One-Dimensional Maps.- Two-Dimensional Maps.- Chaos.- Fractals.- Chaos in Two-Dimensional Maps.- Chaotic Attractors.- Differential Equations.- Periodic Orbits and Limit Sets.- Chaos in Differential Equations.- Stable Manifolds and Crises.- Bifurcations.- Cascades.- State Reconstruction from Data.

Chaos: An Introduction to Dynamical Systems

Chaotic time series data are observed routinely in experiments on physical systems and in observations in the field. The authors review developments in the extraction of information of physical importance from such measurements. They discuss methods for (1) separating the signal of physical interest from contamination ("noise reduction"), (2) constructing an appropriate state space or phase space for the data in which the full structure of the strange attractor associated with the chaotic observations is unfolded, (3) evaluating invariant properties of the dynamics such as dimensions, Lyapunov exponents, and topological characteristics, and (4) model making, local and global, for prediction and other goals. They briefly touch on the effects of linearly filtering data before analyzing it as a chaotic time series. Controlling chaotic physical systems and using them to synchronize and possibly communicate between source and receiver is considered. Finally, chaos in space-time systems, that is, the dynamics of fields, is briefly considered. While much is now known about the analysis of observed temporal chaos, spatio-temporal chaotic systems pose new challenges. The emphasis throughout the review is on the tools one now has for the realistic study of measured data in laboratory and field settings. It is the goal of this review to bring these tools into general use among physicists who study classical and semiclassical systems. Much of the progress in studying chaotic systems has rested on computational tools with some underlying rigorous mathematics. Heuristic and intuitive analysis tools guided by this mathematics and realizable on existing computers constitute the core of this review.

The analysis of observed chaotic data in physical systems

We describe the implementation of methods of nonlinear time series analysis which are based on the paradigm of deterministic chaos. A variety of algorithms for data representation, prediction, noise reduction, dimension and Lyapunov estimation, and nonlinearity testing are discussed with particular emphasis on issues of implementation and choice of parameters. Computer programs that implement the resulting strategies are publicly available as the TISEAN software package. The use of each algorithm will be illustrated with a typical application. As to the theoretical background, we will essentially give pointers to the literature. (c) 1999 American Institute of Physics.

Practical implementation of nonlinear time series methods: The TISEAN package.

Nonlinear time series analysis is becoming a more and more reliable tool for the study of complicated dynamics from measurements. The concept of low-dimensional chaos has proven to be fruitful in the understanding of many complex phenomena despite the fact that very few natural systems have actually been found to be low dimensional deterministic in the sense of the theory. In order to evaluate the long term usefulness of the nonlinear time series approach as inspired by chaos theory, it will be important that the corresponding methods become more widely accessible. This paper, while not a proper review on nonlinear time series analysis, tries to make a contribution to this process by describing the actual implementation of the algorithms, and their proper usage. Most of the methods require the choice of certain parameters for each specific time series application. We will try to give guidance in this respect. The scope and selection of topics in this article, as well as the implementational choices that have been made, correspond to the contents of the software package TISEAN which is publicly available from this http URL . In fact, this paper can be seen as an extended manual for the TISEAN programs. It fills the gap between the technical documentation and the existing literature, providing the necessary entry points for a more thorough study of the theoretical background.

https://arxiv.org/pdf/chao-dyn/9810005.pdf

Practical implementation of nonlinear time series methods: The TISEAN package

A robust method to estimate the maximal Lyapunov exponent of a time series

We apply a recently proposed nonlinear noise-reduction method to time sequences from two different experiments. We demonstrate that it is not difficult to choose the parameters of this algorithm, even though we use no other information about the underlying dynamics than the data themselves. The noise reduction is very robust with respect to changes in the choice of parameters. The reliability of the result is tested by an analysis of the corrections. We discuss the effect of noise reduction on estimates of dimensions, entropies, and Liapunov exponents. For comparison we process one of the sets, densely sampled Taylor-Couette flow data, with a global filter based on singular value decomposition.

Nonlinear noise reduction: A case study on experimental data.

We apply a recently proposed nonlinear noise-reduction method to time sequences from two different experiments. We demonstrate that it is not difficult to choose the parameters of this algorithm, even though we use no other information about the underlying dynamics than the data themselves. The noise reduction is very robust with respect to changes in the choice of parameters. The reliability of the result is tested by an analysis of the corrections. We discuss the effect of noise reduction on estimates of dimensions, entropies, and Liapunov exponents. For comparison we process one of the sets, densely sampled Taylor-Couette flow data, with a global filter based on singular value decomposition

Nonlinear noise reduction : a case study on experimental data

The understanding of chaotic systems can be considerably improved with the knowledge of their periodic-orbit structure. The identification of the low-order unstable periodic orbits embedded in a strange attractor induces a hierarchical organization of the dynamics which is invariant under smooth coordinate changes. The applicability of this technique is by no means limited to analytical or numerical calculations, but has been recently extended to experimental time series. As a specific example, the authors review some of the major results obtained on a nuclear-magnetic-resonance laser which have led to an extension of the conventional (Bloch-Kirchhoff) equations of motion, to the construction of approximate generating partitions, and to an efficient control of the chaotic system around various unstable periodic orbits. The determination of the symbolic dynamics, with the precision achieved by recording all unstable cycles up to order 9, improves the topological and metric characterization of a heteroclinic crisis. The periodic-orbit approach permits detailed study of chaotic motion, thereby leading to an improved classification scheme which subsumes the older ones, based on estimates of scalar quantities such as fractal dimensions and metric entropies.

Progress in the analysis of experimental chaos through periodic orbits

The unstable periodic orbits of a chaotic NMR laser have been identified and classified according to their stability properties. The dynamical control method by Ott, Grebogi, and Yorke (OGY) [Phys. Rev. Lett. 64, 1196 (1990)] has been successfully applied in a six-dimensional embedding space, as required for a proper reconstruction of the attractors. An improved method has then been implemented in order to circumvent the occurrence of several, possibly complex, unstable eigenvalues in the standard OGY procedure and has shown superior performance. Finally, an extension to several control stations per Poincar\'e time is discussed.

Control of NMR-laser chaos in high-dimensional embedding space.

It is shown that the conventional Bloch-Kirchhoff description of NMR-laser activity needs the inclusion of a nonlinear relaxation term for the transverse magnetization in order to account for the experimental observation. The validity of the modified equations is demonstrated by comparing in a two-parameter space the Poincar\'e sections and periodic-orbit structures of model and experiment. All unstable orbits up to order nine have been extracted and employed to obtain an approximation to a generating partition.

R. Badii

Papers

Nonlinear noise reduction: A case study on experimental data.

Nonlinear noise reduction : a case study on experimental data

Progress in the analysis of experimental chaos through periodic orbits

Control of NMR-laser chaos in high-dimensional embedding space.

Model identification by periodic-orbit analysis for NMR-laser chaos.