Bifurcations in piecewise-smooth continuous systems /

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Bibliographic Details
Author / Creator:	Simpson, David John Warwick.
Imprint:	Singapore ; Hackensack, NJ : World Scientific, 2010.
Description:	xv, 238 p. : ill. (some col.) ; 24 cm.
Language:	English
Series:	World Scientific series on nonlinear science. Series A, Monographs and treatises ; v. 70 World Scientific series on nonlinear science. Series A, Monographs and treatises ; v. 70.
Subject:	Bifurcation theory. Differential equations. Saccharomyces cerevisiae. Bifurcation theory. Differential equations. Saccharomyces cerevisiae.
Format:	Print Book
URL for this record:	http://pi.lib.uchicago.edu/1001/cat/bib/8056368

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ISBN:	9814293849 9789814293846
Notes:	Originally presented as: Thesis (Ph.D.)--University of Colorado at Boulder, 2008. Includes bibliographical references and index.
Summary:	Real-world systems that involve some non-smooth change are often well-modeled by piecewise-smooth systems. However there still remain many gaps in the mathematical theory of such systems. This doctoral thesis presents new results regarding bifurcations of piecewise-smooth, continuous, autonomous systems of ordinary differential equations and maps. Various codimension-two, discontinuity induced bifurcations are unfolded in a rigorous manner. Several of these unfoldings are applied to a mathematical model of the growth of Saccharomyces cerevisiae (a common yeast). The nature of resonance near border-collision bifurcations is described; in particular, the curious geometry of resonance tongues in piecewise-smooth continuous maps is explained in detail. Neimark-Sacker-like border-collision bifurcations are both numerically and theoretically investigated. A comprehensive background section is conveniently provided for those with little or no experience in piecewise-smooth systems.

Table of Contents:

Preface
Acknowledgments
1. Fundamentals of Piecewise-Smooth, Continuous Systems
1.1. Applications
1.2. A Framework for Local Behavior
1.3. Existence of Equilibria and Fixed Points
1.4. The Observer Canonical Form
1.5. Discontinuous Bifurcations
1.6. Border-Collision Bifurcations
1.7. Poincaré Maps and Discontinuity Maps
1.8. Period Adding
1.9. Smooth Approximations
2. Discontinuous Bifurcations in Planar Systems
2.1. Periodic Orbits
2.2. The Focus-Focus Case in Detail
2.3. Summary and Classification
3. Codimension-Two, Discontinuous Bifurcations
3.1. A Nonsmooth, Saddle-Node Bifurcation
3.2. A Nonsmooth, Hopf Bifurcation
3.3. A Codimension-Two, Discontinuous Hopf Bifurcation
4. The Growth of Saccharomyces cerevisiae
4.1. Mathematical Model
4.2. Basic Mathematical Observations
4.3. Bifurcation Structure
4.4. Simple and Complicated Stable Oscillations
5. Codimension-Two, Border-Collision Bifurcations
5.1. A Nonsmooth, Saddle-Node Bifurcation
5.2. A Nonsmooth, Period-Doubling Bifurcation
6. Periodic Solutions and Resonance Tongues
6.1. Symbolic Dynamics
6.2. Describing and Locating Periodic Solutions
6.3. Resonance Tongue Boundaries
6.4. Rotational Symbol Sequences
6.5. Cardinality of Symbol Sequences
6.6. Shrinking Points
6.7. Unfolding Shrinking Points
7. Neimark-Sacker-Like Bifurcations
7.1. A Two Dimensional Map
7.2. Basic Dynamics
7.3. Limiting Parameter Values
7.4. Resonance Tongues
7.5. Complex Phenomena Relating to Resonance Tongues
7.6. More Complex Phenomena
Appendix A. Selected Proofs
Lemma 1.3.
Theorem 1.1.
Theorem 2.1.
Theorem 3.1.
Theorem 3.2.
Theorem 3.3.
Theorem 3.4.
Theorem 5.2.
Theorem 5.3.
Lemma 6.9.
Theorem 6.1.
Lemma 7.1.
Appendix B. Additional Figures
Appendix C. Adjugate Matrices
Appendix D. Parameter Values for S. cerevisiae
Bibliography
Index

Bifurcations in piecewise-smooth continuous systems /

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