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Principles of Discontinuous Dynamical Systems
von: Marat Akhmet
Springer-Verlag, 2010
ISBN: 9781441965813 , 176 Seiten
Format: PDF
Kopierschutz: DRM
Preis: 53,49 EUR
Principles of Discontinuous Dynamical Systems
1
Preface
7
Contents
Contents
1 Introduction
13
2 Description of the System with Fixed Moments of Impulses and Its Solutions
19
2.1 Spaces of Piecewise Continuous Functions
19
2.2 Description of the System
21
2.3 Description of Solutions
22
2.4 Equivalent Integral Equations
26
2.5 The Gronwall--Bellman Lemma for Piecewise Continuous Functions
28
2.6 Existence and Uniqueness Theorems
30
2.7 Continuity
32
Notes
35
3 Stability and Periodic Solutions of Systems with Fixed Moments of Impulses
37
3.1 Definitions of Stability
37
3.2 Basics of Periodic Systems
39
Notes
42
4 Basics of Linear Systems
43
4.1 Linear Homogeneous Systems
43
4.2 Linear Nonhomogeneous Systems
53
4.3 Linear Periodic Systems
59
Notes
64
5 Nonautonomous Systems with Variable Moments of Impulses
67
5.1 Description of Systems
67
5.2 Existence, Uniqueness, and Extension
68
5.3 Beating Phenomena and Related Properties
71
5.4 The Topology on the Set of Discontinuous Functions
74
5.5 B-Equivalence: General Case
75
5.6 Continuity Properties
80
5.7 Generalities of Stability
82
5.8 B-Equivalence: Quasilinear Systems
85
5.9 Poincaré Criterion and Periodic Solutions of Quasilinear Systems
90
Notes
91
6 Differentiability Properties of Nonautonomous Systems
93
6.1 Differentiability with Respect to Initial Conditions
94
6.2 Differentiability with Respect to Parameters
100
6.3 Higher Order B-Derivatives
102
6.4 B-Analyticity Property
105
6.5 B-Asymptotic Approximation of Solutions
107
Notes
109
7 Periodic Solutions of Nonlinear Systems
111
7.1 Quasilinear Systems: the Noncritical Case
111
7.2 The Critical Case
118
Notes
122
8 Discontinuous Dynamical Systems
124
8.1 Generalities
124
8.2 Local Existence and Uniqueness
129
8.3 Extension of Solutions
130
8.4 The Group Property
137
8.5 Continuity Properties
139
8.6 B-Equivalence
141
8.7 Differentiability Properties
144
8.8 Conclusion
146
8.9 Examples
146
Notes
148
9 Perturbations and Hopf Bifurcation of a Discontinuous Limit Cycle
149
9.1 The Nonperturbed System
149
9.2 The Perturbed System
152
9.3 Foci of the D-System
154
9.4 The Center and Focus Problem
157
9.5 Bifurcation of a Discontinuous Limit Cycle
159
9.6 Examples
163
Notes
163
10 Chaos and Shadowing
165
10.1 Introduction and Preliminaries
165
10.2 The Devaney's Chaos
167
10.3 Shadowing Property
172
10.4 Simulations
173
Notes
175
References
176
Index
183
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