This book inquires into nonlinear dynamics in mathematical economic modelling. The formation of spatial patterns, like market areas, trade flows, and settlement structures are studied. The basic approach is to study the qualitative features of the structurally stable pattern by means of the generic theory of differential equations, as well as their transformations by means of catastrophe theory. The main part dwells on business cycle theory, using a family of non-linear multiplier accelerator models put in spatial setting by inter-regional trade. The models are capable of producing sustained…mehr
This book inquires into nonlinear dynamics in mathematical economic modelling. The formation of spatial patterns, like market areas, trade flows, and settlement structures are studied. The basic approach is to study the qualitative features of the structurally stable pattern by means of the generic theory of differential equations, as well as their transformations by means of catastrophe theory. The main part dwells on business cycle theory, using a family of non-linear multiplier accelerator models put in spatial setting by inter-regional trade. The models are capable of producing sustained frequency locked oscillation, quasi-periodicity, and chaos. A mixture of analytical methods, such as perturbation analysis, and heuristic simulations is used. The third edition has been thoroughly revised and considerably expanded, in particular with reference to the study of chaotic behaviour.
1 Introduction.- 1.1 Dynamics Versus Equilibrium Analysis.- 1.2 Linear Versus Nonlinear Modelling.- 1.3 Perturbation Methods.- 1.4 Structural Stability.- 1.5 Chaos and Fractals.- 1.6 The Choice of Topics Included.- 2 Differential Equations.- 2.1 The Phase Portrait.- 2.2 Linear Systems.- 2.3 Structural Stability.- 2.4 Limit Cycles.- 2.5 The Hopf Bifurcation.- 2.5 The Saddle-Node Bifurcation.- 2.7 Perturbation Methods: Poincaré-Lindstedt.- 2.8 Perturbation Methods: Two-Timing.- 2.9 Forced Oscillators: van der Pol.- 2.10 Forced Oscillators: Duffing.- 2.11 Chaos.- 2.12 A Short History of Chaos.- 3 Iterated Maps.- 3.1 Introduction.- 3.2 The Logistic Map.- 3.3 The Lyapunov Exponent.- 3.4 Symbolic Dynamics.- 3.5 Sarkovskii's Theorem and the Schwarzian Derivative.- 3.6 The Hénon Model.- 3.7 Lyapunov Exponents in 2D.- 3.8 Fractals and Fractal Dimension.- 3.9 The Mandelbrot Set.- 4 Monopoly.- 4.1 Introduction.- 4.2 The Model.- 4.3 Adaptive Search.- 4.4 Numerical Results.- 4.5 Fixed Points and Cycles.- 4.6 Chaos.- 4.7 A Case with Two Products.- 4.8 Discussion.- 5 Duopoly and Oligopoly.- 5.1 Introduction.- 5.2 The Cournot Model.- 5.3 Stackelberg Equilibria.- 5.4 The Iterative Process.- 5.5 Stability of the Cournot Point.- 5.6 Periodic Points and Chaos.- 5.7 Adaptive Expectations.- 5.8 Adjustments Including Stackelberg Points.- 5.9 Oligopoly with Three Firms.- 5.10 Stackelberg Action Reconsidered.- 5.11 The Iteration with Three Oligopolists.- 5.12 Back to "Duopoly".- 5.13 Changing the Order of Adjustment.- 6 Business Cycles: Continuous Time.- 6.1 The Multiplier-Accelerator Model.- 6.2 The Original Model.- 6.3 Nonlinear Investment Functions and Limit Cycles.- 6.4 Limit Cycles: Existence.- 6.5 Limit Cycles: Asymptotic Approximation.- 6.6 Limit Cycles: Transients andStability.- 6.7 The Two-Region Model.- 6.8 The Persistence of Cycles.- 6.9 Perturbation Analysis of the Coupled Model.- 6.10 The Unstable Zero Equilibrium.- 6.11 Other Fixed Points.- 6.12 Properties of Fixed Points.- 6.13 The Arbitrary Phase Angle.- 6.14 Stability of the Coupled Oscillators.- 6.15 The Forced Oscillator.- 6.16 The World Market.- 6.17 The Small Open Economy.- 6.18 Stability of the Forced Oscillator.- 6.19 Catastrophe.- 6.20 Period Doubling and Chaos.- 6.21 Relaxation Cycles.- 6.22 Relaxation: The Autonomous Case.- 6.23 Relaxation: The Forced Case.- 6.24 Three Identical Regions.- 6.25 On the Existence of Periodic Solutions.- 6.26 Stability of Three Oscillators.- 6.27 Simulations.- 7 Business Cycles: Continuous Space.- 7.1 Introduction.- 7.2 Interregional Trade.- 7.3 The Linear Model.- 7.4 Coordinate Separation.- 7.5 The Square Region.- 7.6 The Circular Region.- 7.7 The Spherical Region.- 7.8 The Nonlinear Spatial Model.- 7.9 Dispersive Waves.- 7.10 Standing Waves.- 8 Business Cycles: Discrete Time.- 8.1 Introduction.- 8.2 Investments.- 8.3 Consumption.- 8.4 The Cubic Iterative Map.- 8.5 Fixed Points, Cycles, and Chaos.- 8.6 Formal Analysis of Chaotic Dynamics.- 8.7 Coordinate Transformation.- 8.8 The Three Requisites of Chaos.- 8.9 Symbolic Dynamics.- 8.10 Brownian Random Walk.- 8.11 Digression on Order and Disorder.- 8.12 The General Model.- 8.13 Relaxation Cycles.- 8.14 The Slow Feed Back.- 8.15 The Autonomous Term: Changes of Fixed Points.- 8.16 The Autonomous Term: Response of the Chaotic Process.- 8.17 Lyapunov Exponents and Fractal Dimensions.- 8.18 Non-Relaxation Cycles.- 8.19 Conclusion.- References.
1 Introduction.- 1.1 Dynamics Versus Equilibrium Analysis.- 1.2 Linear Versus Nonlinear Modelling.- 1.3 Perturbation Methods.- 1.4 Structural Stability.- 1.5 Chaos and Fractals.- 1.6 The Choice of Topics Included.- 2 Differential Equations.- 2.1 The Phase Portrait.- 2.2 Linear Systems.- 2.3 Structural Stability.- 2.4 Limit Cycles.- 2.5 The Hopf Bifurcation.- 2.5 The Saddle-Node Bifurcation.- 2.7 Perturbation Methods: Poincaré-Lindstedt.- 2.8 Perturbation Methods: Two-Timing.- 2.9 Forced Oscillators: van der Pol.- 2.10 Forced Oscillators: Duffing.- 2.11 Chaos.- 2.12 A Short History of Chaos.- 3 Iterated Maps.- 3.1 Introduction.- 3.2 The Logistic Map.- 3.3 The Lyapunov Exponent.- 3.4 Symbolic Dynamics.- 3.5 Sarkovskii's Theorem and the Schwarzian Derivative.- 3.6 The Hénon Model.- 3.7 Lyapunov Exponents in 2D.- 3.8 Fractals and Fractal Dimension.- 3.9 The Mandelbrot Set.- 4 Monopoly.- 4.1 Introduction.- 4.2 The Model.- 4.3 Adaptive Search.- 4.4 Numerical Results.- 4.5 Fixed Points and Cycles.- 4.6 Chaos.- 4.7 A Case with Two Products.- 4.8 Discussion.- 5 Duopoly and Oligopoly.- 5.1 Introduction.- 5.2 The Cournot Model.- 5.3 Stackelberg Equilibria.- 5.4 The Iterative Process.- 5.5 Stability of the Cournot Point.- 5.6 Periodic Points and Chaos.- 5.7 Adaptive Expectations.- 5.8 Adjustments Including Stackelberg Points.- 5.9 Oligopoly with Three Firms.- 5.10 Stackelberg Action Reconsidered.- 5.11 The Iteration with Three Oligopolists.- 5.12 Back to "Duopoly".- 5.13 Changing the Order of Adjustment.- 6 Business Cycles: Continuous Time.- 6.1 The Multiplier-Accelerator Model.- 6.2 The Original Model.- 6.3 Nonlinear Investment Functions and Limit Cycles.- 6.4 Limit Cycles: Existence.- 6.5 Limit Cycles: Asymptotic Approximation.- 6.6 Limit Cycles: Transients andStability.- 6.7 The Two-Region Model.- 6.8 The Persistence of Cycles.- 6.9 Perturbation Analysis of the Coupled Model.- 6.10 The Unstable Zero Equilibrium.- 6.11 Other Fixed Points.- 6.12 Properties of Fixed Points.- 6.13 The Arbitrary Phase Angle.- 6.14 Stability of the Coupled Oscillators.- 6.15 The Forced Oscillator.- 6.16 The World Market.- 6.17 The Small Open Economy.- 6.18 Stability of the Forced Oscillator.- 6.19 Catastrophe.- 6.20 Period Doubling and Chaos.- 6.21 Relaxation Cycles.- 6.22 Relaxation: The Autonomous Case.- 6.23 Relaxation: The Forced Case.- 6.24 Three Identical Regions.- 6.25 On the Existence of Periodic Solutions.- 6.26 Stability of Three Oscillators.- 6.27 Simulations.- 7 Business Cycles: Continuous Space.- 7.1 Introduction.- 7.2 Interregional Trade.- 7.3 The Linear Model.- 7.4 Coordinate Separation.- 7.5 The Square Region.- 7.6 The Circular Region.- 7.7 The Spherical Region.- 7.8 The Nonlinear Spatial Model.- 7.9 Dispersive Waves.- 7.10 Standing Waves.- 8 Business Cycles: Discrete Time.- 8.1 Introduction.- 8.2 Investments.- 8.3 Consumption.- 8.4 The Cubic Iterative Map.- 8.5 Fixed Points, Cycles, and Chaos.- 8.6 Formal Analysis of Chaotic Dynamics.- 8.7 Coordinate Transformation.- 8.8 The Three Requisites of Chaos.- 8.9 Symbolic Dynamics.- 8.10 Brownian Random Walk.- 8.11 Digression on Order and Disorder.- 8.12 The General Model.- 8.13 Relaxation Cycles.- 8.14 The Slow Feed Back.- 8.15 The Autonomous Term: Changes of Fixed Points.- 8.16 The Autonomous Term: Response of the Chaotic Process.- 8.17 Lyapunov Exponents and Fractal Dimensions.- 8.18 Non-Relaxation Cycles.- 8.19 Conclusion.- References.
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