Dominant Lyapunov Exponent Research Articles

Verification of chaotic behavior in an experimental loudspeaker

The dynamics of an experimental electrodynamic loudspeaker is studied by using the tools of chaos theory and time series analysis. Delay time, embedding dimension, fractal dimension, and other empirical quantities are determined from experimental data. Particular attention is paid to issues of stationarity in a system in order to identify sources of uncertainty. Lyapunov exponents and fractal dimension are measured using several independent techniques. Results are compared in order to establish independent confirmation of low dimensional dynamics and a positive dominant Lyapunov exponent. We thus show that the loudspeaker may function as a chaotic system suitable for low dimensional modeling and the application of chaos control techniques.

Does Dormancy Increase Fitness of Bacterial Populations in Time-Varying Environments?

A simple family of models of a bacterial population in a time varying environment in which cells can transit between dormant and active states is constructed. It consists of a linear system of ordinary differential equations for active and dormant cells with time-dependent coefficients reflecting an environment which may be periodic or random, with alternate periods of low and high resource levels. The focus is on computing/estimating the dominant Lyapunov exponent, the fitness, and determining its dependence on various parameters and the two strategies-responsive and stochastic-by which organisms switch between dormant and active states. A responsive switcher responds to good and bad times by making timely and appropriate transitions while a stochastic switcher switches continuously without regard to the environmental state. The fitness of a responsive switcher is examined and compared with fitness of a stochastic switcher, and with the fitness of a dormancy-incapable organism. Analytical methods show that both switching strategists have higher fitness than a dormancy-incapable organism when good times are rare and that responsive switcher has higher fitness than stochastic switcher when good times are either rare or common. Numerical calculations show that stochastic switcher can be most fit when good times are neither too rare or too common.

Role of inducible defenses in the stability of a tritrophic system

Inducible defenses are a form of phenotypic plasticity that potentially modify direct interactions between various members of an ecological community, generating trait-mediated indirect effects. In this work, the hypothesis that inducible defenses increase the stability of tritrophic chains is tested, through the numerical analysis of a continuous-time model that discriminate between defenses affecting attack rate of predators, and defenses affecting predator handling time. In addition, discrimination between feeding costs of defenses affecting attack rate, and metabolic costs affecting feeding requirement for zero growth are considered. System stability was examined by computing dominant Lyapunov exponents, and through continuation routines of bifurcation points. Background parameter values were taken from two published studies. Our results show that a tritrophic system will generally be stabilized by the incorporation of inducible defenses and by their associated costs, but a number of new outcomes were obtained. Different long-term behavior is predicted if either one or two prey populations exhibit defenses. In the latter case, the defense of the basal prey dominates the dynamics. Handling time based inducible defenses exert a stronger stabilizing effect than attack rate based ones, but also impose a higher extinction risk for top predators. Inducible defenses in particular and trait-mediated indirect effects in general can be important sources of stability in natural systems.

Chaotic monetary dynamics with confidence

This paper uses tools from dynamical systems theory to investigate the properties of Canadian and US money and velocity measures. In doing so, we follow the recent contribution by Whang and Linton [Whang, Y.-J., Linton, O., 1999. The asymptotic distribution of nonparametric estimates of the Lyapunov exponent for stochastic time series. Journal of Econometrics 91, 1–42] and construct the standard error for the Nychka et al. [Nychka, D.W., Ellner, S., Gallant, R.A., McCaffrey, D., 1992. Finding chaos in noisy systems. Journal of the Royal Statistical Society B 54, 399–426] dominant Lyapunov exponent. Comparisons are made among simple-sum, Divisia, and currency equivalent aggregates at different levels of monetary aggregation. We find statistically significant evidence against low-dimensional chaos and point to the use of stochastic models and statistical inference in the modeling of these variables.

Chaotic vibration of a nonlinear full-vehicle model

The present work investigates the chaotic responses of a nonlinear seven degree-of-freedom ground vehicle model. The disturbances from the road are assumed to be sinusoid and the time delay between the disturbances is investigated. Numerical results show that the responses of the vehicle model could be chaotic. With the bifurcation phenomenon detected, the chaotic motion is confirmed with the dominant Lyapunov exponent. The results can be useful in dynamic design of a vehicle.

Spatiotemporal complexity of patchy invasion in a predator-prey system with the Allee effect.

Invasion of an exotic species initiated by its local introduction is considered subject to predator–prey interactions and the Allee effect when the prey growth becomes negative for small values of the prey density. Mathematically, the system dynamics is described by two nonlinear diffusion–reaction equations in two spatial dimensions. Regimes of invasion are studied by means of extensive numerical simulations. We show that, in this system, along with well-known scenarios of species spread via propagation of continuous population fronts, there exists an essentially different invasion regime which we call a patchy invasion. In this regime, the species spreads over space via irregular motion and interaction of separate population patches without formation of any continuous front, the population density between the patches being nearly zero. We show that this type of the system dynamics corresponds to spatiotemporal chaos and calculate the dominant Lyapunov exponent. We then show that, surprisingly, in the regime of patchy invasion the spatially average prey density appears to be below the survival threshold. We also show that a variation of parameters can destroy this regime and either restore the usual invasion scenario via propagation of continuous fronts or brings the species to extinction; thus, the patchy spread can be qualified as the invasion at the edge of extinction. Finally, we discuss the implications of this phenomenon for invasive species management and control.

Misfire detection of locomotive diesel engine by non-linear analysis

The paper describes the measurements of vibroacoustic exhaust locomotive engine signals for misfire phenomenon simulation and some results of their non-linear analysis. The misfire was simulated by disconnection of fuel supply of one cylinder of the engine. The analysis consists of non-linear methods that are based on the deterministic chaos theory. The results of the analysis demonstrate the dominant Lyapunov exponents in the case when all the cylinders are in operation to be differing considerably from those obtained when one cylinder does not work. The stimulus for our research is expected during nearest years introduction of regulations on limits of combustion gases emission and obligatory on-board diagnostic systems in exhaust locomotives. This fact induced us to perform some experiments on new methods of misfire detection diagnostic.

Determination of Limit Cycles Using Both the Slope of Correlation Integral and Dominant Lyapunov Methods

A method for nonlinear analysis of instabilities in boiling water reactors (BWRs) is presented. Both the Dominant Lyapunov Exponent method and the Slope of the Correlation Integral (SOCI) method are used to analyze the average power reactor monitor (APRM) signals from a BWR. The main advantage of using the two methods in a complementary manner is that doing so results in an enhancement of the capability to analyze noisy systems, such as the APRM signals in a BWR. Previously, such nonlinear analysis had been performed using independently either the Dominant Lyapunov Exponent Method or the SOCI method. These two methods are sensitive to noise in a signal and normally require large amounts of data for a reliable analysis.This proposed system for nonlinear analysis is composed first of a home-developed computer program called “SLOPE,” which is based on the SOCI method. Then, the signal analysis is also performed by the “LENNS” code, which is used to obtain the dominant Lyapunov exponent. Since only the dominant Lyapunov exponent is computed, there is no need to acquire large amounts of data; thus, computational processing time is greatly reduced, even in the case of noisy data.The system was used to analyze BWR signals containing stationary and nonstationary limit cycles. It was found that this method satisfactorily calculates the limit cycles, extracting useful information from noisy signals.

Computation of the dominant Lyapunov exponent via spatial integration using matrix norms

In a previous paper (Aston, P. J. & Dellnitz, M. 1999 Comput. Meth. Appl. Mech. Engng 170, 223-237) we introduced a new method for computing the dominant Lyapunov exponent of a chaotic map by using spatial integration involving a matrix norm. We conjectured that this sequence of integrals decayed proportional to 1/n. We now prove this conjecture and derive a bound on the next term in the asymptotic expansion of the terms in the sequence. The Henon map and a system of coupled Duffing oscillators are explored in detail in the light of these theoretical results.

Time Series Prediction Based on Chaotic Attractor**The project supported by National Natural Science Foundation of China (60074020), the National Outstanding Young Investigator Grant (70225005) of National Natural Science Foundation of China and Research Award Program (2001) for Outstanding Young Teachers in Higher Education Insitutions of Ministry of Education of China

A new prediction technique is proposed for chaotic time series. The usefulness of the technique is that it can kick off some false neighbor points which are not suitable for the local estimation of the dynamics systems. A time-delayed embedding is used to reconstruct the underlying attractor, and the prediction model is based on the time evolution of the topological neighboring in the phase space. We use a feedforward neural network to approximate the local dominant Lyapunov exponent, and choose the spatial neighbors by the Lyapunov exponent. The model is tested for the Mackey-Glass equation and the convection amplitude of lorenz systems. The results indicate that this prediction technique can improve the prediction of chaotic time series.

Can noise induce chaos?

An important component of the mathematical definition of chaos is sensitivity to initial conditions. Sensitivity to initial conditions is usually measured in a deterministic model by the dominant Lyapunov exponent (LE), with chaos indicated by a positive LE. The sensitivity measure has been extended to stochastic models; however, it is possible for the stochastic Lyapunov exponent (SLE) to be positive when the LE of the underlying deterministic model is negative, and vice versa. This occurs because the LE is a long‐term average over the deterministic attractor while the SLE is the long‐term average over the stationary probability distribution. The property of sensitivity to initial conditions, uniquely associated with chaotic dynamics in deterministic systems, is widespread in stochastic systems because of time spent near repelling invariant sets (such as unstable equilibria and unstable cycles). Such sensitivity is due to a mechanism fundamentally different from deterministic chaos. Positive SLE's should therefore not be viewed as a hallmark of chaos. We develop examples of ecological population models in which contradictory LE and SLE values lead to confusion about whether or not the population fluctuations are primarily the result of chaotic dynamics. We suggest that “chaos” should retain its deterministic definition in light of the origins and spirit of the topic in ecology. While a stochastic system cannot then strictly be chaotic, chaotic dynamics can be revealed in stochastic systems through the strong influence of underlying deterministic chaotic invariant sets.

Quantification of the Spatial Aspect of Chaotic Dynamics in Biological and Chemical Systems

The need to study spatio-temporal chaos in a spatially extended dynamical system which exhibits not only irregular, initial-value sensitive temporal behavior but also the formation of irregular spatial patterns, has increasingly been recognized in biological science. While the temporal aspect of chaotic dynamics is usually characterized by the dominant Lyapunov exponent, the spatial aspect can be quantified by the correlation length. In this paper, using the diffusion-reaction model of population dynamics and considering the conditions of the system stability with respect to small heterogeneous perturbations, we derive an analytical formula for an ‘intrinsic length’ which appears to be in a very good agreement with the value of the correlation length of the system. Using this formula and numerical simulations, we analyze the dependence of the correlation length on the system parameters. We show that our findings may lead to a new understanding of some well-known experimental and field data as well as affect the choice of an adequate model of chaotic dynamics in biological and chemical systems.

No evidence of chaos but some evidence of dependence in the US stock market

This paper uses recent advances in the field of applied econometrics and tools from dynamical systems theory to test for random walks and chaos in the US stock market, using daily observations on the Dow Jones Industrial Average (from January 3, 1928 to October 18, 2000––a total of 18,490 observations). In doing so, we follow the recent contribution by Whang and Linton [J Econometr 91 (1999) 1] and construct the standard error for the Nychka et al. [J Roy Statist Soc B 54 (1992) 399] dominant Lyapunov exponent, thereby providing a statistical test of chaos. We find statistically significant evidence against low-dimensional chaos and point to the use of stochastic models and statistical inference in the modeling of asset markets.

Estimation of the dominant Lyapunov exponent of non-smooth systems on the basis of maps synchronization

A novel method of estimation of the largest Lyapunov exponent for discrete maps is introduced and evaluated for chosen examples of maps described by difference equations or generated from non-smooth dynamical systems. The method exploits the phenomenon of full synchronization of two identical discrete maps when one of them is disturbed. The presented results show that this method can be successfully applied both for discrete dynamical systems described by known difference equations and for discrete maps reconstructed from actual time series. Applications of the method for mechanical systems with discontinuities and examples of classical maps are presented. The comparison between the results obtained by means of the known algorithms and novel method is discussed.

Phase Space Prediction Model Based on the Chaotic Attractor

A new prediction technique is proposed for chaotic time series. The usefulnessof the technique is that it removes some false neighbouring points which are notsuitable for the local estimation of the dynamics systems. We use a feedforwardneural network to approximate the local dominant Lyapunov exponent, and choosethe neighbouring points by the exponent. The model is tested for the convectionamplitude of the Lorenz model, and the results indicate that this predictiontechnique can improve the prediction of chaotic time series.

Estimating Lyapunov exponents in biomedical time series.

Among nonlinear dynamical invariants, determination of the largest Lyapunov exponent is well suited to positive identification of chaos in observed time series. When analyzing the dynamics of biomedical series, such as an electro-encephalogram (EEG), model-based methods should be used. Moreover, in the absence of any well founded theoretical model, and because of unexplained variability in the data, candidate models must provide for a stochastic component. Here we use nonlinear autoregressive stochastic modeling to estimate the dominant Lyapunov exponent in an EEG series and compute confidence intervals from surrogate data. The results are found to differ from those of approaches which aim at deleting noise prior to analysis.

Chaotic properties and pattern competition during the learning phase of back propagation neural networks

This paper examines the chaotic behavior of Back Propagation neural networks during the training phase. The networks are trained using ordinary parameter values, while two different cases are considered. In the first one, the network does not meet desirable convergence within a pre-specified number of epochs. Chaotic behavior of this network is depicted by examining the values of the dominant Lyapunov exponents of the weight data series produced by additional training. For each training epoch, the data series representing input patterns producing the minimum absolute error in output during additional training, is also subjected to Lyapunov exponent investigation. The task of this investigation is to determine whether the network exhibits chaotic pattern competition of the best learned inputs. In the second case, the network is improved and desirable convergence is accomplished. Again, investigation focuses on the series of values representing input patterns that produce outputs with minimum absolute error. The results obtained from dominant Lyapunov exponent estimations show that chaotic pattern competition is still present, despite the fact that the network practically satisfies stability demands within predetermined accuracy limits. The best estimation series consist of the output values corresponding to the best learned input patterns. These series are examined using the theoretical tool of topological conjugacy, in addition to numerical verification of the results.

Symmetry and chaos in the complex Ginzburg–Landau equation. II. Translational symmetries

The complex Ginzburg–Landau (CGL) equation on a one-dimensional domain with periodic boundary conditions has a number of different symmetries, and solutions of the equation may or may not be fixed by the action of these symmetries. We investigate the stability of chaotic solutions that are spatially periodic with period L with respect to subharmonic perturbations that have a spatial period kL for some integer k>1. This is done by considering the isotypic decomposition of the space of solutions and finding the dominant Lyapunov exponent associated with each isotypic component. We find a region of parameter space in which there exist chaotic solutions with spatial period L and homogeneous Neumann boundary conditions that are stable with respect to perturbations of period 2 L. On varying the parameters it is possible to arrange for this solution to become unstable to perturbations of period 2 L while remaining chaotic, leading to a supercritical subharmonic blowout bifurcation. For a large number of parameter values checked, chaotic solutions with spatial period L were found to be unstable with respect to perturbations of period 3 L. We conclude that while periodic boundary conditions are often convenient mathematically, we would not expect to see chaotic, spatially periodic solutions forming starting with an arbitrary, non-periodic initial condition.

Symmetry and chaos in the complex Ginzburg-Landau equation - I. Reflectional symmetries

The complex Ginzburg-Landau (CGL) equation on a one-dimensional domain with periodic boundary conditions has a number of different symmetries. Solutions of the CGL equation may or may not be fixed by the action of these symmetries. We investigate the stability of chaotic solutions with some reflectional symmetry to perturbations which break that symmetry. This can be achieved by considering the isotypic decomposition of the space and finding the dominant Lyapunov exponent associated with each isotypic component. Our numerical results indicate that for most parameter values, chaotic solutions that have been restricted to lie in invariant subspaces are unstable to perturbations out of these subspaces, leading us to conclude that for these parameter values arbitrary initial conditions will generically evolve to a solution with the minimum amount of symmetry allowable. We have also found a small region of parameter space in which chaotic solutions that are even are stable with respect to odd perturbations.

Nonlinear analysis of the fracture surface of a single-crystal brittle solid

The dynamical behavior of a spatial fracture mechanism in a rapid crack in a single-crystal brittle solid was under investigation. Wide thin strips like specimens were fractured under three-point bending. The spatial mechanism was photographed and digitized, and the results were then studied using nonlinear analysis. The nonlinear tools used were correlation dimension, false nearest neighbor, null hypothesis surrogate data, and the dominant Lyapunov exponent. The governing equation of motion in the analyzed region is a differential equations with a minimum of seven independent dynamical variables, in contrast to a single independent variable used in analytical calculations.