Screen LibraryHopf bifurcation analysis : a frequency domain approach /
This book is devoted to the frequency domain approach, for both regular and degenerate Hopf bifurcation analyses. Besides showing that the time and frequency domain approaches are in fact equivalent, the fact that many significant results and computational formulas obtained in the studies of regular...
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| Main Authors: | |
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| Corporate Authors: | |
| Group Author: | |
| Published: |
World Scientific Pub. Co.,
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| Publisher Address: | Singapore : |
| Publication Dates: | 1996. |
| Literature type: | eBook |
| Language: | English |
| Series: |
World Scientific series on nonlinear science. Series A ;
vol. 21 |
| Subjects: | |
| Online Access: |
http://www.worldscientific.com/worldscibooks/10.1142/3070#t=toc |
| Summary: |
This book is devoted to the frequency domain approach, for both regular and degenerate Hopf bifurcation analyses. Besides showing that the time and frequency domain approaches are in fact equivalent, the fact that many significant results and computational formulas obtained in the studies of regular and degenerate Hopf bifurcations from the time domain approach can be translated and reformulated into the corresponding frequency domain setting, and be reconfirmed and rediscovered by using the frequency domain methods, is also explained. The description of how the frequency domain approach can be used to obtain several types of standard bifurcation conditions for general nonlinear dynamical systems is given as well as is demonstrated a very rich pictorial gallery of local bifurcation diagrams for nonlinear systems under simultaneous variations of several system parameters. In conjunction with this graphical analysis of local bifurcation diagrams, the defining and nondegeneracy conditions for several degenerate Hopf bifurcations is presented. With a great deal of algebraic computation, some higher-order harmonic balance approximation formulas are derived, for analyzing the dynamical behavior in small neighborhoods of certain types of degenerate Hopf bifurcations that involve multiple limit cycles and multiple limit points of periodic solutions. In addition, applications in chemical, mechanical and electrical engineering as well as in biology are discussed. This book is designed and written in a style of research monographs rather than classroom textbooks, so that the most recent contributions to the field can be included with references. |
| Carrier Form: | 1 online resource (xv,326pages) : illustrations. |
| Bibliography: | Includes bibliographical references (pages 299-310) and indexes. |
| ISBN: | 9789812798633 |
| CLC: | O175 |
| Contents: | 1. Introduction. 1.1. Stability bifurcations. 1.2. Center manifold theorem. 1.3. Limit cycles and degenerate Hopf bifurcations -- 2. The Hopf bifurcation theorem. 2.1. Introduction. 2.2. The Hopf bifurcation theorem in the time domain. 2.3. The Hopf theorem in the frequency domain. 2.4. Equivalence of the two Hopf theorems. 2.5. Advantages of the frequency domain approach. 2.6. An application of the graphical Hopf theorem -- 3. Continuation of bifurcation curves on the parameter plane. 3.1. Introduction. 3.2. Static and dynamic bifurcations. 3.3. Bifurcation analysis in the frequency domain. 3.4. Degenerate Hopf bifurcations of co-dimension 1. 3.5. Applications and examples -- 4. Degenerate bifurcations in the space of system parameters. 4.1. Introduction. 4.2. Multiplicity of equilibrium solutions. 4.3. Multiple Hopf bifurcation points. 4.4. Degenerate Hopf bifurcations and the singularity theory. 4.5. Degenerate Hopf bifurcations and feedback systems. 4.6. Degenerate Hopf bifurcations and the graphical Hopf theorem. 4.7. Some applications -- 5. High-order Hopf bifurcation formulas. 5.1. Introduction. 5.2. Approximation of periodic solutions by higher-order formulas. 5.3. Continuation of periodic solutions: Degenerate cases. 5.4. Local bifurcation diagrams and the graphical Hopf theorem. 5.5. Algorithms for recovering periodic solutions. 5.6. Multiple limit cycles and numerical problems -- 6. Hopf bifurcation in nonlinear systems with time delays. 6.1. Introduction. 6.2. Conditions for degenerate bifurcations in time-delayed systems. 6.3. Applications in control systems. 6.4. Time-delayed feedback systems: The general case. 6.5. Application examples -- 7. Birth of multiple limit cycles. 7.1. Introduction. 7.2. Harmonic balance and curvature coefficients. 7.3. Some application examples. 7.4. Controlling the multiplicities of limit cycles. |