摘要:

This paper is a brief survey of numerical methods for computing bifurcations of generic families of dynamical systems. Emphasis is placed upon algorithms that reflect the structure of the underlying mathematical theory while retaining numerical efficiency. Significant improvements in the computational analysis of dynamical systems are to be expected from more reliance of geometric insight coming from dynamical systems theory. ©1996 American Institute of Physics.

ISSN:0094-243X    

DOI:10.1063/1.51040

出版商:AIP    

年代:1996

数据来源: AIP

摘要:

Much of the current study of dynamical systems is focused on geometry (e.g., chaos and bifurcations) and ergodic theory. Yet dynamical systems were originally motivated by an attempt to ‘‘solve,’’ or at least understand, a discrete‐time analogue of differential equations. As such, numerical, analytical solution techniques for dynamical systems would seem desirable. We discuss an approach that provides such techniques, the approximation of dynamical systems by suitable finite state Markov Chains. Steady state distributions for these Markov Chains, a straightforward calculation, will converge to the true dynamical system steady state distribution, with appropriate limit theorems indicated. Thus (i) approximation by a computable, linear map holds the promise of vastly faster steady state solutions for nonlinear, multidimensional differential equations; (ii) the solution procedure is unaffected by the presence or absence of a probability density function for theattractor, entirely skirting singularity, fractal/multifractal, and renormalization considerations. The theoretical machinery underpinning this development also implies that under very general conditions, steady state measures are weakly continuous with control parameter evolution. This means that even though a system may change periodicity, or become chaotic in its limiting behavior, such statistical parameters as the mean, standard deviation, and tail probabilities change continuously, not abruptly with system evolution. ©1996 American Institute of Physics.

ISSN:0094-243X    

DOI:10.1063/1.51026

出版商:AIP    

年代:1996

数据来源: AIP

摘要:

Observational data of natural systems, as measured in astrophysical, geophysical or physiological experiments are typically quite different from those obtained in laboratories. Due to the peculiarities with these data, well‐known characteristics processes, such as periodicities or fractal dimension, often do not provide a suitable description. To study such data, we present here the use of measures of complexity, which are mainly basing on symbolic dynamics. We distinguish two types of such quantities: traditional measures (e.g. algorithmic complexity) which are measures of randomness and alternative measures (e.g. &egr;‐complexity) which relate highest complexity to some critical points. It is important to note that there is no optimum measure of complexity. Its choice should depend on the context. Mostly, a combination of some such quantities is appropriate. Applying this concept to three examples in astrophysics, cardiology and cognitive psychology, we show that it can be helpful also in cases where other tools of data analysis fail. ©1996 American Institute of Physics.

ISSN:0094-243X    

DOI:10.1063/1.51037

出版商:AIP    

年代:1996

数据来源: AIP

摘要:

We describe the role of chaotic noise reduction in detecting an underlying smoothness in a dataset. We have described elsewhere a general method for assessing the presence of determinism in a time series, which is to test against the class of datasets producing smoothness (i.e., the null hypothesis is determinism). In order to reduce the likelihood of a false call, we recommend this kind of analysis be applied first to a time series whose deterministic origin is at question. We believe this step should be taken before implementing other methods of dynamical analysis and measurement, such as correlation dimension or Lyapounov spectrum. ©1996 American Institute of Physics.

ISSN:0094-243X    

DOI:10.1063/1.51053

出版商:AIP    

年代:1996

数据来源: AIP

摘要:

Methods for reconstruction chaotic dynamical system structure directly from experimental time series are described. Effectiveness of general methods is illustrated with the results of numerical simulation. It is of common interest that from the single time series it is possible to reconstruct a set of interconnected variables. Predictive power of dynamical models, provided by the nonlinear dynamics inverse problem solution, is limited firstly by the noise level in the system under study and is characterized by the horizon of predictability. New physical results are presented, concerning nonstationary and bifurcation nonlinear systems: 1) algorithms for revealing of nonstationarity in random‐like chaotic time‐series are suggested based on discriminant analysis with nonlinear discriminant function; 2) an opportunity is established to predict the final state in bifurcation system with quickly varying control parameters; 3) hysteresis is founded out in bifurcation system with quickly varying parameters; 4) delayed correlation ⟨noise‐prediction error⟩ in chaotic systems is revealed. ©1996 American Institute of Physics.

ISSN:0094-243X    

DOI:10.1063/1.51009

出版商:AIP    

年代:1996

数据来源: AIP

摘要:

The orbital complexity and exponential sensitivity of chaotic systems has the consequence that such systems offer the possibility of being feedback controlled by use of only small perturbations. The potential consequences of this recent realization are being investigated in a broad range of applications. ©1996 American Institute of Physics.

ISSN:0094-243X    

DOI:10.1063/1.51017

出版商:AIP    

年代:1996

数据来源: AIP

摘要:

A method of control of spatiotemporal chaos in lattices of coupled maps is proposed in this work. Forms of spatiotemporal perturbations of a system parameter are analytically determined for one‐ and two‐dimensional logistic map lattices with different kinds of coupling to stabilize chosen spatiotemporal states previously unstable. The results are illustrated by numerical simulation. Controlled transition from the regime of spatiotemporal chaos to the previously chosen regular spatiotemporal patterns is demonstrated. ©1996 American Institute of Physics.

ISSN:0094-243X    

DOI:10.1063/1.51021

出版商:AIP    

年代:1996

数据来源: AIP

摘要:

A technique to identify the state of a dynamical system is proposed. The technique is based upon an identification of all period‐one orbits present in the system. These orbits can then be classified in a way that permits an organization into a hierarchical ordering. The scheme is applied to time‐series data gathered from a carefully constructed damped driven pendulum. ©1996 American Institute of Physics.

ISSN:0094-243X    

DOI:10.1063/1.51022

出版商:AIP    

年代:1996

数据来源: AIP

摘要:

An important issue in nonlinear dynamics is the optimal estimation and detection of the partially observed states of a system at low signal‐to‐noise ratios. In this paper, we will outline a Bayesian‐based approach that allows for the optimal determination of the state probability density function in time as a function of the observations. This leads to optimal detector designs based on the notion of generalized innovation sequences. Here, the density functions are defined over a computational grid which is designed to capture the phase space dynamics of the nonlinear system. Partial measurements are used to update the projected system state and density function. Estimation and detection decisions are based on the propagated density functions. ©1996 American Institute of Physics.

ISSN:0094-243X    

DOI:10.1063/1.51023

出版商:AIP    

年代:1996

数据来源: AIP

摘要:

In recent years, several specific advances in the study of chaotic processes have been made which appear to have immediate applicability to signal processing. This paper describes two applications of one of these advances, nonlinear modeling, to signal detection & classification, in particular for short‐lived or transient or signals. The first method uses the coefficients from an adaptively fit model as a set of features for signal detection and classification. In the second method, a library of predictive nonlinear dynamic equations is used as a filter bank, and statistics on the prediction residuals are used to form feature vectors for input data segments. These feature vectors provide a mechanism for detecting and classifying model transients at signal‐to‐noise ratios as low as −10 dB, even when the generating dynamics of the transient signals are not present in the filter bank. The second method and some validating experiments are described in detail. ©1996 American Institute of Physics.

ISSN:0094-243X    

DOI:10.1063/1.51024

出版商:AIP    

年代:1996

数据来源: AIP