Nonlinearity, Chaos, and Complexity The Dynamics of Natural and Social Systems

Name: Nonlinearity, Chaos, and Complexity The Dynamics of Natural and Social Systems | The Dynamics of Natural and Social Systems
Brand: Oxford University Press
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The Dynamics of Natural and Social Systems

BERTUGLIA CRISTOFOR(Autor*in)

Oxford University Press

1. Auflage

Erschienen am 12. Mai 2005

404 Seiten

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978-0-19-152444-8 (ISBN)

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Inhalt

Intro
Contents
PART 1 Linear and Nonlinear Processes
1 Introduction
What we mean by 'system'
Physicalism: the first attempt to describe social systems using the methods of natural systems
2 Modelling
A brief introduction to modelling
Direct problems and inverse problems in modelling
The meaning and the value of models
3 The origins of system dynamics: mechanics
The classical interpretation of mechanics
The many-body problem and the limitations of classical mechanics
4 Linearity in models
5 One of the most basic natural systems: the pendulum
The linear model (Model 1)
The linear model of a pendulum in the presence of friction (Model 2)
Autonomous systems
6 Linearity as a first, often insufficient approximation
The linearization of problems
The limitations of linear models
7 The nonlinearity of natural processes: the case of the pendulum
The nonlinear pendulum (Model 3 without friction, and Model 3' with friction)
Non-integrability, in general, of nonlinear equations
8 Dynamical systems and the phase space
What we mean by dynamical system
The phase space
Oscillatory dynamics represented in the phase space
9 Extension of the concepts and models used in physics to economics
Jevons, Pareto and Fisher: from mathematical physics to mathematical economics
Schumpeter and Samuelson: the economic cycle
Dow and Elliott: periodicity in financial markets
10 The chaotic pendulum
The need for models of nonlinear oscillations
The case of a nonlinear forced pendulum with friction (Model 4)
11 Linear models in social processes: the case of two interacting populations
Introduction
The linear model of two interacting populations
Some qualitative aspects of linear model dynamics
The solutions of the linear model
Complex conjugate roots of the characteristic equation: the values of the two populations fluctuate
12 Nonlinear models in social processes: the model of Volterra-Lotka and some of its variants in ecology
Introduction
The basic model
A non-punctiform attractor: the limit cycle
Carrying capacity
Functional response and numerical response of the predator
13 Nonlinear models in social processes: the Volterra-Lotka model applied to urban and regional science
Introduction
Model of joint population-income dynamics
The population-income model applied to US cities and to Madrid
The symmetrical competition model and the formation of niches
PART 2 From Nonlinearity to Chaos
14 Introduction
15 Dynamical systems and chaos
Some theoretical aspects
Two examples: calculating linear and chaotic dynamics
The deterministic vision and real chaotic systems
The question of the stability of the solar system
16 Strange and chaotic attractors
Some preliminary concepts
Two examples: Lorenz and Rössler attractors
A two-dimensional chaotic map: the baker's map
17 Chaos in real systems and in mathematical models
18 Stability in dynamical systems
The concept of stability
A basic case: the stability of a linear dynamical system
Poincaré and Lyapunov stability criteria
Application of Lyapunov's criterion to Malthus' exponential law of growth
Quantifying a system's instability: the Lyapunov exponents
Exponential growth of the perturbations and the predictability horizon of a model
19 The problem of measuring chaos in real systems
Chaotic dynamics and stochastic dynamics
A method to obtain the dimension of attractors
An observation on determinism in economics
20 Logistic growth as a population development model
Introduction: modelling the growth of a population
Growth in the presence of limited resources: Verhulst equation
The logistic function
21 A nonlinear discrete model: the logistic map
Introduction
The iteration method and the fixed points of a function
The dynamics of the logistic map
22 The logistic map: some results of numerical simulations and an application
The Feigenbaum tree
An example of the application of the logistic map to spatial interaction models
23 Chaos in systems: the main concepts
PART 3 Complexity
24 Introduction
25 Inadequacy of reductionism
Models as portrayals of reality
Reductionism and linearity
A reflection on the role of mathematics in models
A reflection on mathematics as a tool for modelling
The search for regularities in social science phenomena
26 Some aspects of the classical vision of science
Determinism
The principle of sufficient reason
The classical vision in social sciences
Characteristics of systems described by classical science
27 From determinism to complexity: self-organization, a new understanding of system dynamics
Introduction
The new conceptions of complexity
Self-organization
28 What is complexity?
Adaptive complex systems
Basic aspects of complexity
An observation on complexity in social systems
Some attempts at defining a complex system
The complexity of a system and the observer
The complexity of a system and the relations between its parts
29 Complexity and evolution
Introduction
The three ways in which complexity grows according to Brian Arthur
The Tierra evolutionistic model
The appearance of life according to Kauffman
30 Complexity in economic processes
Complex economic systems
Synergetics
Two examples of complex models in economics
A model of the complex phenomenology of the financial markets
31 Some thoughts on the meaning of 'doing mathematics'
The problem of formalizing complexity
Mathematics as a useful tool to highlight and express recurrences
A reflection on the efficacy of mathematics as a tool to describe the world
32 Digression into the main interpretations of the foundations of mathematics
Introduction
Platonism
Formalism and 'les Bourbaki'
Constructivism
Experimental mathematics
The paradigm of the cosmic computer in the vision of experimental mathematics
A comparison between Platonism, formalism, and constructivism in mathematics
33 The need for a mathematics of (or for) complexity
The problem of formulating mathematical laws for complexity
The description of complexity linked to a better understanding of the concept of mathematical infinity: some reflections
References
Subject index
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Name index
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