Covering a broad range of topics, this text provides a comprehensive survey of the modeling 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 a unique text in the midst of many current books on chaos and complexity. Including an extensive index and bibliography along with numerous examples and simplified models, this is an ideal course text.
Author(s): Cristoforo Sergio Bertuglia, Franco Vaio
Publisher: Oxford University Press, USA
Year: 2005
Language: English
Pages: 404
Contents......Page 12
PART 1 Linear and Nonlinear Processes......Page 18
What we mean by ‘system’......Page 20
Physicalism: the first attempt to describe social systems using the methods of natural systems......Page 22
A brief introduction to modelling......Page 27
Direct problems and inverse problems in modelling......Page 29
The meaning and the value of models......Page 31
The classical interpretation of mechanics......Page 36
The many-body problem and the limitations of classical mechanics......Page 39
4 Linearity in models......Page 44
The linear model (Model 1)......Page 49
The linear model of a pendulum in the presence of friction (Model 2)......Page 52
Autonomous systems......Page 54
The linearization of problems......Page 56
The limitations of linear models......Page 59
The nonlinear pendulum (Model 3 without friction, and Model 3’ with friction)......Page 63
Non-integrability, in general, of nonlinear equations......Page 64
What we mean by dynamical system......Page 66
The phase space......Page 67
Oscillatory dynamics represented in the phase space......Page 71
Jevons, Pareto and Fisher: from mathematical physics to mathematical economics......Page 77
Schumpeter and Samuelson: the economic cycle......Page 79
Dow and Elliott: periodicity in financial markets......Page 81
The need for models of nonlinear oscillations......Page 84
The case of a nonlinear forced pendulum with friction (Model 4)......Page 85
Introduction......Page 88
The linear model of two interacting populations......Page 89
Some qualitative aspects of linear model dynamics......Page 90
The solutions of the linear model......Page 93
Complex conjugate roots of the characteristic equation: the values of the two populations fluctuate......Page 101
Introduction......Page 110
The basic model......Page 111
A non-punctiform attractor: the limit cycle......Page 115
Carrying capacity......Page 118
Functional response and numerical response of the predator......Page 120
Introduction......Page 125
Model of joint population–income dynamics......Page 126
The population–income model applied to US cities and to Madrid......Page 130
The symmetrical competition model and the formation of niches......Page 135
PART 2 From Nonlinearity to Chaos......Page 140
14 Introduction......Page 142
Some theoretical aspects......Page 144
Two examples: calculating linear and chaotic dynamics......Page 148
The deterministic vision and real chaotic systems......Page 152
The question of the stability of the solar system......Page 154
Some preliminary concepts......Page 158
Two examples: Lorenz and Rössler attractors......Page 163
A two-dimensional chaotic map: the baker’s map......Page 167
17 Chaos in real systems and in mathematical models......Page 171
The concept of stability......Page 176
A basic case: the stability of a linear dynamical system......Page 178
Poincaré and Lyapunov stability criteria......Page 180
Application of Lyapunov’s criterion to Malthus’ exponential law of growth......Page 185
Quantifying a system’s instability: the Lyapunov exponents......Page 188
Exponential growth of the perturbations and the predictability horizon of a model......Page 193
Chaotic dynamics and stochastic dynamics......Page 196
A method to obtain the dimension of attractors......Page 200
An observation on determinism in economics......Page 203
Introduction: modelling the growth of a population......Page 207
Growth in the presence of limited resources: Verhulst equation......Page 208
The logistic function......Page 211
Introduction......Page 216
The iteration method and the fixed points of a function......Page 218
The dynamics of the logistic map......Page 221
The Feigenbaum tree......Page 231
An example of the application of the logistic map to spatial interaction models......Page 241
23 Chaos in systems: the main concepts......Page 248
PART 3 Complexity......Page 254
24 Introduction......Page 256
Models as portrayals of reality......Page 257
Reductionism and linearity......Page 258
A reflection on the role of mathematics in models......Page 260
A reflection on mathematics as a tool for modelling......Page 263
The search for regularities in social science phenomena......Page 266
Determinism......Page 270
The principle of sufficient reason......Page 274
The classical vision in social sciences......Page 276
Characteristics of systems described by classical science......Page 278
Introduction......Page 283
The new conceptions of complexity......Page 285
Self-organization......Page 288
Adaptive complex systems......Page 292
Basic aspects of complexity......Page 294
An observation on complexity in social systems......Page 297
Some attempts at defining a complex system......Page 298
The complexity of a system and the observer......Page 302
The complexity of a system and the relations between its parts......Page 303
The three ways in which complexity grows according to Brian Arthur......Page 308
The Tierra evolutionistic model......Page 312
The appearance of life according to Kauffman......Page 314
Complex economic systems......Page 318
Synergetics......Page 321
Two examples of complex models in economics......Page 324
A model of the complex phenomenology of the financial markets......Page 326
The problem of formalizing complexity......Page 332
Mathematics as a useful tool to highlight and express recurrences......Page 337
A reflection on the efficacy of mathematics as a tool to describe the world......Page 340
Introduction......Page 346
Platonism......Page 347
Formalism and ‘les Bourbaki’......Page 348
Constructivism......Page 353
Experimental mathematics......Page 357
The paradigm of the cosmic computer in the vision of experimental mathematics......Page 358
A comparison between Platonism, formalism, and constructivism in mathematics......Page 360
The problem of formulating mathematical laws for complexity......Page 365
The description of complexity linked to a better understanding of the concept of mathematical infinity: some reflections......Page 368
References......Page 373
E......Page 392
L......Page 393
M......Page 394
P......Page 395
U......Page 396
B......Page 397
D......Page 398
G......Page 399
K......Page 400
N......Page 401
R......Page 402
S......Page 403
Z......Page 404
