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 -- A -- B -- C -- D -- E -- F -- H -- I -- J -- K -- L -- M -- N -- O -- P -- R -- S -- T -- U -- Name index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- X -- Y -- Z.