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.

A Terminal Sliding Mode Control of Disturbed Nonlinear Second-Order Dynamical Systems

2016 ◽  
Vol 11 (5) ◽  
Author(s):  
Pawel Skruch
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Sliding Mode ◽  
Second Order ◽  
Control Law ◽  
Sliding Surface ◽  
Terminal Sliding Mode ◽  
Nonlinear Dynamical ◽  
Second Order Differential Equations ◽  
Bounded Disturbance

The paper presents a terminal sliding mode controller for a certain class of disturbed nonlinear dynamical systems. The class of such systems is described by nonlinear second-order differential equations with an unknown and bounded disturbance. A sliding surface is defined by the system state and the desired trajectory. The control law is designed to force the trajectory of the system from any initial condition to the sliding surface within a finite time. The trajectory of the system after reaching the sliding surface remains on it. A computer simulation is included as an example to verify the approach and to demonstrate its effectiveness.

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.

scholarly journals Existence of Perpetual Points in Nonlinear Dynamical Systems and Its Applications

2015 ◽  
Vol 25 (02) ◽  
pp. 1530005 ◽  
Author(s):  
Awadhesh Prasad
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Critical Points ◽  
Nonlinear Dynamical Systems ◽  
Transient Dynamics ◽  
Coexisting Attractors ◽  
Nonlinear Dynamical ◽  
New Class ◽  
Perpetual Points ◽  
Bifurcation Behavior

A new class of critical points, termed as perpetual points, where acceleration becomes zero but the velocity remains nonzero, are observed in dynamical systems. The velocity at these points is either maximum or minimum or of inflection behavior. These points also show the bifurcation behavior as the parameters of the system vary. These perpetual points are useful for locating the hidden oscillating attractors as well as coexisting attractors. Results show that these points are important for a better understanding of transient dynamics in the phase space. The existence of these points confirms whether a system is dissipative or not. Various examples are presented, and the results are discussed analytically as well as numerically.

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.

CLOSURES OF FRACTAL SETS IN NONLINEAR DYNAMICAL SYSTEMS WITH SWITCHED INPUTS

2001 ◽  
Vol 11 (08) ◽  
pp. 2205-2215 ◽  
Author(s):  
RYOICHI WADA ◽  
KAZUTOSHI GOHARA
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Dynamical Systems ◽  
Limit Cycle ◽  
Nonlinear Dynamical Systems ◽  
Duffing Oscillator ◽  
Nonlinear Dynamical ◽  
Fractal Sets ◽  
Fractal Set

This paper studies closures of fractal sets observed in nonlinear dynamical systems excited stochastically by switched inputs. The Duffing oscillator and the forced dumped pendulum are analyzed as examples. The dynamics of the system is characterized by a fractal set in the phase space. We can numerically construct a closure that encloses the fractal set. Furthermore, it is shown that the closure is a limit cycle attractor of a dynamical system defined by the switching manifold.

NONEXISTENCE OF PERIODIC SOLUTIONS IN A CLASS OF DYNAMICAL SYSTEMS WITH CYLINDRICAL PHASE SPACE

2005 ◽  
Vol 15 (04) ◽  
pp. 1423-1431 ◽  
Author(s):  
YING YANG ◽  
ZHISHENG DUAN ◽  
LIN HUANG
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Periodic Solutions ◽  
Nonlinear Dynamical Systems ◽  
Dynamical Behavior ◽  
Linear Matrix ◽  
Specific Kind ◽  
Nonlinear Dynamical ◽  
Cylindrical Phase ◽  
Cylindrical Phase Space

This paper investigates the nonexistence of a specific kind of periodic solutions in a class of nonlinear dynamical systems with cylindrical phase space. Those types of systems can be viewed as an interconnection of several simpler subsystems with the interconnecting structure specified by a permutation matrix. Frequency-domain conditions as well as linear matrix inequalities conditions for nonexistence of limit cycles of the second kind are established. The main results also define the frequency range on which cycles of the second kind of the system cannot exist. Based on this LMI approach, an estimate of the frequency of cycles of the second kind can be explicitly computed by solving a generalized eigenvalue minimization problem. Numerical results demonstrate the applicability and validity of the proposed method and show the effect of nonlinear interconnections on dynamical behavior of entire interconnected systems.

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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