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

by Bertuglia, Cristoforo Sergio; Vaio, Franco

• ISBN13: 9780198567905
• ISBN10: 0198567901
• Format: Hardcover
• Copyright: 2005-08-04
• Publisher: Oxford University Press

Summary

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 philosophi

Table of Contents

PART 1 Linear and Nonlinear Processes	1	(122)
1 Introduction	3	(7)
What we mean by 'system'	3	(2)
Physicalism: the first attempt to describe social systems using the methods of natural systems	5	(5)
2 Modelling	10	(9)
A brief introduction to modelling	10	(2)
Direct problems and inverse problems in modelling	12	(2)
The meaning and the value of models	14	(5)
3 The origins of system dynamics: mechanics	19	(8)
The classical interpretation of mechanics	19	(3)
The many-body problem and the limitations of classical mechanics	22	(5)
4 Linearity in models	27	(5)
5 One of the most basic natural systems: the pendulum	32	(7)
The linear model (Model 1)	32	(3)
The linear model of a pendulum in the presence of friction (Model 2)	35	(2)
Autonomous systems	37	(2)
6 Linearity as a first, often insufficient approximation	39	(7)
The linearization of problems	39	(3)
The limitations of linear models	42	(4)
7 The nonlinearity of natural processes: the case of the pendulum	46	(3)
The nonlinear pendulum (Model 3 without friction, and Model 3' with friction)	46	(1)
Non-integrability, in general, of nonlinear equations	47	(2)
8 Dynamical systems and the phase space	49	(11)
What we mean by dynamical system	49	(1)
The phase space	50	(4)
Oscillatory dynamics represented in the phase space	54	(6)
9 Extension of the concepts and models used in physics to economics	60	(7)
Jevons, Pareto and Fisher: from mathematical physics to mathematical economics	60	(2)
Schumpeter and Samuelson: the economic cycle	62	(2)
Dow and Elliott: periodicity in financial markets	64	(3)
10 The chaotic pendulum	67	(4)
The need for models of nonlinear oscillations	67	(1)
The case of a nonlinear forced pendulum with friction (Model 4)	68	(3)
11 Linear models in social processes: the case of two interacting populations	71	(22)
Introduction	71	(1)
The linear model of two interacting populations	72	(1)
Some qualitative aspects of linear model dynamics	73	(3)
The solutions of the linear model	76	(8)
Complex conjugate roots of the characteristic equation: the values of the two populations fluctuate	84	(9)
12 Nonlinear models in social processes: the model of Volterra-Lotka and some of its variants in ecology	93	(15)
Introduction	93	(1)
The basic model	94	(4)
A non-punctiform attractor: the limit cycle	98	(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 Rossler 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