Nonlinear Economic Dynamics

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1 Introduction1.1 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 and Stability -- 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.

10.1007/978-3-642-60775-2 [DOI]

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