Akhmet Principles of Discontinuous Dynamical Systems

1. Auflage 2010
ISBN: 978-1-4419-6581-3
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 176 Seiten

Discontinuous dynamical systems have played an important role in both theory and applications during the last several decades. This is still an area of active research and techniques to make the applications more effective are an ongoing topic of interest. Principles of Discontinuous Dynamical Systems is devoted to the theory of differential equations with variable moments of impulses. It introduces a new strategy of implementing an equivalence to systems whose solutions have prescribed moments of impulses and utilizing special topologies in spaces of piecewise continuous functions. The achievements obtained on the basis of this approach are described in this book. The text progresses systematically, by covering preliminaries in the first four chapters. This is followed by more complex material and special topics such as Hopf bifurcation, Devaney's chaos, and the shadowing property are discussed in the last two chapters. This book is suitable for researchers and graduate students in mathematics and also in diverse areas such as biology, computer science, and engineering who deal with real world problems.

Autoren/Hrsg.

Akhmet, Marat

Inhaltsverzeichnis

1;Principles of Discontinuous Dynamical Systems;1
1.1;Preface;7
1.2;Contents
;9
1.3;1 Introduction;13
1.4;2 Description of the System with Fixed Moments of Impulses and Its Solutions;19
1.4.1;2.1 Spaces of Piecewise Continuous Functions;19
1.4.2;2.2 Description of the System;21
1.4.3;2.3 Description of Solutions;22
1.4.4;2.4 Equivalent Integral Equations;26
1.4.5;2.5 The Gronwall--Bellman Lemma for Piecewise Continuous Functions;28
1.4.6;2.6 Existence and Uniqueness Theorems;30
1.4.7;2.7 Continuity;32
1.4.8;Notes;35
1.5;3 Stability and Periodic Solutions of Systems with Fixed Moments of Impulses;37
1.5.1;3.1 Definitions of Stability;37
1.5.2;3.2 Basics of Periodic Systems;39
1.5.3;Notes;42
1.6;4 Basics of Linear Systems;43
1.6.1;4.1 Linear Homogeneous Systems;43
1.6.2;4.2 Linear Nonhomogeneous Systems;53
1.6.3;4.3 Linear Periodic Systems;59
1.6.4;Notes;64
1.7;5 Nonautonomous Systems with Variable Moments of Impulses;67
1.7.1;5.1 Description of Systems;67
1.7.2;5.2 Existence, Uniqueness, and Extension;68
1.7.3;5.3 Beating Phenomena and Related Properties;71
1.7.4;5.4 The Topology on the Set of Discontinuous Functions;74
1.7.5;5.5 B-Equivalence: General Case;75
1.7.6;5.6 Continuity Properties;80
1.7.7;5.7 Generalities of Stability;82
1.7.8;5.8 B-Equivalence: Quasilinear Systems;85
1.7.9;5.9 Poincaré Criterion and Periodic Solutions of Quasilinear Systems;90
1.7.10;Notes;91
1.8;6 Differentiability Properties of Nonautonomous Systems;93
1.8.1;6.1 Differentiability with Respect to Initial Conditions;94
1.8.2;6.2 Differentiability with Respect to Parameters;100
1.8.3;6.3 Higher Order B-Derivatives;102
1.8.4;6.4 B-Analyticity Property;105
1.8.5;6.5 B-Asymptotic Approximation of Solutions;107
1.8.6;Notes;109
1.9;7 Periodic Solutions of Nonlinear Systems;111
1.9.1;7.1 Quasilinear Systems: the Noncritical Case;111
1.9.2;7.2 The Critical Case;118
1.9.3;Notes;122
1.10;8 Discontinuous Dynamical Systems;124
1.10.1;8.1 Generalities;124
1.10.2;8.2 Local Existence and Uniqueness;129
1.10.3;8.3 Extension of Solutions;130
1.10.4;8.4 The Group Property;137
1.10.5;8.5 Continuity Properties;139
1.10.6;8.6 B-Equivalence;141
1.10.7;8.7 Differentiability Properties;144
1.10.8;8.8 Conclusion;146
1.10.9;8.9 Examples;146
1.10.10;Notes;148
1.11;9 Perturbations and Hopf Bifurcation of a Discontinuous Limit Cycle;149
1.11.1;9.1 The Nonperturbed System;149
1.11.2;9.2 The Perturbed System;152
1.11.3;9.3 Foci of the D-System;154
1.11.4;9.4 The Center and Focus Problem;157
1.11.5;9.5 Bifurcation of a Discontinuous Limit Cycle ;159
1.11.6;9.6 Examples;163
1.11.7;Notes;163
1.12;10 Chaos and Shadowing;165
1.12.1;10.1 Introduction and Preliminaries;165
1.12.2;10.2 The Devaney's Chaos;167
1.12.3;10.3 Shadowing Property;172
1.12.4;10.4 Simulations;173
1.12.5;Notes;175
1.13;References;176
1.14;Index;183