Positive Dynamical Systems in Discrete Time: Theory, Models, and Applications

✍ Scribed by Ulrich Krause

• Publisher: De Gruyter
• Year: 2015
• Tongue: English
• Leaves: 366
• Series: De Gruyter Studies in Mathematics; 62
• Category: Library

✦ Synopsis

This book provides a systematic, rigorous and self-contained treatment of positive dynamical systems. A dynamical system is positive when all relevant variables of a system are nonnegative in a natural way. This is in biology, demography or economics, where the levels of populations or prices of goods are positive. The principle also finds application in electrical engineering, physics and computer sciences.

"The author has greatly expanded the field of positive systems in surprising ways." - Prof. Dr. David G. Luenberger, Stanford University(USA)

✦ Table of Contents

Preface
Notation
List of Figures
1 How positive discrete dynamical systems do arise
1.1 Non-linear population dynamics in one dimension
Exercises
1.2 The density dependent Leslie model
Exercises
1.3 Non-linear price dynamics in one dimension
Exercises
1.4 The Leontief model with choice of techniques
Exercises
1.5 Opinion dynamics under bounded confidence
Exercises
Bibliography
2 Concave Perron–Frobenius theory
2.1 Iteration of normalized concave operators
Exercises
2.2 Indecomposability and primitivity for ray-preserving concave operators
Exercises
2.3 Concave operators which are positively homogeneous
Exercises
2.4 A special case: Linear Perron–Frobenius theory
Exercises
2.5 Applications to difference equations of concave type
Exercises
2.6 Relative stability in the concave Leslie model
Exercises
2.7 Price setting and balanced growth in a concave Leontief model
Exercises
Bibliography
3 Internal metrics on convex cones
3.1 Extraction within convex cones
Exercises
3.2 Internal metrics
Exercises
3.3 Geometrical properties
Exercises
3.4 Completeness for internal metrics
Exercises
Bibliography
4 Contractive dynamics on metric spaces
4.1 Iteration of contractive selfmappings
Exercises
4.2 Non-autonomous discrete systems
Exercises
4.3 A local-global stability principle for power-lipschitzian mappings
Exercises
Bibliography
5 Ascending dynamics in convex cones of infinite dimension
5.1 Definition and examples of ascending operators
Exercises
5.2 Relative stability for ascending operators by Hilbert’s projective metric
Exercises
5.3 Absolute stability for weakly ascending operators by the part metric
Exercises
5.4 Applications to nonlinear difference equations and to nonlinear integral operators
Exercises
Bibliography
6 Limit set trichotomy
6.1 Weak and strong forms of limit set trichotomy in Banach spaces
6.2 Differentiability criteria for non-expansiveness and contractivity
6.3 Applications to nonlinear difference equations and cooperative systems of differential equations
Exercises
Bibliography
7 Non-autonomous positive systems
7.1 The concepts of path stability, asymptotic proportionality, weak and strong ergodicity
7.2 Path stability and weak ergodicity for ascending operators
7.3 Strong ergodicity for ascending operators
7.4 A nonlinear version of Poincaré’s theorem on nonautonomous difference equations
7.5 Price setting in case of technical change
7.6 Populations under bounded and periodic enforcement
Exercises
Bibliography
8 Dynamics of interaction: opinions, mean maps, multi-agent coordination, and swarms
8.1 Scrambling matrices
8.2 Consensus formation and opinion dynamics under bounded confidence
8.3 Mean processes, mean structures and the iteration of mean maps
8.4 Infinite products of stochastic matrices: path stability, convergence and a generalized theorem of Wolfowitz
8.5 Multi-agent coordination and opinion dynamics
8.6 Swarm dynamics
Exercises
Bibliography
Index