Nonlinear Dynamical Economics and Chaotic Motion

ISBN-13: 9783642783265 / Angielski / Miękka / 2011 / 320 str.

Hans-Walter Lorenz

Opis

Usually, the first edition of a book still contains a multiplicity of typographic, con ceptional, and computational errors even if one believes the opposite at the time of publication. As this book did not represent a counterexample to this rule, the current second edition offers a chance to remove at least the known shortcomings. The book has been partly re-organized. The previously rather long Chapter 4 has been split into two separate chapters dealing with discrete-time and continuous time approaches to nonlinear economic dynamics. The short summary of basic properties of linear dynamical systems has been banned to an appendix because the line of thought in the chapter seems to have been unnecessarily interrupted by these technical details and because the book concentrates on nonlinear systems. This appendix, which mainly deals with special formal properties of dynamical sys tems, also contains some new material on invariant subspaces and center-manifold reductions. A brief introduction into the theory of lags and operators is followed by a few remarks on the relation between the 'true' properties of dynamical systems and their behavior observable in numerical experiments. Additional changes in the main part of the book include a re-consideration of Popper's determinism vs. inde terminism discussion in the light of chaotic properties of deterministic, nonlinear systems in Chapter 1. An investigation of a simultaneous price-quantity adjustment process, a more detailed inquiry into the uniqueness property of limit cycles, and a short presentation of relaxation oscillations are included in Chapter 2."

Spis treści

1. Economic Dynamics, Linearities, and the Classical Mechanistic Worldview.- 1.1. Some Reflections on the Origin of Economic Dynamics.- 1.2. The Deterministic Worldview and Deterministic Theories.- 1.3. The Dominance of Linear Dynamical Systems in Economics.- 2. Nonlinearities and Economic Dynamics.- 2.1. Preliminary Concepts.- 2.2. The Poincaré-Bendixson Theorem.- 2.2.1. The Existence of Limit Cycles.- 2.2.2. The Kaldor Model as a Prototype Model in Nonlinear Economic Dynamics.- 2.2.3. A Classical Cross-Dual Adjustment Process.- 2.3. The Uniqueness of Limit Cycles.- 2.3.1. The Liénard Equation and Related Tools.- 2.3.2. The Symmetric Case: Unique Cycles in a Modified Phillips Model.- 2.3.3. The Asymmetric Case: Unique Cycles in a Kaldor Model.- 2.4. Predator-Prey Models.- 2.4.1. The Dynamics of Conservative Dynamical Systems.- 2.4.2. Goodwin’s Predator-Prey Model of the Class Struggle.- 2.4.3. Other Examples and Predator-Prey Structures in Dissipative Systems.- 2.5. Relaxation Oscillations.- 2.6. Irreversibility and Determinism in Dynamical Systems.- 3. Bifurcation Theory and Economic Dynamics.- 3.1. Preliminaries and Different Concepts of Structural Stability.- 3.2. Local Bifurcations in Continuous-Time Dynamical Systems.- 3.2.1. Fold, Transcritical, and Pitchfork Bifurcations.- 3.2.2. The Hopf Bifurcation in Continuous-Time Dynamical Systems.- 3.2.2.1. The Hopf Bifurcation in Business-Cycle Theory.- 3.2.2.2. Closed Orbits in Optimal Economic Growth.- 3.3. Local Bifurcations in Discrete-Time Dynamical Systems.- 3.3.1. Fold, Transcritical, Pitchfork, and Flip Bifurcations.- 3.3.2. The Hopf Bifurcation in Discrete-Time Dynamical Systems.- 4. Chaotic Dynamics in Discrete-Time Economic Models.- 4.1. Chaos in One-Dimensional, Discrete-Time Dynamical Systems.- 4.1.1. Basic Concepts.- 4.1.2. Chaos in Descriptive Growth Theory.- 4.1.3. Chaos in Discrete-Time Models of Optimal Economic Growth.- 4.1.4. Other Economic Examples.- 4.2. Chaos in Higher-Dimensional Discrete-Time Systems.- 4.2.1. Some Basic Ideas.- 4.2.2. An Economic Example.- 4.3. Complex Transients in Discrete-Time Dynamical Systems.- 4.3.1. Complex Transient Behavior in One-Dimensional Systems.- 4.3.2. Horseshoes, Homoclinic Orbits, and Complicated Invariant Sets.- 5. Chaotic Dynamics in Continuous-Time Economic Models.- 5.1. Basic Ideas.- 5.2. The Coupling of Oscillators.- 5.2.1. Toroidal Motion.- 5.2.2. International Trade as the Coupling of Oscillators.- 5.3. The Forced Oscillator.- 5.3.1. Forced Oscillator Systems and Chaotic Motion.- 5.3.2. Goodwins’s Nonlinear Accelerator as a Forced Oscillator.- 5.3.3. Keynesian Demand Policy as the Source of Chaotic Motion.- 5.3.4. Conclusion.- 5.4. Homoclinic Orbits and Spiral-Type Attractors.- 5.4.1. The Shil’nikov Scenario.- 5.4.2. Spiral-Type Chaos in a Business Cycle Model with Inventories.- 6. Numerical Tools.- 6.1. Spectral Analysis.- 6.2. Dimension, Entropy, and Lyapunov Exponents.- 6.2.1. Phase Space Embedding.- 6.2.2. Fractal Dimensions.- 6.2.3. Correlation Dimension.- 6.2.4. Lyapunov Exponents.- 6.2.5. Kolmogorov Entropy.- 6.2.6. Summary.- 6.3. Are Economic Time Series Chaotic?.- 6.4. Predictability in the Face of Chaotic Dynamics.- 7. Catastrophe Theory and Economic Dynamics.- 7.1. Basic Ideas.- 7.2. The Kaldor Model in the Light of Catastrophe Theory.- 7.3. A Catastrophe-Theoretical Approach to Stagflation.- 8. Concluding Remarks.- A.1. Basic Properties of Linear Dynamical Systems.- A.2. Center Manifolds and the Reduction of (Effective) Dimensions.- A.3. A Brief Introduction to the Theory of Lags and Operators.- A.4. Numerical Simulations and Chaotic Dynamics in Theoretical Economics.- References.- Name Index.