NONLINEAR DYNAMICS AND CHAOS PART II: ERGODIC APPROACH

This is the second part of a two-part survey of the modern theory of nonlinear dynamical systems. We focus on the study of statistical properties of orbits generated by maps, a field of research known as ergodic theory. After introducing some basic concepts of measure theory, we discuss the notions of invariant and ergodic measures and provide examples of economic applications. The question of attractiveness and observability, already considered in Part I, is revisited and the concept of natural, or physical, measure is explained. This theoretical apparatus then is applied to the question of predictability of dynamical systems, and the notion of metric entropy is discussed. Finally, we consider the class of Bernoulli dynamical systems and discuss the possibility of distinguishing orbits of deterministic chaotic systems and realizations of stochastic processes.

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Nonlinear Dynamical Systems, Their Stability, and Chaos

Applied Mechanics Reviews ◽

10.1115/1.4026864 ◽

2014 ◽

Vol 66 (2) ◽

Cited By ~ 1

Author(s):

Amol Marathe ◽

Rama Govindarajan

Keyword(s):

Fluid Flow ◽

Dynamical Systems ◽

Nonlinear Systems ◽

Fluid Mechanics ◽

Fixed Points ◽

Chaotic Systems ◽

Nonlinear Dynamical Systems ◽

Not Given ◽

Nonlinear Dynamical ◽

Detailed Explanation

This introduction to nonlinear systems is written for students of fluid mechanics, so connections are made throughout the text to familiar fluid flow systems. The aim is to present how nonlinear systems are qualitatively different from linear and to outline some simple procedures by which an understanding of nonlinear systems may be attempted. Considerable attention is paid to linear systems in the vicinity of fixed points, and it is discussed why this is relevant for nonlinear systems. A detailed explanation of chaos is not given, but a flavor of chaotic systems is presented. The focus is on physical understanding and not on mathematical rigor.

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Controlling nonlinear dynamical systems into arbitrary states using machine learning

Scientific Reports ◽

10.1038/s41598-021-92244-6 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Alexander Haluszczynski ◽

Christoph Räth

Keyword(s):

Machine Learning ◽

Dynamical Systems ◽

Chaotic Systems ◽

Nonlinear Dynamical Systems ◽

Chaotic Behavior ◽

Large Data ◽

Data Sets ◽

Initial State ◽

Nonlinear Dynamical ◽

Phase Space Methods

AbstractControlling nonlinear dynamical systems is a central task in many different areas of science and engineering. Chaotic systems can be stabilized (or chaotified) with small perturbations, yet existing approaches either require knowledge about the underlying system equations or large data sets as they rely on phase space methods. In this work we propose a novel and fully data driven scheme relying on machine learning (ML), which generalizes control techniques of chaotic systems without requiring a mathematical model for its dynamics. Exploiting recently developed ML-based prediction capabilities, we demonstrate that nonlinear systems can be forced to stay in arbitrary dynamical target states coming from any initial state. We outline and validate our approach using the examples of the Lorenz and the Rössler system and show how these systems can very accurately be brought not only to periodic, but even to intermittent and different chaotic behavior. Having this highly flexible control scheme with little demands on the amount of required data on hand, we briefly discuss possible applications ranging from engineering to medicine.

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CONTROLLING CHAOTIC SYSTEMS VIA PHASE SPACE RECONSTRUCTION TECHNIQUE

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127403006649 ◽

2003 ◽

Vol 13 (02) ◽

pp. 467-471 ◽

Cited By ~ 7

Author(s):

Y. J. CAO ◽

P. X. ZHANG ◽

S. J. CHENG

Keyword(s):

Phase Space ◽

Dynamical Systems ◽

Chaotic Systems ◽

Nonlinear Dynamical Systems ◽

Linear Part ◽

Phase Space Reconstruction ◽

Control Law ◽

Reconstruction Technique ◽

Nonlinear Dynamical ◽

Novel Approach

A novel approach to control chaotic systems has been developed. The approach employs the technique of phase space reconstruction in nonlinear dynamical systems theory to construct a linear part in the reconstructed system and design a feedback control law. The effectiveness of the proposed approach has been demonstrated through its applications to two well-known chaotic systems: Lorenz chaos and Rössler chaos.

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