Screen LibraryBifurcations and chaos in piecewise-smooth dynamical systems /
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 va...
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| Main Authors: | |
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| Corporate Authors: | |
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| Published: |
World Scientific Pub. Co.,
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| Publisher Address: | Singapore ; River Edge, N.J. : |
| Publication Dates: | 2003. |
| Literature type: | eBook |
| Language: | English |
| Series: |
World Scientific series on nonlinear science. Series A ;
v. 44 |
| Subjects: | |
| Online Access: |
http://www.worldscientific.com/worldscibooks/10.1142/5313#t=toc |
| Summary: |
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. |
| Carrier Form: | 1 online resource (xii,363pages) : illustrations. |
| Bibliography: | Includes bibliographical references and index. |
| ISBN: | 9789812564436 (electronic bk.) |
| CLC: | O177.91 |
| Contents: | 1. On the dynamics of nonlinear systems. 1.1. Unpredictability and chaos. 1.2. Oscillations and chaos in engineering systems. 1.3. Simple one-dimensional maps. 1.4. Border-collision bifurcations. 1.5. The sewing approach -- 2. Basic concepts and methods. 2.1. Bifurcation analysis for the H non map. 2.2. Division of the parameter plane. 2.3. Further analysis of the H non map. 2.4. Piecewise-smooth two-dimensional maps. 2.5. Some remaining problems -- 3. Relay control systems. 3.1. Bifurcations and chaos in relay systems. 3.2. Modeling relay systems with hysteresis. 3.3. Algorithms for determining limit cycles. 3.4. Local stability of periodic solutions. 3.5. Application to relay systems. 3.6. Model of a DC/DC converter with relay feedback control. 3.7. DC electric drive with relay control -- 4. Bifurcations and chaotic oscillations in relay systems. 4.1. Relay control systems with complex dynamics. 4.2. Two-parameter analysis of dynamical modes. 4.3. Analysis of the branching patterns. 4.4. Chaotization of oscillations in relay systems. 4.5. Complex oscillations in a DC electric drive. 4.6. On the concept of normal structures -- 5. Chaotic oscillations in pulse-width modulated systems. 5.1. Application of pulse-width modulation in power electronics. 5.2. Voltage converter with pulse-width modulation. 5.3. Periodic solutions and their local stability. 5.4. Transitions to chaos via local bifurcations. 5.5. Border-collision bifurcations and transitions to chaos -- 6. Border-collision bifurcations on a two-dimensional torus. 6.1. Model of a switching power converter. 6.2. Two-parameter diagram of dynamical modes. 6.3. Resonance tongues and border-collision bifurcations. 7. Border-collision bifurcations in a management system. 7.1. Structure produces behavior. 7.2. A cascaded production-distribution system. 7.3. The BEER model. 7.4. Transitions to chaos. 7.5. Hyperchaotic phenomena. 7.6. Border-collisions and resonance phenomena. 7.7 Conclusions. |