Showing papers on "Chaotic published in 1991"

Synchronizing chaotic circuits

Abstract: The authors describe the conditions necessary for synchronizing a subsystem of one chaotic system with a separate chaotic system by sending a signal from the chaotic system to the subsystem. The general scheme for creating synchronizing systems is to take a nonlinear system, duplicate some subsystem of this system, and drive the duplicate and the original subsystem with signals from the unduplicated part. This is a generalization of driving or forcing a system. The process can be visualized with ordinary differential equations. The authors have build a simple circuit based on chaotic circuits described by R. W. Newcomb et al. (1983, 1986), and they use this circuit to demonstrate this chaotic synchronization. >

Chaos and Nonlinear Dynamics: Application to Financial Markets

Abstract: After the stock market crash of October 19, 1987, interest in nonlinear dynamics, especially deterministic chaotic dynamics, has increased in both the financial press and the academic literature. This has come about because the frequency of large moves in stock markets is greater than would be expected under a normal distribution. There are a number of possible explanations. A popular one is that the stock market is governed by chaotic dynamics. What exactly is chaos and how is it related to nonlinear dynamics? How does one detect chaos? Is there chaos in financial markets? Are there other explanations of the movements of financial prices other than chaos? The purpose of this paper is to explore these issues. CHAOS HAS CAPTURED THE fancy of many macroeconomists and financial economists. The attractiveness of chaotic dynamics is its ability to generate large movements which appear to be random, with greater frequency than linear models. As a result, there has been an explosion of papers searching for chaotic behavior in macroeconomic and financial time series. The purpose of this paper is to discuss some of the methodological issues in detecting chaotic and nonlinear behavior. Section I provides a description of the key features of deterministic chaotic systems via a number of examples. Section II shows how deterministic chaos can, in principle, be detected using the method of correlation dimension proposed by Grassberger and Procaccia (1983). Section III deals with some limitations of this method. The Grassberger/Procaccia method requires a substantial number of data points, which is difficult to obtain in standard economic and financial time series. It also lacks a statistical theory for hypothesis testing. A different but related method has been proposed by Brock, Dechert, and Scheinkman (1987). Under the null hypothesis of independence and identical distribution (IID), the Brock, Dechert, and Scheinkman statistic has been shown to have good finite sample properties and good power against departures from IID. Some Monte Carlo evidence is

Driving systems with chaotic signals.

Abstract: We generalize the idea of driving a stable system to the situation when the drive signal is chaotic. This leads to the concept of conditional Lyapunov exponents and also generalizes the usual criteria of the linear stability theorem. We show that driving with chaotic signals can be done in a robust fashion, rather insensitive to changes in system parameters. The calculation of the stability criteria leads naturally to an estimate for the convergence of the driven system to its stable state. We focus on a homogeneous driving situation that leads to the construction of synchronized chaotic subsystems. We apply these ideas to the Lorenz and R\"ossler systems, as well as to an electronic circuit and its numerical model.

Optimal Embeddings of Chaotic Attractors from Topological Considerations

Abstract: Guided by topological considerations, a new method is introduced to obtain optimal delay coordinates for data from chaotic dynamic systems. By determining simultaneously the minimal necessary embedding dimension as well as the proper delay time we achieve optimal reconstructions of attractors. This can be demonstrated, e.g., by reliable dimension estimations from limited data series.

Symmetry breaking bifurcation for coupled chaotic attractors

Abstract: The authors consider transitions from synchronous to asynchronous chaotic motion in two identical dissipatively coupled one-dimensional mappings. They show that the probability density of the asymmetric component satisfies a scaling law. The exponent in this scaling law varies continuously with the distance from the bifurcation point, and is determined by the spectrum of local Lyapunov exponents of the uncoupled map. Finally they show that the topology of the invariant set is rather unusual: though the attractor for supercritical coupling is a line, it is surrounded by a strange invariant set which is dense in a two-dimensional neighbourhood of the attractor.

Control of a chaotic system

Abstract: The Lorenz equations are well known for their ability to produce chaotic motion. We investigate here the Lorenz system subject to a control input. Two different controllers are designed for this system, one based on linear methods and one based on a nonlinear analysis. The objective of the controller is to drive the system to one of the unstable equilibrium points associated with uncontrolled chaotic motion. Each controller is able to produce stable motion. However, the character of this motion may differ considerably, depending on adjustment of “Gains” used in the controller. In particular, the motion may contain chaotic transients. It is possible to create a system with intermediate-term-sensitive dependence on initial conditions, but with no such long-term dependence.

Calculating the rate of loss of information from chaotic time series by forecasting

Abstract: DETERMINING whether time series of data from dynamical systems exhibit regular, stochastic or chaotic behaviour is a goal in a wide variety of problems. For sparse time series (those containing only of the order of 1,000 data points), the goal may simply be to discover whether the series are chaotic or not. Examples are case rates for infectious diseases1 and proxy palaeoclimatic records from deep-sea cores2. Sugihara and May3 have recently extended previous work4 aimed at distinguishing chaos from noise in sparse time series. Their approach is based on a comparison of future predictions of terms in the time series—derived using a data base of information from another part of the series—with the known terms. Here I present a method for estimating from such forecasting the largest Liapunov exponent of the dynamics, which provides a measure of how chaotic the system is—that is, how rapidly information is lost from the system.

Chaos in the sunspot cycle: Analysis and Prediction

Abstract: The variability of solar activity over long time scales, given semiquantitatively by measurements of sunspot numbers, is examined as a nonlinear dynamical system. First, a discussion of the data set used and the techniques utilized to reduce the noise and capture the long-term dynamics inherent in the data is presented. Subsequently, an attractor is reconstructed from the data set using the method of time delays. The reconstructed attractor is then used to determine both the dimension of the underlying system and also the largest Lyapunov exponent, which together indicate that the sunspot cycle is indeed chaotic and also low dimensional. In addition, recent techniques of exploiting chaotic dynamics to provide accurate, short-term predictions are utilized in order to improve upon current forecasting methods and also to place theoretical limits on predictability extent. The results are compared to chaotic solar-dynamo models as a possible physically motivated source of this chaotic behavior.

Chaotic Evolution and Strange Attractors

Abstract: In this 17th volume of the \"Springer Series in Surface Sciences\" the editors publish the 10 lectures, given on the third workshop on interface phenomena at the Dalhousie University. The topic was this time \"Adhesion and Friction: Microscopic Concepts\". This very important and interesting theme for many scientists, who are working in this field, had the aim to achieve more and better understanding of complicated issues such

Statistical properties of chaotic systems

Abstract: The theory of smooth dynamical systems and the theory of abstract dynamical systems (ergodic theory), although having the same roots, have for many years developed quite independently of one another. These theories have now matured to the point where they can be combined to shed light on the nature of chaotic behavior

Non-linear and linear forecasting of the EEG time series

Abstract: The method of non-linear forecasting of time series was applied to different simulated signals and EEG in order to check its ability of distinguishing chaotic from noisy time series. The goodness of prediction was estimated, in terms of the correlation coefficient between forecasted and real time series, for non-linear and autoregressive (AR) methods. For the EEG signal both methods gave similar results. It seems that the EEG signal, in spite of its chaotic character, is well described by the AR model.

An investigation of chaotic Kolmogorov flows

Abstract: A two‐dimensional flow governed by the incompressible Navier–Stokes equations with a steady spatially periodic forcing (known as the Kolmogorov flow) is numerically simulated. The behavior of the flow and its transition states as the Reynolds number Re varies is investigated in detail, as well as a number of the flow features. A sequence of bifurcations is shown to take place in the flow as Re varied. Two main regimes of the flow have been observed: small and large scale structure regimes corresponding to different ranges of Re. Each of the regimes includes a number of periodic, chaotic, and relaminarization windows. In addition, each range contains a chaotic window with nonunique chaotic attractors. Spatially disordered, but temporally steady states have been discovered in the large scale structure regime. Features of the diverse cases are displayed in terms of the temporal power spectrum, Poincare sections and, where possible, Lyapunov exponents and Lyapunov dimension.

The Art of Modeling Dynamic Systems: Forecasting for Chaos, Randomness and Determinism

Abstract: A Brief History of Dynamics and Computing. A THUMBNAIL SKETCH OF APPLIED MATHEMATICS. Foundations and Abstract Entities. Classical Analysis. Numerical Analysis and Approximation Theory. Statistical Methods. Classical Modeling Techniques. CLASSICAL MODELS AND DYNAMICAL CONCEPTS. Dynamics Without Calculus. Basic Models. Cycles. Analysis of Mathematical Models. THE HIERARCHY OF DYNAMIC SYSTEMS. A Classification Scheme for Dynamic Systems. Static Systems--Type Zero. Solvable Systems--Type I. Perturbation Theory--Type II. Chaotic Systems--Type III. Stochastic Systems--Type IV. THE ART OF MODEL MAKING. Qualitative Analysis. Quantitative Analysis. Model Validation. References--The Modeler's Library. Index.

Amplification of intrinsic fluctuations by chaotic dynamics in physical systems

Abstract: A quantitative method for the treatment of large-scale intrinsic fluctuations amplified by chaotic trajectories in macrovariable physical systems is presented. Paradigmatic results for the Rossler model and preliminary computational results for chaotic Josephson junctions and for chaotic multimode Nd:YAG (yttrium aluminum garnet) lasers are described. These studies are directed towards identification of a real physical system in which experimental confirmation may be realized. The probability distribution on the intrinsic-noise-modified, chaotic attractor is identified as a likely candidate for comparison of experiment and theory.

Chaotic behaviour of deterministic dissipative systems

Abstract: Introduction 1. Differential equations, maps and asymptotic behaviour 2. Transition from order to chaos 3. Numerical methods for studies of parametric dependences, bifurcations and chaos 4. Chaotic dynamics in experiments 5. Forced and coupled chemical oscillators: a case study of chaos 6. Chaos in distributed systems Appendices Bibliography Index.

Time series prediction by adaptive networks: a dynamical systems perspective

Abstract: The links between adaptive layered networks, functional interpolation and dynamical systems are considered and applied to the nonlinear predictive analysis of time series. The ability of networks to produce interpolation surfaces to generators of data (i.e. differential equations, iterative maps) is used to analyse a variety of time series. If network may be trained to approximate a static) generator of data, the network may be iterated on its own output to produce a time series with the same characteristics as the training waveform. However, since iterated networks are one example of nonlinear dynamical systems, this raises problems of sensitive dependence upon initial conditions leading ultimately to deterministic chaos. An introduction to the relevant concepts is presented and illustrations are provided from simple chaotic maps, nonlinear differential equations, and stock-market prediction. The latter example is included to illustrate the problems which often occur in real-world data due to noise, undersampling, high dimensionality and insufficient data.

Searching for chaotic dynamics in snowmelt runoff

Abstract: Chaos analysis has altered the way we view natural systems. Complex or random-appearing phenomena may be chaotic and thus deterministic, rather than random. In this study, we used the Grassberger-Procaccia algorithm (GPA) to evaluate a runoff time series from a second-order catchment in southwestern Idaho for chaotic dynamics. GPA can identify the presence of low-dimensional chaotic dynamics for experimental time series. A daily runoff record, 8800 days in length, was examined. We found no evidence of chaotic dynamics in snowmelt runoff. Snowmelt runoff measured at a daily time step has a large number of degrees of freedom, which is characteristic of a random rather than chaotic process. These results suggest that the random-appearing behavior of snowmelt runoff is generated from the complex interactions of many factors, rather than low-dimensional chaotic dynamics.

Nonlinear impact and chaotic response of slender rocking objects

Abstract: This investigation focuses on the influence of nonlinearities associated with impact on the behavior of the rocking response of free‐standing rigid objects subjected to horizontal base excitations. The object is to identify the causes that make the rocking response difficult to predict and that have made past experiments with identical setups and excitations unrepeatable. The impact nonlinearities examined are: (1) The transition of governing equations; and (2) the abrupt reduction in angular velocity associated with impact at the base. In addition to the periodic and overturning responses, the existence of two new types of (bounded) response not previously revealed in literature—quasi‐periodic and chaotic are discovered in this study. An approximate method based on the Melnikov function to analytically predict the existence of chaotic response is derived. Modern geometric and numerical identification techniques are employed. The accuracy of the method is assessed by numerical results. The relationship be...

Small-amplitude periodic and chaotic solutions of the complex Ginzburg-Landau equation for a subcritical bifurcation.

Abstract: We preent numerically obtained bounded solutions of the one-dimensional complex Ginzburg-Landau equation with a destabilizing cubic term and no stabilizing higher-order contributions. The boundedness results from competition between dispersion and nonlinear frequency renormalization. We find chaotic and also stationary and time-periodic states with spatial structure corresponding to a periodic array of pulses. An analytical description is presented. Possibly experimental results connected with the dispersive chaos found in binary-fluid mixtures can be explained.

Chaos prediction in nonlinear feedback systems

Abstract: Investigates the chaotic behaviour of nonlinear feedback systems. A heuristic model of this phenomenon is proposed and applied. Conditions for the existence and the location of chaotic motions are derived in terms of simple relations among the parameters of the system. Two examples show the application of the method and its approximation is discussed.

Likelihood and Bayesian Prediction of Chaotic Systems

Abstract: There has recently been considerable interest in both applied disciplines and in mathematics, as well as in the popular science literature, in the areas of nonlinear dynamical systems and chaotic processes. By a nonlinear, deterministic dynamical system, we mean a time series in which, starting at some initial condition, the values of the series are some fixed, nonlinear function of the previous states. One of the more intriguing aspects of these models is their propensity for displaying very complex, apparently random behavior, even when simple models are analyzed. A consequence of such chaotic behavior is that it is difficult to predict the exact behavior of a chaotic system. The difficulty in prediction stems from the fact that even the tiniest of errors, including computer roundoff, in either the specification of the function or the initial condition, can lead to huge errors in prediction. After a brief review of dynamical systems and the role of probability in dealing with uncertainty, a com...

On chaos of digital filters in the real world

Abstract: Numerical simulations are presented showing that over certain parameter ranges, the asymptotic behavior of a real-world digital filter having a wordlength of only 16 b is virtually indistinguishable from that of an infinite wordlength digital filter. It is suggested that for all practical purposes, a finite-state machine can behave in a chaotic way if its wordlength is sufficiently large. It is concluded that the chaotic nature of a real digital filter may be hidden because of short wordlengths, but the chaotic behavior must be considered in a real digital filter when the wordlength exceeds 16 b. >