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

, by Bertuglia, Cristoforo Sergio; Vaio, Franco

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ISBN: 9780198567912 | 019856791X
Cover: Paperback
Copyright: 7/28/2005

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Covering a broad range of topics, this text provides a comprehensive survey of the modelling of chaotic dynamics and complexity in the natural and social sciences. Its attention to models in both the physical and social sciences and the detailed philosophical approach make this an unique textin the midst of many current books on chaos and complexity.Part 1 deals with the mathematical model as an instrument of investigation. The general meaning of modelling and, more specifically, questions concerning linear modelling are discussed. Part 2 deals with the theme of chaos and the origin of chaotic dynamics. Part 3 deals with the theme ofcomplexity: a property of the systems and of their models which is intermediate between stability and chaos.Including an extensive index and bibliography along with numerous examples and simplified models, this is an ideal course text.

PART 1 Linear and Nonlinear Processes

(122)

1 Introduction

(7)

What we mean by 'system'

(2)

Physicalism: the first attempt to describe social systems using the methods of natural systems

(5)

2 Modelling

(9)

A brief introduction to modelling

(2)

Direct problems and inverse problems in modelling

(2)

The meaning and the value of models

(5)

3 The origins of system dynamics: mechanics

(8)

The classical interpretation of mechanics

(3)

The many-body problem and the limitations of classical mechanics

(5)

4 Linearity in models

(5)

5 One of the most basic natural systems: the pendulum

(7)

The linear model (Model 1)

(3)

The linear model of a pendulum in the presence of friction (Model 2)

(2)

Autonomous systems

(2)

6 Linearity as a first, often insufficient approximation

(7)

The linearization of problems

(3)

The limitations of linear models

(4)

7 The nonlinearity of natural processes: the case of the pendulum

(3)

The nonlinear pendulum (Model 3 without friction, and Model 3' with friction)

(1)

Non-integrability, in general, of nonlinear equations

(2)

8 Dynamical systems and the phase space

(11)

What we mean by dynamical system

(1)

The phase space

(4)

Oscillatory dynamics represented in the phase space

(6)

9 Extension of the concepts and models used in physics to economics

(7)

Jevons, Pareto and Fisher: from mathematical physics to mathematical economics

(2)

Schumpeter and Samuelson: the economic cycle

(2)

Dow and Elliott: periodicity in financial markets

(3)

10 The chaotic pendulum

(4)

The need for models of nonlinear oscillations

(1)

The case of a nonlinear forced pendulum with friction (Model 4)

(3)

11 Linear models in social processes: the case of two interacting populations

(22)

Introduction

(1)

The linear model of two interacting populations

(1)

Some qualitative aspects of linear model dynamics

(3)

The solutions of the linear model

(8)

Complex conjugate roots of the characteristic equation: the values of the two populations fluctuate

(9)

12 Nonlinear models in social processes: the model of Volterra-Lotka and some of its variants in ecology

(15)

Introduction

(1)

The basic model

(4)

A non-punctiform attractor: the limit cycle

(3)

Carrying capacity

101

(2)

Functional response and numerical response of the predator

103

(5)

13 Nonlinear models in social processes: the Volterra-Lotka model applied to urban and regional science

108

(15)

Introduction

108

(1)

Model of joint population-income dynamics

108

(5)

The population-income model applied to US cities and to Madrid

113

(5)

The symmetrical competition model and the formation of niches

118

(5)

PART 2 From Nonlinearity to Chaos

123

(114)

14 Introduction

125

(2)

15 Dynamical systems and chaos

127

(14)

Some theoretical aspects

127

(4)

Two examples: calculating linear and chaotic dynamics

131

(4)

The deterministic vision and real chaotic systems

135

(2)

The question of the stability of the solar system

137

(4)

16 Strange and chaotic attractors

141

(13)

Some preliminary concepts

141

(5)

Two examples: Lorenz and Rössler attractors

146

(4)

A two-dimensional chaotic map: the baker's map

150

(4)

17 Chaos in real systems and in mathematical models

154

(5)

18 Stability in dynamical systems

159

(20)

The concept of stability

159

(2)

A basic case: the stability of a linear dynamical system

161

(2)

Poincaré and Lyapunov stability criteria

163

(5)

Application of Lyapunov's criterion to Malthus' exponential law of growth

168

(3)

Quantifying a system's instability: the Lyapunov exponents

171

(5)

Exponential growth of the perturbations and the predictability horizon of a model

176

(3)

19 The problem of measuring chaos in real systems

179

(11)

Chaotic dynamics and stochastic dynamics

179

(4)

A method to obtain the dimension of attractors

183

(3)

An observation on determinism in economics

186

(4)

20 Logistic growth as a population development model

190

(9)

Introduction: modelling the growth of a population

190

(1)

Growth in the presence of limited resources: Verhulst equation

191

(3)

The logistic function

194

(5)

21 A nonlinear discrete model: the logistic map

199

(15)

Introduction

199

(2)

The iteration method and the fixed points of a function

201

(3)

The dynamics of the logistic map

204

(10)

22 The logistic map: some results of numerical simulations and an application

214

(17)

The Feigenbaum tree

214

(10)

An example of the application of the logistic map to spatial interaction models

224

(7)

23 Chaos in systems: the main concepts

231

(6)

PART 3 Complexity

237

(119)

24 Introduction

239

(1)

25 Inadequacy of reductionism

240

(13)

Models as portrayals of reality

240

(1)

Reductionism and linearity

241

(2)

A reflection on the role of mathematics in models

243

(3)

A reflection on mathematics as a tool for modelling

246

(3)

The search for regularities in social science phenomena

249

(4)

26 Some aspects of the classical vision of science

253

(13)

Determinism

253

(4)

The principle of sufficient reason

257

(2)

The classical vision in social sciences

259

(2)

Characteristics of systems described by classical science

261

(5)

27 From determinism to complexity: self-organization, a new understanding of system dynamics

266

(9)

Introduction

266

(2)

The new conceptions of complexity

268

(3)

Self-organization

271

(4)

28 What is complexity?

275

(16)

Adaptive complex systems

275

(2)

Basic aspects of complexity

277

(3)

An observation on complexity in social systems

280

(1)

Some attempts at defining a complex system

281

(4)

The complexity of a system and the observer

285

(1)

The complexity of a system and the relations between its parts

286

(5)

29 Complexity and evolution

291

(10)

Introduction

291

(1)

The three ways in which complexity grows according to Brian Arthur

291

(4)

The Tierra evolutionistic model

295

(2)

The appearance of life according to Kauffman

297

(4)

30 Complexity in economic processes

301

(14)

Complex economic systems

301

(3)

Synergetics

304

(3)

Two examples of complex models in economics

307

(2)

A model of the complex phenomenology of the financial markets

309

(6)

31 Some thoughts on the meaning of 'doing mathematics'

315

(14)

The problem of formalizing complexity

315

(5)

Mathematics as a useful tool to highlight and express recurrences

320

(3)

A reflection on the efficacy of mathematics as a tool to describe the world

323

(6)

32 Digression into the main interpretations of the foundations of mathematics

329

(19)

Introduction

329

(1)

Platonism

330

(1)

Formalism and 'les Bourbaki'

331

(5)

Constructivism

336

(4)

Experimental mathematics

340

(1)

The paradigm of the cosmic computer in the vision of experimental mathematics

341

(2)

A comparison between Platonism, formalism, and constructivism in mathematics

343

(5)

33 The need for a mathematics of (or for) complexity

348

(8)

The problem of formulating mathematical laws for complexity

348

(3)

The description of complexity linked to a better understanding of the concept of mathematical infinity: some reflections

351

(5)

References

356

(19)

Subject index

375

(5)

Name index

380

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

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Table of Contents

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