Bifurcations And Chaos In Piecewise-smooth Dynamical Systems: Applications To Power Converters, Relay And Pulse-width Modulated Control Systems, And Human Decision-making Behavior

by Zhanybai T Zhusubaliyev, Erik Mosekilde

Overview

Technical problems often lead to differential equations with piecewise-smooth right-hand sides. Problems in mechanical engineering, for instance, violate the requirements of smoothness if they involve collisions, finite clearances, or stick-slip phenomena. Systems of this type can display a large variety of complicated bifurcation scenarios that still lack a detailed description.This book presents some of the fascinating new phenomena that one can observe in piecewise-smooth dynamical systems. The practical significance of these phenomena is demonstrated through a series of well-documented and realistic applications to switching power converters, relay systems, and different types of pulse-width modulated control systems. Other examples are derived from mechanical engineering, digital electronics, and economic business-cycle theory.The topics considered in the book include abrupt transitions associated with modified period-doubling, saddle-node and Hopf bifurcations, the interplay between classical bifurcations and border-collision bifurcations, truncated bifurcation scenarios, period-tripling and -quadrupling bifurcations, multiple-choice bifurcations, new types of direct transitions to chaos, and torus destruction in nonsmooth systems.In spite of its orientation towards engineering problems, the book addresses theoretical and numerical problems in sufficient detail to be of interest to nonlinear scientists in general.

Product Details

ISBN-13:	9789812384201
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	06/27/2003
Series:	World Scientific Series On Nonlinear Science Series A , #44
Pages:	376
Product dimensions:	6.20(w) x 9.10(h) x 1.10(d)

Table of Contents

	Preface	v
1	On the Dynamics of Nonlinear Systems	1
1.1	Unpredictability and Chaos	1
1.2	Oscillations and Chaos in Engineering Systems	12
1.3	Simple One-Dimensional Maps	24
1.4	Border-Collision Bifurcations	37
1.5	The Sewing Approach	43
1.5.1	Existence of a periodic solution	51
1.5.2	Global bifurcations	54
2	Basic Concepts and Methods	67
2.1	Bifurcation Analysis for the Henon Map	67
2.2	Division of the Parameter Plane	71
2.3	Further Analysis of the Henon Map	80
2.4	Piecewise-Smooth Two-Dimensional Maps	92
2.5	Some Remaining Problems	99
3	Relay Control Systems	113
3.1	Bifurcations and Chaos in Relay Systems	113
3.2	Modeling Relay Systems with Hysteresis	118
3.3	Algorithms for Determining Limit Cycles	121
3.4	Local Stability of Periodic Solutions	123
3.4.1	Nonautonomous dynamical systems	124
3.4.2	Autonomous dynamical systems	131
3.5	Application to Relay Systems	132
3.5.1	Relay system with hysteresis	132
3.5.2	Relay system without hysteresis	133
3.6	Model of a DC/DC Converter with Relay Feedback Control	138
3.7	DC Electric Drive with Relay Control	141
3.7.1	Practical realization of relay control systems	141
3.7.2	Model of electric drive with relay control	153
4	Bifurcations and Chaotic Oscillations in Relay Systems	167
4.1	Relay Control Systems with Complex Dynamics	167
4.2	Two-Parameter Analysis of Dynamical Modes	170
4.3	Analysis of the Branching Patterns	178
4.4	Chaotization of Oscillations in Relay Systems	197
4.5	Complex Oscillations in a DC Electric Drive	206
4.5.1	Mathematical model of a relay system with hysteresis and time deadzone	207
4.5.2	Analysis of the complex behavior	210
4.6	On the Concept of Normal Structures	225
5	Chaotic Oscillations in Pulse-Width Modulated Systems	231
5.1	Application of Pulse-Width Modulation in Power Electronics	231
5.2	Voltage Converter with Pulse-Width Modulation	239
5.3	Periodic Solutions and Their Local Stability	246
5.4	Transitions to Chaos via Local Bifurcations	251
5.5	Border-Collision Bifurcations and Transitions to Chaos	254
6	Border-Collision Bifurcations on a Two-Dimensional Torus	279
6.1	Model of a Switching Power Converter	279
6.2	Two-Parameter Diagram of Dynamical Modes	288
6.3	Resonance Tongues and Border-Collision Bifurcations	301
7	Border-Collision Bifurcations in a Management System	321
7.1	Structure Produces Behavior	321
7.2	A Cascaded Production-Distribution System	325
7.3	The BEER Model	327
7.4	Transitions to Chaos	330
7.5	Hyperchaotic Phenomena	337
7.6	Border-Collisions and Resonance Phenomena	344
7.7	Conclusions	350
	Index	359