Dimension Of Strange Attractor Research Articles

Predicting the dimension of strange attractors

The correlation dimension was calculated for a collection of 6080 strange attractors obtained numerically from low-degree polynomial, low-dimensional maps and flows. It was found that the average correlation dimension scales approximately as the square root of the dimension of the system with a surprisingly small variation. This result provides an estimate of the number of dynamical variables required to characterize an experiment in which a strange attractor has been found as well as an estimate of the dimension of attractors produced by chaotic systems in which the dimension of the state space is known.

Strange attractor dimension as a new measure of seismotectonics around Koyna reservoir, India

The fractal dimension of a strange attractor based on 44,792 earthquakes around Koyna reservoir in the period 1967–1981 has been determined for three periods during which the main shock of 1967 ( M 6.5) and the largest aftershocks of magnitude 5.0 in 1973 and 1980 occurred. It was found that the fractal dimension remains almost the same and that at least five parameters are needed for earthquakes to be predictable in the region. The non-integral dimension obtained and the positive value of the largest Lyapunov Exponent show that the system is chaotic. The phase space portrait of the Koyna earthquake also shows that the trajectories lie on a strange attractor. It is surmised that the fractal dimension is a new measure of seismotectonics around Koyna region.

Hausdorff dimension from the minimal spanning tree.

A technique to estimate the Hausdorff dimension of strange attractors, based on the minimal spanning tree of the point distribution is extensively tested in this work. This method takes into account in some sense the infimum requirement appearing in the definition of the Hausdorff dimension. It provides accurate estimates even for a low number of data points and it is especially suited to high-dimensional systems.

Fractal dimensions of strange attractors obtained from the Taylor-Couette experiment

We present scenarios from the rotational Taylor-Couette flow, which has been implemented as a high precision hydrodynamic experiment. The system shows a rich variety of routes to chaos, e.g. the period doubling cascade, quasiperiodic and intermittent transitions. From a scalar time series, that is the axial velocity component of the flow field measured with Laser-Doppler velocimetry, strange attractors are reconstructed using Takens' delay time coordinates. To obtain an estimate of the fractal dimensions of these structures in phase space we calculate the correlation dimension from the reconstructed attractors. We discuss the fractal dimension as a function of Reynolds number and geometry of the experiment.

Lyapunov's direct method in the estimation of the Hausdorff dimension of attractors

This paper surveys results of the authors and others conceming estimates for the Hausdorff dimension of strange attractors, particularly in the case of (generalized) Lorenz systems and Rossler systems. A key idea is the interpretation of Hausdorff measure as an analogue of a Lyapunov function.

Dimension of strange attractors in four-dimensional maps

The method of Tsang for calculating the Hausdorff dimension of a strange attractor in two-dimensional maps is generalized to a four-dimensional map for the double rotor system. The computed dimension compares well with previous work, and is much faster computationally.

Chaotic Vibrations: An Introduction for Applied Scientists and Engineers

1. Introduction: A New Age of Dynamics. 1.1 What Is Chaotic Dynamics? 1.2 Classical Nonlinear Vibration Theory: A Brief Review. 1.3 Maps and Flows. 2. How to Identify Chaotic Vibrations. 3. A Survey of Systems with Chaotic Vibrations. 3.1 New Paradigms in Dynamics. 3.2 Mathematical Models of Chaotic Physical Systems. 3.3 Physical Experiments in Chaotic Systems. 4. Experimental Methods in Chaotic Vibrations. 4.1 Introduction: Experimental Goals. 4.2 Nonlinear Elements in Dynamical Systems. 4.3 Experimental Controls. 4.4 Phase Space Measurements. 4.5 Bifurcation Diagrams. 4.6 Experimental Poincare Maps. 4.7 Quantitative Measures of Chaotic Vibrations. 5. Criteria for Chaotic Vibrations. 5.1 Introduction. 5.2 Introduction Empirical Criteria for Chaos. 5.3 Theoretical Predictive Criteria. 5.4 Lyapunov Exponents. 6. Fractal Concepts in Nonlinear Dynamics. 6.1 Introduction. 6.2 Measures of Fractal Dimension. 6.3 Fractal Dimension of Strange Attractors. 6.4 Optical Measurement of Fractal Dimension. 6.5 Fractal Basin Boundaries. 6.6 complex Maps and the Mandelbrot Set. Appendix A. Glossary of Terms in Chaotic and Nonlinear Vibrations. Appendix B. Appendix C. Numerical Experiments in Chaos. Appendix C. Chaotic Toys. References. Author Index. Subject Index.

Evaluation of dimensions and entropies of chaotic systems

Accurate evaluations of fractal dimensions are limited by intrinsic properties of strange attractors (lacunarity, nonuniformity) and by statistical effects (finite number of data points). The application of the fixed-mass (nearest-neighbor distance) method is studied in view of these limitations, showing that the method is also well suited for the measurement of large dimensions. The tuning of the algorithm and the problems inherent in the analysis of experimental systems (small number of data points, noise, and drifts) are analyzed with reference to several computer experiments. The evaluation of the metric entropies of experimental signals by means of the same algorithm is discussed.

Topological and statistical properties of strange attractors in lasers with periodically modulated parameters

Topological and statistical properties of strange attractors in lasers with periodically modulated parameters

On calculating the dimension of strange attractors

Numerical algorithms for calculating the capacity, information dimension, and correlation exponent of strange attractors are considered.

Spatial and Temporal Scaling Properties of Strange Attractors and Their Representations by Unstable Periodic Orbits

Scaling properties of chaos are studied from a dynamic viewpoint by taking invertible two· dimensional maps. A fundamental role is played by an evolution equation for the probability of a small box along a chaotic orbit. A new relation between the partial dimen~ions of strange attractors in the expanding and contracting direction is derived in terms of the local expansion rates of nearby orbits. When the lacobians of the maps are constant, this relation can be written in terms of a potential rJJ(q) for the fluctuations of the local expansion rate in the expanding direction. For hyperbolic attractors, this potential rJJ(q) is related to the generalized entropies K(q) by rJJ(q)=(q -l)K(q), and the above relation reduces to a simple relation between the generalized dimensions and the generalized entropies. Furthermore, the generalized dimensions, the potential rJJ(q) and the probability densities are expressed in terms of the local expansion rates of unstable periodic orbits within the attractors, leading to a new method of studying chaos.

Correlations in one-dimensional fully developed chaos

Correlations in the baker map and the tent map as examples of one-dimensional, fully developed chaos are considered. It is shown, utilizing symbolic dynamical systems derived from these maps, that the vanishing second-order correlation function is not sufficient to guarantee uncorrelatedness. Importance of the higher-order, especially third-order, correlation functions is emphasized for chaotic systems. In search of the quantities that grasp correlational behaviors as a whole in chaotic systems, it is proposed to use the fixed-separation correlation integral, which is a modified quantity of the usual correlation integral devised to calculate the fractal dimension of strange attractors, for these maps. It is shown that the new quantity contains all the even-number orders of autocorrelation function that are commonly considered.

Strange attractors in weakly turbulent Couette-Taylor flow.

We have conducted an experiment on the transition from quasiperiodic to weakly turbulent flow of a fluid contained between concentric cylinders with the inner cylinder rotating and the outer cylinder at rest. Power spectra, phase-space portraits, and circle maps obtained from velocity time-series data indicate that the nonperiodic behavior which we have observed is deterministic, that is, it is described by strange attractors. We discuss various problems that arise in computing the dimension of strange attractors constructed from experimental data and show that these problems impose severe requirements on the quantity and accuracy of data necessary for determining dimensions greater than about 5. In the present experiment the attractor dimension increases from 2 at the onset of turbulence to about 4 at a Reynolds number 50% above the onset of turbulence.

Strange Attractor of Chaotic Magnons Observed in Ferromagnetic (CH3NH3)2CuCl4

By parallel pumping experiments in ferromagnetic (CH 3 NH 3 ) 2 CuCl 4 chaotic auto-oscillations of magnon amplitude are observed at the pumping frequency of 9.39 GHz and the temperature of 1.65 K. Strange attractors constructed in three-dimensional phase space from time series data of chaotic oscillations display properties of stretching and folding of the sheet of trajectories. Fractal dimensions of strange attractors are obtained by the method of the correlation integral as a function of driving microwave power. With increasing driving power fractal dimension increases from 1.6 to 3.4. The largest Lyapunov exponent is obtained as 0.43±0.05 by using empirical return maps. The chaotic state arises directly from period-two oscillations without cascade of period doubling bifurcations.

Chaos and neutron scattering

Abstract Employing a real-time method, neutron scattering has been used in the investigation of nonlinear convection and the development of chaos in a nematic liquid crystal. Features of the experimental method are explained. The data processing involved in obtaining the autocorrelation function, the power spectrum, the phase portrait and the fractal dimension of strange attractors is demonstrated. The convective system under study is shown to be a rich nonequilibrium system, displaying multicriticality, multistability, period doubling and chaos.

Properties of strange attractors in yeast glycolysis

The properties of periodic and aperiodic glycolytic oscillations observed in yeast extracts under sinusoidal glucose input were analyzed by the following methods. (1) Spectral analysis, rendering sharp peaks for periodic responses and enhanced broad-band noise for aperiodic oscillations. (2) Phase plane analysis, leading to closed and to open trajectories for periodic and aperiodic oscillations, respectively. (3) Rotation of a phase plane proportionally to time, revealing strange attractors associated with the aperiodic oscillations. (4) Stroboscopic plot on the phase plane, showing that the strange attractors follow a stretch-fold-press process, if the stroboscoping phase is varied. (5) Stroboscopic transfer plot, admitting a period of three transfer processes and thus implying chaos according to the Li-Yorke theorem. (6) Determination of the rate of information production by differentiation of the transfer plot, yielding approx. 0.21 bits per min for the chaotically glycolyzing yeast extract.

Universal Power Law for the Dimension of Strange Attractors near the Onset of Chaos

Received 27 January 1984DOI:https://doi.org/10.1103/PhysRevLett.53.1704©1984 American Physical Society

On the use of perturbation theory in the calculation of the fractal dimension of strange attractors

For certain classes of difference equations it is possible to use perturbation theory to obtain a prescription that recursively generates the structure of the strange attractor. We describe the use of this technique in the calculation of the fractal dimension of the attractor for certain two and three dimensional maps. The results are compared with published numberical work and relationships with Liapunov numbers conjectured by Kaplan and Yorke and Mori and Fujisaka. Where there is discrepancy between these conjectures our results agree with the work of the former authors. In these examples the source of the discrepancy can be identified.

Fractal Dimension of Strange Attractors from Radius versus Size of Arbitrary Clusters

On donne une estimation de la dimension fractale D F ' des attracteurs etranges. Des amas de n points plus proches voisins sont echantillonnes a partir d'une serie temporelle; on trouve D F ' a partir de R DF' ∼n ou R est le rayon d'amas moyen

Generalized dimensions of strange attractors

It is pointed out that there exists an infinity of generalized dimensions for strange attractors, related to the order- q Renyi entropies. They are monotonically decreasing with q. For q = 0, 1 and 2, they are the capacity, the information dimension, and the correlation exponent, respectively. For all q, they are measurable from recurrence times in a time series, without need for a box-counting algorithm. For the Feigenbaum map and for the generalized Baker transformation, all generalized dimensions are finite and calculable, and depend non-trivially on q.