Dynamical Systems. Pierre N.V. Tu. An Introduction with Applications in Economics and Biology Second Revised and Enlarged Edition.

Transcription

1 Pierre N.V. Tu Dynamical Systems An Introduction with Applications in Economics and Biology Second Revised and Enlarged Edition With 105 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

2 Contents Preface v 1 Introduction 1 2 Review of Ordinary Differential Equations First Order Linear Differential Equations First Order Constant Coefficient Linear Differential Equations Variable Coefficient First Order Linear Differential Equations Equations Reducible to Linear Differential Equations Qualitative Solution: Phase Diagrams Some Economic Applications Walrasian Tätonnement Process The Keynesian Model Harrod Domar's Economic Growth Model Domar's Debt Model (1944) Profit and Investment The Neo-Classical Model of Economic Growth Second and Higher Order Linear Differential Equations Particular Integral (x p or x e ) where d(i) = d Constants Particular Integral (x p ) when d = g(t) is some Function of t The Undetermined Coefficients Method Inverse Operator Method Laplace Transform Method Higher Order Linear Differential Equations with Constant Coefficients Stability Conditions Some Economic Applications The IS-LM Model of the Economy A Continuous Multiplier-Acceleration Model Stabilization Policies Equilibrium Models with Stock Conclusion 38 3 Review of Difference Equations Introduction First Order Difference Equations Linear Difference Equations 40

3 X Non-linear DiflFerence Equations and Phase Diagram Some Economic Applications The Cobweb Cycle The Dynamic Multiplier Model The Overlapping Generations Model Second Order Linear DifFerence Equations Particular Integral The Complementary Functions x c (t) Complete Solution and Examples Higher Order DiflFerence Equations Stability Conditions Stability of First Order DifFerence Equations Stability of Second Order DifFerence Equations Stability of Higher Order DifFerence Equations Economic Applications Samuelson's (1939) Business Cycle Hick's (1950) Contribution to the Theory of Trade Cycle Concluding Remarks 57 4 Review of Some Linear Algebra Vector and Vector Spaces Vector Spaces Inner Product Space Null Space and Range, Rank and Kernel Matrices Some Special Matrices Matrix Operations Determinant Functions Properties of Determinants : Computations of Determinants Matrix Inversion and Applications Eigenvalues and Eigenvectors Similar Matrices Real Symmetrie Matrices Quadratic Forms Diagonalization of Matrices Real Eigenvalues Complex Eigenvalues and Eigenvectors Jordan Canonical Form Idempotent Matrices and Projection Conclusion 82

4 XI 5 First Order Differential Equations Systems Introduction Constant Coefficient Linear Differential Equation (ODE) Systems Case (i). Real and Distinct Eigenvalues Case (ii). Repeated Eigenvalues Case (iii). Complex Eigenvalues Jordan Canonical Form of ODE Systems 89 Case (i) Real Distinct Eigenvalues 90 Case (ii) Multiple Eigenvalues 91 Case (iii) Complex Eigenvalues Alternative Methods for Solving x = Ax Sylvester's Method Putzer's Methods (Putzer 1966) A Direct Method of Solving x = Ax Reduction to First Order of ODE Systems Fundamental Matrix Stability Conditions of ODE Systems Asymptotic Stability Global Stability: Liapunov's Second Method Qualitative Solution: Phase Portrait Diagrams Some Economic Applications Dynamic IS-LM Keynesian Model Dynamic Leontief Input-Output Model Multimarket Equilibrium Walras-Cassel-Leontief General Equilibrium Model First Order Difference Equations Systems First Order Linear Systems Jordan Canonical Form 117 Case (i). Real Distinct Eigenvalues 118 Case (ii). Multiple Eigenvalues 119 Case (iii). Complex Eigenvalues Reduction to First Order Systems Stability Conditions Local Stability Global Stability Qualitative Solutions: Phase Diagrams Some Economic Applications A Multisectoral Multiplier-Accelerator Model Capital Stock Adjustment Model Distributed Lags Model Dynamic Input-Output Model 130

5 Xll 7 Nonlinear Systems Introduction Linearization Theory Linearization of Dynamic Systems in the Plane Linearization Theory in Three Dimensions Linearization Theory in Higher Dimensions Qualitative Solution: Phase Diagrams Limit Cycles 149 Economic Application I: Kaldor's Trade Cycle Model The Lienard-Van der Pol Equations and the Uniqueness of Limit Cycles 154 Economic Application II: Kaldor's Model as a Lienard Equation Linear and Nonlinear Maps Stability of Dynamical Systems Asymptotic Stability Structural Stability Conclusion Gradient Systems, Lagrangean and Hamiltonian Systems Introduction The Gradient Dynamic Systems (GDS) Lagrangean and Hamiltonian Systems Hamiltonian Dynamics Conservative Hamiltonian Dynamic Systems (CHDS) Perturbed Hamiltonian Dynamic Systems (PHDS) Economic Applications Hamiltonian Dynamic Systems (HDS) in Economics Gradient (GDS) vs Hamiltonian (HDS) Systems in Economics Economic Applications: Two-State-Variables Optimal Economic Control Models Conclusion Simplifying Dynamical Systems Introduction Poincare Map Floquet Theory Centre Manifold Theorem (CMT) Normal Forms Elimination of Passive Coordinates Liapunov-Schmidt Reduction Economic Applications and Conclusions Bifurcation, Chaos and Catastrophes in Dynamical Systems Introduction Bifurcation Theory (BT) 195

6 One Dimensional Bifurcations Hopf Bifurcation Some Economic Applications The Keynesian IS-LM Model Hopf Bifurcation in an Advertising Model A Dynamic Demand Supply Model Generalized Tobin's Model of Money and Economic Growth Bifurcations in Discrete Dynamical Systems The Fold of Saddle Node Bifurcation Transcritical Bifurcation Flip Bifurcation Logistic System Chaotic or Complex Dynamical Systems (DS) Chaos in Unimodal Maps in Discrete Systems Chaos in Higher Dimensional Discrete Systems Chaos in Continuous Systems Routes to Chaos Period Doubling and Intermittency Horseshoe and Homoclinic Orbits Liapunov Characteristic Exponent (LCE) and Attractor's Dimension Some Economic Applications Chaotic Dynamics in a Macroeconomic Model Erratic Demand of the Rieh Structure and Stability of Inventory Cycles Chaotic economic Growth with Pollution Chaos in Business Cycles Catastrophe Theory (CT.) Some General Concepts The Morse and Splitting Lemma Codimension and Unfolding Classification of Singularities Some Elementary Catastrophes The Fold Catastrophe The Cusp Catastrophe Some Economic Applications The Shutdown of the Firm (Tu 1982) Kaldor's Trade Cycle A Catastrophe Theory of Defence Expenditure Innovation, Industrial Evolution and Revolution Comparative Statics (CS.), Singularities and Unfolding Concluding Remarks 243 Xlll

7 XIV 11 Optimal Dynamical Systems Introduction Pontryagin's Maximum Principle First Variations and Necessary Conditions Second Variations and Sufficient Conditions Stabilization Control Models Some Economic Applications Intergenerational Distribution of Non-renewable Resources Optimal Harvesting of Renewable Resources Multiplier-Accelerator Stabilization Model Optimal Economic Growth (OEG) Asymptotic Stability of Optimal Dynamical Systems (ODS) Structural Stability of Optimal Dynamical Systems Hopf Bifurcation in Optimal Economic Control Models and Optimal Limit Cycles 263 Two-State-Variable Models 264 Multisectoral OEG Models Chaos in Optimal Dynamical Systems (ODS) Conclusion Some Applications in Economics and Biology Introduction Economic Applications of Dynamical Systems Business Cycles Theories Linear Multiplier-Accelerator Models Nonlinear Models Flexible Multiplier-Accelerator Models Kaldor's Type of Flexible Accelerator Models Goodwin's Class Struggle Model Optimal Economic Fluctuations and Chaos General Equilibrium Dynamics 276 Tätonnement Adjustment Process 277 Non-Tätonnement Models Economic Growth Theories Harrod-Domar's Models Neo-Classical Models Two Sector Models Economic Growth with Money Optimal Economic Growth Models Endogenous Economic Growth Models Dynamical Systems in Biology One Species Growth Models Two Species Models Predation Models Competition Models The Dynamics of a Heartbeat 288

8 XV 12.4 Bioeconomics and Natural Resources Optimal Management of Renewable and Exhaustible Resources Optimal Control of Prey-Predator Models 292 (i) Control by an Ideal Pesticide 292 (ii) Biological Control Conclusion 294 Bibliography 295