Bifurcations in piecewise-smooth continuous systems

✍ Scribed by Simpson D.J.W.

• Publisher: WS
• Year: 2010
• Tongue: English
• Leaves: 255
• Series: World Scientific Series on Nonlinear Science Series a
• Category: Library

✦ Synopsis

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......Page 6
Acknowledgments......Page 12
Contents......Page 14
1. Fundamentals of Piecewise-Smooth, Continuous Systems......Page 18
1.1 Applications......Page 20
1.2 A Framework for Local Behavior......Page 25
1.3 Existence of Equilibria and Fixed Points......Page 29
1.4 The Observer Canonical Form .......Page 31
1.5 Discontinuous Bifurcations......Page 35
1.6 Border-Collision Bifurcations......Page 37
1.7 Poincare Maps and Discontinuity Maps......Page 41
1.8 Period Adding......Page 46
1.9 Smooth Approximations......Page 48
2. Discontinuous Bifurcations in Planar Systems......Page 50
2.1 Periodic Orbits......Page 51
2.2 The Focus-Focus Case in Detail......Page 58
2.3 Summary and Classification......Page 63
3. Codimension-Two, Discontinuous Bifurcations......Page 70
3.1 A Nonsmooth, Saddle-Node Bifurcation......Page 73
3.2 A Nonsmooth, Hopf Bifurcation .......Page 76
3.3 A Codimension-Two, Discontinuous Hopf Bifurcation......Page 87
4. The Growth of Saccharomyces cerevisiae......Page 92
4.1 Mathematical Model......Page 94
4.2 Basic Mathematical Observations......Page 99
4.3 Bifurcation Structure......Page 100
4.4 Simple and Complicated Stable Oscillations......Page 106
5. Codimension-Two, Border-Collision Bifurcations......Page 112
5.1 A Nonsmooth, Saddle-Node Bifurcation......Page 113
5.2 A Nonsmooth, Period-Doubling Bifurcation......Page 116
6. Periodic Solutions and Resonance Tongues......Page 124
6.1 Symbolic Dynamics......Page 126
6.2 Describing and Locating Periodic Solutions......Page 129
6.3 Resonance Tongue Boundaries......Page 133
6.4 Rotational Symbol Sequences .......Page 139
6.5 Cardinality of Symbol Sequences......Page 142
6.6 Shrinking Points......Page 143
6.7 Unfolding Shrinking Points......Page 148
7. Neimark-Sacker-Like Bifurcations......Page 152
7.1 A Two-Dimensional Map......Page 154
7.2 Basic Dynamics......Page 156
7.3 Limiting Parameter Values......Page 159
7.4 Resonance Tongues......Page 162
7.5 Complex Phenomena Relating to Resonance Tongues......Page 170
7.6 More Complex Phenomena......Page 177
Lemma 1.3......Page 182
Theorem 1.1......Page 183
Theorem 2.1......Page 184
Theorem 3.1......Page 188
Theorem 3.2......Page 190
Theorem 3.3......Page 198
Theorem 3.4......Page 200
Theorem 5.2......Page 204
Theorem 5.3......Page 211
Lemma 6.9......Page 214
Theorem 6.1......Page 216
Lemma 7.1......Page 218
Appendix B Additional Figures......Page 222
Appendix C Adjugate Matrices......Page 228
Appendix D Parameter Values for S. cerevisiae......Page 230
Bibliography......Page 232
Index......Page 254