Bifurcations In Piecewise-smooth Continuous Systems

by David John Warwick Simpson

Overview

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.

Product Details

ISBN-13:	9789814293846
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	01/15/2010
Series:	World Scientific Series On Nonlinear Science Series A , #70
Pages:	256
Product dimensions:	6.10(w) x 9.00(h) x 0.80(d)

Table of Contents

Preface v

Acknowledgments xi

1 Fundamentals of Piecewise-Smooth, Continuous Systems 1

1.1 Applications 3

1.2 A Framework for Local Behavior 8

1.3 Existence of Equilibria and Fixed Points 12

1.4 The Observer Canonical Form 14

1.5 Discontinuous Bifurcations 18

1.6 Border-Collision Bifurcations 20

1.7 Poincaré Maps and Discontinuity Maps 24

1.8 Period Adding 29

1.9 Smooth Approximations 31

2 Discontinuous Bifurcations in Planar Systems 33

2.1 Periodic Orbits 34

2.2 The Focus-Focus Case in Detail 41

2.3 Summary and Classification 46

3 Codimension-Two, Discontinuous Bifurcations 53

3.1 A Nonsmooth, Saddle-Node Bifurcation 56

3.2 A Nonsmooth, Hopf Bifurcation 59

3.3 A Codimension-Two, Discontinuous Hopf Bifurcation 70

4 The Growth of Saccharomyces cerevisiae 75

4.1 Mathematical Model 77

4.2 Basic Mathematical Observations 82

4.3 Bifurcation Structure 83

4.4 Simple and Complicated Stable Oscillations 89

5 Codimension-Two, Border-Collision Bifurcations 95

5.1 A Nonsmooth, Saddle-Node Bifurcation 96

5.2 A Nonsmooth, Period-Doubling Bifurcation 99

6 Periodic Solutions and Resonance Tongues 107

6.1 Symbolic Dynamics 109

6.2 Describing and Locating Periodic Solutions 112

6.3 Resonance Tongue Boundaries 116

6.4 Rotational Symbol Sequences 122

6.5 Cardinality of Symbol Sequences 125

6.6 Shrinking Points 126

6.7 Unfolding Shrinking Points 131

7 Neimark-Sacker-Like Bifurcations 135

7.1 A Two Dimensional Map 137

7.2 Basic Dynamics 139

7.3 Limiting Parameter Values 142

7.4 Resonance Tongues 145

7.5 Complex Phenomena Relating to Resonance Tongues 153

7.6 More Complex Phenomena 160

Appendix A Selected Proofs 165

Lemma 1.3 165

Theorem 1.1 166

Theorem 2.1 167

Theorem 3.1 171

Theorem 3.2 173

Theorem 3.3 181

Theorem 3.4 183

Theorem 5.2 187

Theorem 5.3 194

Lemma 6.9 197

Theorem 6.1 199

Lemma 7.1 201

Appendix B Additional Figures 211

Appendix C Adjugate Matrices 213

Appendix D Parameter Values for S. cerevisiae 213

Bibliography 215

Index 237