Nonlinear Dynamical Systems, Their Stability, and Chaos

2014 ◽  
Vol 66 (2) ◽  
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.

Carathe´odory Controllability Criterion for Nonlinear Dynamical Systems

1978 ◽  
Vol 100 (3) ◽  
pp. 209-213 ◽  
Author(s):  
G. Langholz ◽  
M. Sokolov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Control Theory ◽  
Linear Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
Modern Control Theory ◽  
Open Question ◽  
Modern Control

The question of whether a system is controllable or not is of prime importance in modern control theory and has been actively researched in recent years. While it is a solved problem for linear systems, it is still an open question when dealing with bilinear and nonlinear systems. In this paper, a controllability criterion is established based on a theorem by Carathe´odory. By associating a given dynamical system with a certain Pfaffian equation, it is argued that the system is controllable (uncontrollable) if its associated Pfaffian form is nonintegrable (integrable).

scholarly journals On diagrammatic technique for nonlinear dynamical systems

2014 ◽  
Vol 29 (35) ◽  
pp. 1430039
Author(s):  
Mykola Semenyakin
Keyword(s):  
Vector Field ◽  
Dynamical Systems ◽  
Fixed Points ◽  
Vector Fields ◽  
Nonlinear Dynamical Systems ◽  
Radius Of Convergence ◽  
Evolution Operators ◽  
Finite Degree ◽  
Nonlinear Dynamical ◽  
Diagrammatic Technique

In this paper, we investigate phase flows over ℂn and ℝn generated by vector fields V = ∑ Pi∂i where Pi are finite degree polynomials. With the convenient diagrammatic technique, we get expressions for evolution operators ev {V|t} : x(0) ↦ x(t) through the series in powers of x(0) and t, represented as sum over all trees of a particular type. Estimates are made for the radius of convergence in some particular cases. The phase flows behavior in the neighborhood of vector field fixed points are examined. Resonance cases are considered separately.

scholarly journals Dynamical Movement Primitives: Learning Attractor Models for Motor Behaviors

2013 ◽  
Vol 25 (2) ◽  
pp. 328-373 ◽  
Author(s):  
Auke Jan Ijspeert ◽  
Jun Nakanishi ◽  
Heiko Hoffmann ◽  
Peter Pastor ◽  
Stefan Schaal
Keyword(s):  
Motor Control ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Nonlinear Dynamical Systems ◽  
Emergent Behavior ◽  
Nonlinear Dynamical ◽  
Movement Primitives ◽  
Model Complex ◽  
Linear Differential ◽  
Goal Directed Behavior

Nonlinear dynamical systems have been used in many disciplines to model complex behaviors, including biological motor control, robotics, perception, economics, traffic prediction, and neuroscience. While often the unexpected emergent behavior of nonlinear systems is the focus of investigations, it is of equal importance to create goal-directed behavior (e.g., stable locomotion from a system of coupled oscillators under perceptual guidance). Modeling goal-directed behavior with nonlinear systems is, however, rather difficult due to the parameter sensitivity of these systems, their complex phase transitions in response to subtle parameter changes, and the difficulty of analyzing and predicting their long-term behavior; intuition and time-consuming parameter tuning play a major role. This letter presents and reviews dynamical movement primitives, a line of research for modeling attractor behaviors of autonomous nonlinear dynamical systems with the help of statistical learning techniques. The essence of our approach is to start with a simple dynamical system, such as a set of linear differential equations, and transform those into a weakly nonlinear system with prescribed attractor dynamics by means of a learnable autonomous forcing term. Both point attractors and limit cycle attractors of almost arbitrary complexity can be generated. We explain the design principle of our approach and evaluate its properties in several example applications in motor control and robotics.

scholarly journals Symmetries and itineracy in nonlinear systems with many degrees of freedom

2001 ◽  
Vol 24 (5) ◽  
pp. 813-813 ◽  
Author(s):  
Michael Breakspear ◽  
Karl Friston
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Brain Function ◽  
Degrees Of Freedom ◽  
Nonlinear Dynamical Systems ◽  
Research Direction ◽  
Potential Contribution ◽  
Nonlinear Dynamical

Tsuda examines the potential contribution of nonlinear dynamical systems, with many degrees of freedom, to understanding brain function. We offer suggestions concerning symmetry and transients to strengthen the physiological motivation and theoretical consistency of this novel research direction: Symmetry plays a fundamental role, theoretically and in relation to real brains. We also highlight a distinction between chaotic “transience” and “itineracy.”

Periodic Flows to Chaos Based on Discrete Implicit Mappings of Continuous Nonlinear Systems

2015 ◽  
Vol 25 (03) ◽  
pp. 1550044 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Time Delay ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Nonlinear Dynamical Systems ◽  
Duffing Oscillator ◽  
Single Step ◽  
Nonlinear Dynamical ◽  
Periodic Flows ◽  
Mapping Structures ◽  
Stability And Bifurcations

This paper presents a semi-analytical method for periodic flows in continuous nonlinear dynamical systems. For the semi-analytical approach, differential equations of nonlinear dynamical systems are discretized to obtain implicit maps, and a mapping structure based on the implicit maps is employed for a periodic flow. From mapping structures, periodic flows in nonlinear dynamical systems are predicted analytically and the corresponding stability and bifurcations of the periodic flows are determined through the eigenvalue analysis. The periodic flows predicted by the single-step implicit maps are discussed first, and the periodic flows predicted by the multistep implicit maps are also presented. Periodic flows in time-delay nonlinear dynamical systems are discussed by the single-step and multistep implicit maps. The time-delay nodes in discretization of time-delay nonlinear systems were treated by both an interpolation and a direct integration. Based on the discrete nodes of periodic flows in nonlinear dynamical systems with/without time-delay, the discrete Fourier series responses of periodic flows are presented. To demonstrate the methodology, the bifurcation tree of period-1 motion to chaos in a Duffing oscillator is presented as a sampled problem. The method presented in this paper can be applied to nonlinear dynamical systems, which cannot be solved directly by analytical methods.

scholarly journals Controlling nonlinear dynamical systems into arbitrary states using machine learning

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.

CONTROLLING CHAOTIC SYSTEMS VIA PHASE SPACE RECONSTRUCTION TECHNIQUE

2003 ◽  
Vol 13 (02) ◽  
pp. 467-471 ◽  
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.

ALGORITHMIC COMPLEXITY OF SOME PHYSICAL PROCESSES

1994 ◽  
Vol 05 (02) ◽  
pp. 421-424
Author(s):  
B.S. SAGINTAEV
Keyword(s):  
Time Series ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Temporal Variation ◽  
Nonlinear Dynamical Systems ◽  
Algorithmic Complexity ◽  
Physical Processes ◽  
Nonlinear Dynamical ◽  
The Relationship

Measure of algorithmic complexity c(n) which has been suggested by Lempel and Ziv is one of important quantities for characterizing properties of nonlinear dynamical systems. The temporal variation of c(n) is investigated for time series generated by some physical processes. The relationship between algorithmic complexity and other characteristics of nonlinear systems is discussed.

Using Strange Attractors to Control Sound Processing in Live Electroacoustic Composition

2015 ◽  
Vol 39 (3) ◽  
pp. 25-45
Author(s):  
Miroslav Spasov
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Nonlinear Dynamical Systems ◽  
Chaotic Attractors ◽  
Live Electronics ◽  
Tenor Saxophone ◽  
Sound Processing ◽  
Nonlinear Dynamical ◽  
Interactive Composition ◽  
Mathematical Equations

This article explores the possibility of using chaotic attractors to control sound processing with software instruments in live electroacoustic composition. The practice-led investigation involves the Attractors Library, a collection of Max/MSP externals based on iterative mathematical equations representing nonlinear dynamical systems; Attractors Player, a Max/MSP patch that controls the attractors' performance and live processing; and the two compositions based on the software: Strange Attractions for flute, clarinet, horn, and live electronics, and Sabda Vidya No. 2 for flute, tenor saxophone, and live electronics. In the article I discuss some specific attractors' characteristics and their use in interactive composition, relying on the experience from the performances of these two compositions. The idea is to highlight the experience with these nonlinear systems and to encourage other composers to use them in their own works.

NONLINEAR DYNAMICS AND CHAOS PART II: ERGODIC APPROACH

1999 ◽  
Vol 3 (1) ◽  
pp. 84-114 ◽  
Author(s):  
Alfredo Medio
Keyword(s):  
Dynamical Systems ◽  
Chaotic Systems ◽  
Nonlinear Dynamical Systems ◽  
Measure Theory ◽  
Modern Theory ◽  
Physical Measure ◽  
Nonlinear Dynamical ◽  
Nonlinear Dynamics And Chaos ◽  
Ergodic Measures ◽  
Theoretical Apparatus

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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