Dynamical Systems (eBook)

Stefan Pickl ist seit 1980 Professor für Operations Research an der Universität der Bundeswehr München. Werner Krabs war Professor am Fachbereich Mathematik an der TU Darmstadt. Er wurde 1999 emeritiert.

Dynamical Systems 4
Foreword 6
Preface 8
Contents 10
1Uncontrolled Systems 12
1.1 Abstract Definition of Dynamical Systems 12
1.2 Time-Continuous Dynamical Systems 15
1.2.1 Elementary Properties 15
1.2.2 Systems in the Plane 19
1.2.3 Stability: The Direct Method of Lyapunov 25
1.2.4 Application to a General Predator-Prey Model 32
1.2.5 Application to Evolution Matrix Games 43
1.3 Time-Discrete Dynamical Systems 47
1.3.1 The Autonomous Case:Definitions and Elementary Properties 47
1.3.2 Localization of Limit Sets with the Aid of LyapunovFunctions 51
1.3.3 Stability Based on Lyapunov's Method 53
1.3.4 Stability of Fixed Points via Linearisation 58
1.3.5 Linear Systems 61
1.3.6 Discretization of Time-Continuous Dynamical Systems 66
1.3.7 Applications 68
1.3.8 The Non-Autonomous Case: De¯nitions and ElementaryProperties 73
1.3.9 Stability Based on Lyapunov's Method 75
1.3.10 Linear Systems 78
1.3.11 Application to a Model for the Process of Hemo-Dialysis 84
2 Controlled Systems 87
2.1 The Time-Continuous Case 87
2.1.1 The Problem of Controllability 87
2.1.2 Controllability of Linear Systems 88
2.1.3 Restricted Null-Controllability of Linear Systems 92
2.1.4 Controllability of Nonlinear Systems into Rest Points 95
2.1.5 Approximate Solution of the Problem of RestrictedNull-Controllability 99
2.1.6 Time-Minimal Restricted Null-Controllability of LinearSystems 101
2.2 The Time-Discrete Autonomous Case 111
2.2.1 The Problem of Fixed Point Controllability 111
2.2.2 Null-Controllability of Linear Systems 121
2.2.3 A Method for Solving the Problem of Null-Controllability 128
2.2.4 Stabilization of Controlled Systems 134
2.2.5 Applications 136
2.3 The Time-Discrete Non-Autonomous Case 144
2.3.1 The Problem of Fixed Point Controllability 144
2.3.2 The General Problem of Controllability 148
2.3.3 Stabilization of Controlled Systems 151
2.3.4 The Problem of Reachability 153
3 Chaotic Behavior of AutonomousTime-Discrete Systems 158
3.1 Chaos in the Sense of Devaney 158
3.2 Topological Conjugacy 163
3.3 The Topological Entropy as a Measure for Chaos 172
3.3.1 Definition and Invariance 172
3.3.2 The Topological Entropy of the Shift-Mapping 174
3.3.3 Disorder-Chaos for One-Dimensional Mappings 176
3.4 Chaos in the Sense of Li and Yorke 181
3.5 Strange (or Chaotic) Attractors 189
3.6 Bibliographical Remarks 199
A A Dynamical Method for the Calculation of Nash-Equilibria in n–Person Games 203
A.1 The Fixed Point Theorems of Brouwer andKakutani 203
A.2 Nash Equilibria as Fixed Points of Mappings 205
A.3 Bi-Matrix Games 210
A.4 Evolution Matrix Games 217
B Optimal Control in Chemotherapy of Cancer 224
B.1 The Mathematical Model and Two ControlProblems 224
B.2 Solution of the First Control Problem 227
B.3 Solution of the Second Control Problem 231
B.4 Pontryagin's Maximum Principle 236
C List of Authors 238
References 240
Index 242