’s group (Burkardt, Gunzburger & Lee,Comput. Meth. Appl. Mech. Engng, vol. 196, 2006a, pp. 337–355) and transition matrix models introduced in fluid dynamics in Eckhardt’s group (Schneider, Eckhardt & Vollmer,Phys. Rev. E, vol. 75, 2007, art. 066313). CROM constitutes a potential alternative to POD models and generalises the Ulam–Galerkin method classically used in dynamical systems to determine a finite-rank approximation of the Perron–Frobenius operator. The proposed strategy processes a time-resolved sequence of flow snapshots in two steps. First, the snapshot data are clustered into a small number of representative states, called centroids, in the state space. These centroids partition the state space in complementary non-overlapping regions (centroidal Voronoi cells). Departing from the standard algorithm, the probabilities of the clusters are determined, and the states are sorted by analysis of the transition matrix. Second, the transitions between the states are dynamically modelled using a Markov process. Physical mechanisms are then distilled by a refined analysis of the Markov process, e.g. using finite-time Lyapunov exponent (FTLE) and entropic methods. This CROM framework is applied to the Lorenz attractor (as illustrative example), to velocity fields of the spatially evolving incompressible mixing layer and the three-dimensional turbulent wake of a bluff body. For these examples, CROM is shown to identify non-trivial quasi-attractors and transition processes in an unsupervised manner. CROM has numerous potential applications for the systematic identification of physical mechanisms of complex dynamics, for comparison of flow evolution models, for the identification of precursors to desirable and undesirable events, and for flow control applications exploiting nonlinear actuation dynamics.

The Application of Random Noise Reduction By Nearest Neighbor Method To Forecasting of Economic Time Series

Abstract Since the deterministic chaos appeared in the literature, we have observed a huge increase in interest in nonlinear dynamic systems theory among researchers, which has led to the creation of new methods of time series prediction, e.g. the largest Lyapunov exponent method and the nearest neighbor method. Real time series are usually disturbed by random noise, which can complicate the problem of forecasting of time series. Since the presence of noise in the data can significantly affect the quality of forecasts, the aim of the paper will be to evaluate the accuracy of predicting the time series filtered using the nearest neighbor method. The test will be conducted on the basis of selected financial time series.

Separated Flow Response to Single Pulse Actuation

The response of a separated flow over a two-dimensional wing to short-duration disturbances from a Lorentz force leading-edge actuator is presented. At a chord Reynolds number of and an angle of attack of , the flow is initially laminar but undergoes transition within the separated shear layer.The transient flow structures and lift force measurements were obtained with varying actuator pulse duration, pulse amplitude, and direction of actuation. The peak amplitude of the lift is shown to depend on the pulse duration when the pulse duration is less than 0.5 convective times, after which it saturates. Saturation of the lift amplitude also occurs when the effective actuator pulse amplitude exceeds . Detailed flow structures that develop in the separated shear layer were identified using the finite-time Lyapunov exponent method. The direction of the actuator pulse has a significant influence on the initial development of the shear layer, but the larger-scale envelope of the separated flow has essentially the same response, irrespective of the direction of actuation. A reconstruction of the velocity field using only three proper orthogonal decomposition modes is sufficient to reproduce the dominant features of the flow response to a single pulse. The second proper orthogonal decomposition mode correlates with the transient lift signal.

INDUCING CHAOS BY BREAKING AXIAL SYMMETRY IN A BLACK HOLE MAGNETOSPHERE

While the motion of particles near a rotating, electrically-neutral (Kerr), and charged (Kerr--Newman) black hole is always strictly regular, a perturbation in the gravitational or the electromagnetic field generally leads to chaos. The transition from regular to chaotic dynamics is relatively gradual if the system preserves axial symmetry, whereas non-axisymmetry induces chaos more efficiently. Here we study the development of chaos in an oblique (electro-vacuum) magnetosphere of a magnetized black hole. Besides the strong gravity of the massive source represented by the Kerr metric we consider the presence of a weak, ordered, large-scale magnetic field. An axially-symmetric model consisting of a rotating black hole embedded in an aligned magnetic field is generalized by allowing an oblique direction of the field having a general inclination, with respect to the rotation axis of the system. The inclination of the field acts as an additional perturbation to the motion of charged particles as it breaks the axial symmetry of the system and cancels the related integral of motion. The axial component of angular momentum is no longer conserved and the resulting system thus has three degrees of freedom. Our primary concern within this contribution is to find out how sensitive the system of bound particles is to the inclination of the field. We employ the method of the maximal Lyapunov exponent to distinguish between regular and chaotic orbits and to quantify their chaoticity. We find that even a small misalignment induces chaotic motion.

Application of Vector Control to Permanent Magnet Synchronous Motors Using Chaos Control

This paper presents the idea of vector control for permanent magnet synchronous motor (PMSM) based on chaos theorem, using chaos controller. PMSM will demonstrate chaotic phenomena when its parameters fall into a certain area. To achieve this aim, the sub-system of controller has been designed by considering block diagram structure of vector control for PMSM and by applying the setting of Lyapunov exponents method. Also, asymptotical stability of closed loop system with given controller is shown, using the direct Lyapunov method. The performance of designed controller in chaotic mode is compared with conventional vector control methods. Also, the normal mode for PMSM is considered and the performance of controller is compared. Simulation results indicate that not only does this controller eliminate the chaos in chaotic mode and have good performance but also is able to control the system in normal mode by using almost the smaller control signal effort.

Forecasting available parking space with largest Lyapunov exponents method

The techniques to forecast available parking space (APS) are indispensable components for parking guidance systems (PGS). According to the data collected in Newcastle upon Tyne, England, the changing characteristics of APS were studied. Thereafter, aiming to build up a multi-step APS forecasting model that provides richer information than a conventional one-step model, the largest Lyapunov exponents (largest LEs) method was introduced into PGS. By experimental tests conducted using the same dataset, its prediction performance was compared with traditional wavelet neural network (WNN) method in both one-step and multi-step processes. Based on the results, a new multi-step forecasting model called WNN-LE method was proposed, where WNN, which enjoys a more accurate performance along with a better learning ability in short-term forecasting, was applied in the early forecast steps while the Lyapunov exponent prediction method in the latter steps precisely reflect the chaotic feature in latter forecast period. The MSE of APS forecasting for one hour time period can be reduced from 83.1 to 27.1 (in a parking building with 492 berths) by using largest LEs method instead of WNN and further reduced to 19.0 by conducted the new method.

Dynamics of pressure oscillations in flow boiling and condensation in the minichannel

The dynamics of pressure and temperature fluctuations occurring in flow boiling in a minichannel of 1mm diameter, which consists of the heated section and glass condenser have been experimentally investigated and modelled. The amount of vapour in the condenser, the wall temperature of heated channel outlet and pressure drop were simultaneously measured. Data has been analyzed using the Fourier and wavelet methods. The dynamics of pressure and temperature fluctuations has been investigated using the attractor reconstruction, correlation dimension and largest Lyapunov exponent methods. A high speed digital video camera has been used to identify two-phase flow patterns in the glass condenser. The high frequency pressure fluctuations with relatively small amplitude have been observed. It has been found that during those fluctuations the dominant frequency of pressure and temperature fluctuations changes chaotically in time. Such pressure oscillations are accompanied by appearance and disappearance of slug flow in the condenser. It has been shown that dynamics of appearance and disappearance of slug flow has a multifractal character. The model of the oscillations between quasi steady two-phase flow patterns has been presented. It has been shown that the necessary condition for the appearance of chaotic fluctuations of parameters describing the system is the slope of function Δp(G). The qualitative agreement between shapes of return maps obtained in the model and in the experiment has been obtained.

Sources of irreversibilities in a perfectly Axisymmetric geometry at low Reynolds numbers

In plastic industry, a large variety of polymeric materials are obtained by liquid–liquid mixing. Recently, a new type of mixer (later called RMX)—based on a convergent/divergent unit which favors elongational flow—was developed with improved efficiency in polymer blending compared to traditional shear flow‐based devices such as internal mixers or extruders. The present study aims at identifying the possible origins of the higher dispersive and distributive mixing capability of this device. An experimental study, including flow visualization, and numerical simulations of creeping flows induced by this original device was carried out for Newtonian, shear‐thinning and viscoelastic fluids. The predictedmixing capability is based on length stretch and Lyapunov exponent methods. Calculation results show a good agreement with experiments and highlight some fundamental mixing mechanisms, such as striation, for viscoelatic fluids. POLYM. ENG. SCI., 54:2046–2056, 2014. © 2013 Society of Plastics Engineers

To What Extent Does Not Wearing Shoes Affect the Local Dynamic Stability of Walking?: Effect Size and Intrasession Repeatability

Local dynamic stability (stability) quantifies how a system responds to small perturbations. Several experimental and clinical findings have highlighted the association between gait stability and fall risk. Walking without shoes is known to slightly modify gait parameters. Barefoot walking may cause unusual sensory feedback to individuals accustomed to shod walking, and this may affect stability. The objective was therefore to compare the stability of shod and barefoot walking in healthy individuals and to analyze the intrasession repeatability. Forty participants traversed a 70 m indoor corridor wearing normal shoes in one trial and walking barefoot in a second trial. Trunk accelerations were recorded with a 3D-accelerometer attached to the lower back. The stability was computed using the finite-time maximal Lyapunov exponent method. Absolute agreement between the forward and backward paths was estimated with the intraclass correlation coefficient (ICC). Barefoot walking did not significantly modify the stability as compared with shod walking (average standardized effect size: +0.11). The intrasession repeatability was high (ICC: 0.73-0.81) and slightly higher in barefoot walking condition (ICC: 0.81-0.87). Therefore, it seems that barefoot walking can be used to evaluate stability without introducing a bias as compared with shod walking, and with a sufficient reliability.

Wave Propagation and Localization in a Randomly Disordered Periodic Piezoelectric Axial-Bending Coupled Beam

The wave propagation and localization in periodic and randomly disordered periodic piezoelectric axial-bending coupled beams are studied in this paper. The dynamic stiffness matrix is derived based on the finite element model and the transfer matrix between two adjacent cells is obtained by using the continuity conditions. In addition, the Lyapunov exponent method is employed to calculate the localization factor characterizing the average exponential rate of decay of the wave amplitude in the disordered periodic structure. Through the comprehensive numerical simulations, the effects of different material and geometric properties as well as disordered parameters on the localization factors are investigated. The results show that wave propagation and localization in the periodic and disordered periodic piezoelectric beam can be altered by properly adjusting the structural parameters.

VANTAGE: A Lyapunov exponents based technique for identification of coherent groups of generators in power systems

Recent trends of deregulation and expansion in power systems have greatly increased their stress levels and hence, ensuring their safety has become a critical challenge. In the event of disturbances, detecting the groups of coherent generators is important to study the characteristics of the system. We propose VANTAGE, a Lyapunov exponents based technique for identification of coherent groups of generators in power systems. VANTAGE performs time domain simulation of contingencies and uses this data to analyze rotor angle stability. VANTAGE offers an efficient algorithmic procedure for identification of coherent groups of generators using Lyapunov exponents method (LEM). VANTAGE does not require a priori estimation of number of groups of coherent generators. We evaluate VANTAGE using a 39-bus system and a 13029-bus system. The results demonstrate the effectiveness of VANTAGE in correctly identifying the coherent groups of generators.

Wave propagation in a periodic elastic-piezoelectric axial-bending coupled beam

The wave propagation in a periodic elastic-piezoelectric axial-bending coupled beam is investigated in this paper by considering the mechanical–electrical coupling behavior. The strain energy and kinetic energy of each sub-cell are first formulated to extract the dynamic stiffness matrices, and then the compatibility and continuity conditions at the interface between the adjacent cells are utilized to derive the transfer matrix that governs the propagation of the wave along the periodic piezoelectric beam. By employing the Lyapunov exponent method, the dynamic behaviors of the periodic beam structure are evaluated with different base beam materials, dimension ratios, piezoelectric constants and elastic stiffness. The results indicate that regardless of the length ratio, there exist certain frequency intervals, where the width and magnitude of the prominent stop band of the aluminum beam with periodic piezoelectric patches are always broader and larger than those of the steel base system. In addition, as the thickness ratio decreases, the location of the stop band tends to move toward a higher frequency. Numerical studies also demonstrate that different piezoelectric constants and elastic stiffness affect the characteristics of wave propagation in completely different fashions. The investigation in the present study provides basic guidelines to design periodic elastic-piezoelectric laminate structures in order to achieve desired filtering characteristics.

基于Lyapunov-Elman的网络控制系统时延预测方法

The time-delay in the networked control system is an important parameter affecting the performance of a system. A maximum Lyapunov exponent and Elman neural network prediction method is presented according to the time-delay prediction in the networked control system based on the Internet. Firstly, time-delay sequences are reconstructed by phase space reconstruction, embedding dimension and time delayed variable is obtained. Then one step time-delay is forecasted by maximum Lyapunov exponent and Elman neural network. A final time delay prediction value is obtained after two prediction methods are multiplied by different weight coefficients. Finally, based on the weight coefficient optimization, this paper presents an improved free search algorithm, whose convergence accuracy and speed are superior to those of the standard free search algorithm. Simulation results show that, time-delay prediction method based on maximum Lyapunov exponent and Elman neural network in the paper has higher prediction accuracy and smaller prediction error than other prediction methods.

Multiple time scales analysis of runoff series based on the Chaos Theory

The significance of accepting runoff processes as nonlinear has been gaining considerable in recent times. However, it is hard to explore the types of nonlinearity acting underlying the runoff processes and the intensity of the nonlinearity at different timescales. Daily runoff time series observed at the Pingshan hydrometric station are used for this study. An attempt is made to identify the existence of chaos and the intensity of nonlinear behavior at three characteristic time scales (one day, 1/3 month, and one month). Six nonlinear dynamic methods are used: (1) phase space reconstruction and the delay time is estimated using average mutual information; (2) the sufficient embedding dimension is estimated using the false nearest neighbor algorithm; (3) correlation dimension method; (4) Lyapunov exponent method; (5) 0–1 test algorithm for chaos; and (6) the multi-step Volterra adaptive method. A comparison of results reveals the presence of low-dimensional chaos in the runoff dynamics at the various time scales and the time scales composes only a limit fraction of the intensity of nonlinear behavior. The reasonably good predictions indicate the efficiency of the nonlinear prediction method for predicting the runoff series.

Experimental investigation of the wall shear stress in a circular impinging jet

The influence of the large-scale vortical structures on the wall shear stress in a circular impinging jet is investigated experimentally for a Reynolds number of 1260. Time-resolved particle image velocimetry and polarographic measurements are performed simultaneously. It is found that the instantaneous wall shear stress is strongly dependent on the vortex dynamics, particularly for different parts of the transverse vortex. The influence of the vortex ring, the secondary and tertiary vortices on the ejection/sweep process near the wall is the main mechanism involved in the wall shear stress variation. In the region of the boundary layer separation, the wall shear stress amplitude increases just upstream of the separation and dramatically decreases in the recirculation zone downstream from the separation. The interaction between primary and secondary structures and their pairing process with the tertiary structure affects the sweep/ejection process near the wall and subsequently the wall shear stress variation. A comparison between the Finite Time Lyapunov Exponent (FTLE) method and the phase average technique is performed. It is shown that both methods describe the flow dynamics in the impinging region of the vortex ring. However, the FTLE method is more suitable for describing the unsteady separation of the boundary layer.

Lyapunov function construction for nonlinear stochastic dynamical systems

Though the Lyapunov function method is more efficient than the largest Lyapunov exponent method in evaluating the stochastic stability of multi-degree-of-freedom (MDOF) systems, the construction of Lyapunov function is a challenging task. In this paper, a specific linear combination of subsystems’ energies is proposed as Lyapunov function for MDOF nonlinear stochastic dynamical systems, and the corresponding sufficient condition for the asymptotic Lyapunov stability with probability one is then determined. The proposed procedure to construct Lyapunov function is illustrated and validated with several representative examples, where the influence of coupled/uncoupled dampings and excitation intensities on stochastic stability is also investigated.

Chaotic Response and Bifurcation Analysis of Rotor-Bearing System with Nonlinear Suspension and Roughness Effect under Turbulent Flow

This research aims to study a numerical analysis to investigate the dynamic orbits of a rotor supported by two turbulent journal bearings considering longitudinal and transverse roughness effect under nonlinear suspension. The dynamic response of the rotor center and bearing center are studied. The dynamic trajectories of the rotor center and bearing center, Poincaré maps, bifurcation diagrams and the maximum Lyapunov exponent method are employed to analyze dynamic responses in this study. The results show that either the longitudinal and transverse or the values of roughness parameters c1 would not affect strongly nonlinear rotor-bearing system very much but the difference of dynamic response between considering with roughness effect and without roughness effect are very obvious. The modeling results provide some theoretical references for further researching of rotor-bearing system for rotating machinery that operate in highly rotational speed, especially it considers roughness effect, turbulent flow and nonlinear suspension of bearing system.

Chaos Many-Body Engine v03: A new version of code C# for chaos analysis of relativistic many-body systems with reactions

In this paper we present a new version of the Chaos Many-Body Engine C# application (Grossu et al. 2012 [1]). In order to benefit from the latest technological advantages, we migrated the application from .Net Framework 2.0 to .Net Framework 4.0. New tools were implemented also. Trying to estimate the particle interactions dependence on initial conditions, we considered a new distance, which takes into account only the structural differences between two systems. We used this distance for implementing the “Structural Lyapunov” function. We propose also a new precision test based on temporal reversed simulations. New version program summaryProgram title: Chaos Many-Body Engine v03Catalogue identifier: AEGH_v3_0Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEGH_v3_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 214429No. of bytes in distributed program, including test data, etc.: 9512380Distribution format: tar.gzProgramming language: Visual C# .Net 2010Computer: PCOperating system: .Net Framework 4.0 running on MS WindowsRAM: 128 MBClassification: 24.60.Lz, 05.45.aCatalogue identifier of previous version: AEGH_v2_0Journal reference of previous version: Computer Physics Communications 183 (2012) 1055–1059Does the new version supersede the previous version?: YesNature of problem: Chaos analysis of three-dimensional, relativistic many-body systems with reactions.Solution method: Second order Runge–Kutta algorithm. Implementation of temporal reversed simulation precision test, and “Structural Lyapunov” function.Reasons for new version:1.In order to benefit from the advantages involved in the latest technologies (e.g. LINQ Queries [2]), Chaos Many-Body Engine was migrated from .Net Framework 2.0 to .Net Framework 4.0.2.In addition to existing energy conservation assessment [3], we propose also a reverse simulation precision test. Thus, for a regular simulation, we considered the corresponding reversed process: initial time equals the end time of regular simulation, and temporal resolution dt<0. One can compare the initial state of the regular system, and the final state of the reversed one (t=0) using, for example, the phase-space distance.3.Trying to measure particle interactions dependence on initial conditions, we considered the following distance, which takes into account only the structure differences between two many-body systems with reactions: (1)ds=∑i=1n(Ni1−Ni2)2 where Ni1 represents the number of particles of type “i” from the first system, and Ni2 is the corresponding number for the second system. We sum over all particle types.Inspired by the Lyapunov Exponent method [4], we implemented the evolution in time of the “Structural Lyapunov” function, for two identical systems with slightly different initial conditions: (2)Ls(t)=lnds(t)ds(0).Summary of revisions:1.Migration from .Net Framework 2.0 to .Net Framework 4.02.Implementation of new chaos analysis tools: a.Reverse simulation precision testb.“Structural Lyapunov” function.Additional comments:1.In [1] we applied the Chaos Many-Body Engine to some nuclear relativistic collisions at 4.5 A GeV/c (SKM 200 collaboration [5,6]). We considered also some first tests on He+He head-on collisions at 1 A TeV/c (choose the Simulation∖Collision menu, and set the appropriate parameters Fig. 1). However, in this case, more complex reaction schemas should be considered. Further investigation on higher energies is currently in progress. [Display omitted] Restrictions: The reverse simulation precision test does not apply for: systems with reactions, parallel simulations, and Monte Carlo simulations.Running time: quadratic complexity.

An analytic criterion for generalized synchronization in unidirectionally coupled systems based on the auxiliary system approach

An analytic criterion is developed to investigate generalized synchronization (GS) in unidirectionally coupled systems based on the auxiliary system approach. The criterion is derived by transforming the existence problem of generalized synchronization into an eigenvalue problem. Numerical simulations show that the analytic criterion is almost as accurate as the response Lyapunov exponents method, and may provide an estimation of the threshold of strong generalized synchronization. A significant result can be deduced from our analysis that the more the number of equilibria of the unidirectionally coupled systems, the greater the chance of generalized synchronization, but the harder it may be for strong generalized synchronization to occur.

On the stability of columns under stochastic loading taking into account nonlocal damping

Using an example of a column clamped at both ends, the stability of columns made of a material characterized by nonlocal damping is investigated. The column is subjected to axial forces that are deterministic and randomly vary in time. In the deterministic case, the stability is understood in Lyapunov’s sense; in the case of the stochastic formulation of the problem, the almost sure stability is considered. In either case, the method of the top Lyapunov exponent is used to analyze the stability.