Prediction Of Chaotic System Research Articles

ON THE RELATION BETWEEN PREDICTABILITY AND HOMOCLINIC TANGENCIES

The predictability of chaotic systems is investigated using paradigmatic models for the conservative and the dissipative cases. Local Lyapunov exponents are used to quantify predictability for short time scales. It is shown that, in both cases, regions of enhanced predictability have been found around homoclinic tangencies. In the dissipative case, we demonstrate that the length of these regions shrinks exponentially with increasing time of prediction.

On the predictability of chaotic systems with respect to maximally effective computation time

The round-off error introduces uncertainty in the numerical solution. A computational uncertainty principle is explained and validated by using chaotic systems, such as the climatic model, the Rossler and super chaos system. Maximally effective computation time (MECT) and optimal stepsize (OS) are discussed and obtained via an optimal searching method. Under OS in solving nonlinear ordinary differential equations, the self-memorization equations of chaotic systems are set up, thus a new approach to numerical weather forecast is described.

Fostering conceptual change by analogies—between Scylla and Charybdis

A growing body of research shows that analogies may be powerful tools for guiding students from their pre-instructional conceptions towards science concepts. But it has also become apparent that analogies may deeply mislead students' learning processes. Conceptual change, to put it into other words, may be both supported and hampered by the same analogy. The study presented here was designed to investigate the processes of analogy generation and development and to reveal the microstructure of analogical reasoning. Analogical reasoning was investigated during a grade 10 physics unit on the limited predictability of chaotic systems. Analogies played a key role in the instructional module to make explicit function and structure of certain chaotic systems. Analogies were also used to introduce the notion of “chaotic systems” embracing prototypical examples of chaotic systems studied. The theoretical orientation of the study merges key features of conceptual change approaches and social-constructivist studies on communities of learners. Hence, students were provided with substantial periods of time to generate their own analogies or employ analogies provided by the teacher to understand chaotic systems. There are many cases in our study that illustrate the affordances and pitfalls of analogies in promoting conceptual change. Hence the use of analogies as teaching and learning aids may always be a delicate course between Scylla and Charybdis.

Uncertainty dynamics and predictability in chaotic systems

An initial uncertainty in the state of a chaotic system is expected to grow even under a perfect model; the dynamics of this uncertainty during the early stages of its evolution are investigated. A variety of ‘error growth’ statistics are contrasted, illustrating their relative strengths when applied to chaotic systems, all within a perfect-model scenario. A procedure is introduced which can establish the existence of regions of a strange attractor within which all infinitesimal uncertainties decrease with time. It is proven that such regions exist in the Lorenz attractor, and a number of previous numerical observations are interpreted in the light of this result; similar regions of decreasing uncertainty exist in the Ikeda attractor. It is proven that no such regions exist in either the Rössler system or the Moore-Spiegel system. Numerically, strange attractors in each of these systems are observed to sample regions of state space where the Jacobians have eigenvalues with negative real parts, yet when the Jacobians are not normal matrices this does not guarantee that uncertainties will decrease. Discussions of predictability often focus on the evolution of infinitesimal uncertainties; clearly, as long as an uncertainty remains infinitesimal it cannot pose a limit to predictability. to reflect realistic boundaries, any proposed ‘limit of predictability’ must be defined with respect to the nonlinear behaviour of perfect ensembles. Such limits may vary significantly with the initial state of the system, the accuracy of the observations, and the aim of the forecaster. Perfect-model analogues of operational weather forecasting ensemble schemes with finite initial uncertainties are contrasted both with perfect ensembles and uncertainty statistics based upon the dynamics infinitesimal uncertainties.

Uncertainty dynamics and predictability in chaotic systems

AbstractAn initial uncertainty in the state of a chaotic system is expected to grow even under a perfect model; the dynamics of this uncertainty during the early stages of its evolution are investigated. A variety of ‘error growth’ statistics are contrasted, illustrating their relative strengths when applied to chaotic systems, all within a perfect‐model scenario. A procedure is introduced which can establish the existence of regions of a strange attractor within which all infinitesimal uncertainties decrease with time. It is proven that such regions exist in the Lorenz attractor, and a number of previous numerical observations are interpreted in the light of this result; similar regions of decreasing uncertainty exist in the Ikeda attractor. It is proven that no such regions exist in either the Rössler system or the Moore‐Spiegel system. Numerically, strange attractors in each of these systems are observed to sample regions of state space where the Jacobians have eigenvalues with negative real parts, yet when the Jacobians are not normal matrices this does not guarantee that uncertainties will decrease. Discussions of predictability often focus on the evolution of infinitesimal uncertainties; clearly, as long as an uncertainty remains infinitesimal it cannot pose a limit to predictability. to reflect realistic boundaries, any proposed ‘limit of predictability’ must be defined with respect to the nonlinear behaviour of perfect ensembles. Such limits may vary significantly with the initial state of the system, the accuracy of the observations, and the aim of the forecaster. Perfect‐model analogues of operational weather forecasting ensemble schemes with finite initial uncertainties are contrasted both with perfect ensembles and uncertainty statistics based upon the dynamics infinitesimal uncertainties.

Predictability in chaotic systems and turbulence

A method for characterizing the predictability of complex chaotic systems based on a generalization of the Lyapunov exponent is introduced. The method is illustrated on a toy system with two time scales and on a model of fully developed turbulence where universal features are found

Studies on educational reconstruction of chaos theory

The studies presented here comprise a project on the educational significance of chaos theory. Or to put it in a nutshell, we attempt to analyse critically whether the core ideas of chaos theory (as, for instance, limited predictability of chaotic systems despite deterministic laws governing them) are worth teaching and learning and furthermore we follow up the question of accessibility of these ideas for students. Subject matter structure clarification (i.e., construction of the mentioned key ideas) analyses of educational significance on the basis of widely accepted aims of teaching science, empirical studies on students' learning processes, and finally, development and evaluation of pilot instructional modules are closely interrelated. To put it into our terminology, the studies are embedded in a Model of Educational Reconstruction which may be seen as an approach to curriculum development that, on the one hand, takes into account the major insights provided by the constructivist view of the past two decades resting, on the other hand, on the tradition of German pedagogical theories on framing educational issues.

Journal of the American statistics association: “Likelihood and bayesian prediction of chaotic systems”, 86 (1991) 938–952.

Journal of the American statistics association: “Likelihood and bayesian prediction of chaotic systems”, 86 (1991) 938–952.

The Atmospheric Sciences in the 1990s: Accomplishments, Challenges, and Imperatives

The atmospheric sciences, along with other disciplines, are today contemplating strong competition for resources, exciting scientific challenges, and the need to assess opportunities and priorities. This review is intended to stimulate thought, discussion, and the active participation of the community in charting a course into the next century. Current efforts concentrate on mesoscale phenomena and severe weather, climate dynamics and prediction, the interplay of chemical and physical processes, the interactions of planetary atmospheres with ionospheres and solar processes, and the predictability of chaotic systems. These efforts are the foundation for initiatives involving global and regional climate change, mesoscale research and prediction, and the modernization of the National Weather Service. Contemporary imperatives include developing leadership and management capabilities, making the requisite investments in advanced data and information systems, enabling the prediction of predictability, and developing appropriate approaches to education and to cooperation with other disciplines. Most of all, the atmospheric sciences community must learn to be more effective in determining priorities and in developing compelling rationales for obtaining the resources necessary to pursue signal opportunities for scientific advance and service to the nation.

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 common example of a simple dynamical system, the logistic map, is introduced. In Section 2 statistical prediction for a dynamical system, based on observing a portion of a realization of the system with error, is considered. The emphasis is on likelihood and Bayesian approaches to the problem of prediction. Example computations, based on simulated data, suggest some novel characteristics of chaotic data analysis. Section 3 is an excursus into the relationships between ergodic theory and chaos, with specific application to Bayesian forecasting. In Section 4 a common numerical approach, which generates discrete-time dynamical systems, to the solution of differential equations is described briefly. An example involving the so-called Duffing equation is presented.

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...

Hierarchical training of neural networks and prediction of chaotic time series

We present a new procedure for hierarchical training of multilayer perceptrons to outputs of high precision. It achieves a dramatic increase in accuracy, e.g. by three orders of magnitude, and can reduce training time considerably. The method is applied to the prediction of chaotic systems where we obtain the optimum error evolution for iterated predictions as well as a substantial reduction of the absolute prediction error.

Information content and predictability of lumped and distributed dynamical systems

We discuss several questions related to the predictability of chaotic systems. We first review the information flow through systems with few degrees of freedom and with sensitive dependence on initial conditions, and ways of measuring this flow. We then ask how the knolwedge obtained in this way can be used in improving forecasts, and propose one way of forecasting suggested by dynamical systems concepts. On the more fundamental level, we point out the difference between difficulty and possibility of forecasting, illustrating it with quadratic maps. Next, we ask ourselves how this should be generalized to distributed (i.e., spatially extended) and homogeneous systems. We point out that even the basic concepts of how information is processed by such systems are unknown. Finally, we discuss some intermittency-like effects in coupled maps and cellular automata.

The decay of correlations in chaotic maps

We investigate correlation functions for several chaotic maps. For the one-dimensional Belousov-Zhabotinsky map noise leads to a slower decay of correlations. In order to eliminate periodic components in a continuous three-dimensional model the second iterate of a next-amplitude map is used. Furthermore we discuss information-theoretical correlation measures, which describe the limited predictability of chaotic systems.