Dynamical Systems: An Introduction with Applications in Economics and Biology - Brossura

Tu, Pierre N. V.

Brossura

ISBN 10: 3540576614 ISBN 13: 9783540576617

Casa editrice: Springer-Verlag, 1994

Riassunto

The favourable reception of the first edition and the encouragement received from many readers have prompted the author to bring out this new edition. This provides the opportunity for correcting a number of errors, typographical and others, contained in the first edition and making further improvements. This second edition has a new chapter on simplifying Dynamical Systems covering Poincare map, Floquet theory, Centre Manifold Theorems, normal forms of dynamical systems, elimination of passive coordinates and Liapunov-Schmidt reduction theory. It would provide a gradual transition to the study of Bifurcation, Chaos and Catastrophe in Chapter 10. Apart from this, most others - in fact all except the first three and last chapters - have been revised and enlarged to bring in some new materials, elaborate some others, especially those sections which many readers felt were rather too concise in the first edition, by providing more explana­ tion, examples and applications. Chapter 11 provides some good examples of this. Another example may be found in Chapter 4 where the review of Linear Algebra has been enlarged to incorporate further materials needed in this edition, for example the last section on idempotent matrices and projection would prove very useful to follow Liapunov-Schmidt reduction theory presented in Chapter 9.

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Recensione

"This book is the most useful recent review of differential equation systems that I have seen. The exposition is clear, explicit and accurate. It strikes an effective balance between rigor and readability, arguing proofs for the simpler cases, but citing publications for the more complex." The Quarterly Review of Biology

Contenuti

1 Introduction.- 2 Review of Ordinary Differential Equations.- 2.1 First Order Linear Differential Equations.- 2.1.1 First Order Constant Coefficient Linear Differential Equations.- 2.1.2 Variable Coefficient First Order Linear Differential Equations.- 2.1.3 Equations Reducible to Linear Differential Equations.- 2.1.4 Qualitative Solution: Phase Diagrams.- 2.1.5 Some Economic Applications.- 1. Walrasian Tâtonnement Process.- 2. The Keynesian Model.- 3. Harrod Domar’s Economic Growth Model.- 4. Domar’s Debt Model (1944).- 5. Profit and Investment.- 6. The Neo-Classical Model of Economic Growth.- 2.2 Second and Higher Order Linear Differential Equations.- 2.2.1 Particular Integral (xp or xe) where d(t) = d Constants.- 2.2.2 Particular Integral (xp) when d = g(t) is some Function of t.- 1. The Undetermined Coefficients Method.- 2. Inverse Operator Method.- 3. Laplace Transform Method.- 2.3 Higher Order Linear Differential Equations with Constant Coefficients.- 2.4 Stability Conditions.- 2.5 Some Economic Applications.- 1. The IS-LM Model of the Economy.- 2. A Continuous Multiplier-Acceleration Model.- 3. Stabilization Policies.- 4. Equilibrium Models with Stock.- 2.6 Conclusion.- 3 Review of Difference Equations.- 3.1 Introduction.- 3.2 First Order Difference Equations.- 3.2.1 Linear Difference Equations.- 3.2.2 Non-linear Difference Equations and Phase Diagram.- 3.2.3 Some Economic Applications.- 1. The Cobweb Cycle.- 2. The Dynamic Multiplier Model.- 3. The Overlapping Generations Model.- 3.3 Second Order Linear Difference Equations.- 3.3.1 Particular Integral.- 3.3.2 The Complementary Functions xc(t).- 3.3.3 Complete Solution and Examples.- 3.4 Higher Order Difference Equations.- 3.5 Stability Conditions.- 3.5.1 Stability of First Order Difference Equations.- 3.5.2 Stability of Second Order Difference Equations.- 3.5.3 Stability of Higher Order Difference Equations.- 3.6 Economic Applications.- 3.6.1 Samuelson’s (1939) Business Cycle.- 3.6.2 Hick’s (1950) Contribution to the Theory of Trade Cycle.- 3.7 Concluding Remarks.- 4 Review of Some Linear Algebra.- 4.1 Vector and Vector Spaces.- 4.1.1 Vector Spaces.- 4.1.2 Inner Product Space.- 4.1.3 Null Space and Range, Rank and Kernel.- 4.2 Matrices.- 4.2.1 Some Special Matrices.- 4.2.2 Matrix Operations.- 4.3 Determinant Functions.- 4.3.1 Properties of Determinants.- 4.3.2 Computations of Determinants.- 4.4 Matrix Inversion and Applications.- 4.5 Eigenvalues and Eigenvectors.- 4.5.1 Similar Matrices.- 4.5.2 Real Symmetric Matrices.- 4.6 Quadratic Forms.- 4.7 Diagonalization of Matrices.- 4.7.1 Real Eigenvalues.- 4.7.2 Complex Eigenvalues and Eigenvectors.- 4.8 Jordan Canonical Form.- 4.9 Idempotent Matrices and Projection.- 4.10 Conclusion.- 5 First Order Differential Equations Systems.- 5.1 Introduction.- 5.2 Constant Coefficient Linear Differential Equation (ODE) Systems.- 5.2.1 Case (i). Real and Distinct Eigenvalues.- 5.2.2 Case (ii). Repeated Eigenvalues.- 5.2.3 Case (iii). Complex Eigenvalues.- 5.3 Jordan Canonical Form of ODE Systems.- Case (i) Real Distinct Eigenvalues.- Case (ii) Multiple Eigenvalues.- Case (iii) Complex Eigenvalues.- 5.4 Alternative Methods for Solving ? = Ax.- 5.4.1 Sylvester’s Method.- 5.4.2 Putzer’s Methods (Putzer 1966).- 5.4.3 A Direct Method of Solving ? = Ax.- 5.5 Reduction to First Order of ODE Systems.- 5.6 Fundamental Matrix.- 5.7 Stability Conditions of ODE Systems.- 5.7.1 Asymptotic Stability.- 5.7.2 Global Stability: Liapunov’s Second Method.- 5.8 Qualitative Solution: Phase Portrait Diagrams.- 5.9 Some Economic Applications.- 5.9.1 Dynamic IS-LM Keynesian Model.- 5.9.2 Dynamic Leontief Input-Output Model.- 5.9.3 Multimarket Equilibrium.- 5.9.4 Walras-Cassel-Leontief General Equilibrium Model.- 6 First Order Difference Equations Systems.- 6.1 First Order Linear Systems.- 6.2 Jordan Canonical Form.- Case (i). Real Distinct Eigenvalues.- Case (ii). Multiple Eigenvalues.- Case (iii). Complex Eigenvalues.- 6.3 Reduction to First Order Systems.- 6.4 Stability Conditions.- 6.4.1 Local Stability.- 6.4.2 Global Stability.- 6.5 Qualitative Solutions: Phase Diagrams.- 6.6 Some Economic Applications.- 1. A Multisectoral Multiplier-Accelerator Model.- 2. Capital Stock Adjustment Model.- 3. Distributed Lags Model.- 4. Dynamic Input-Output Model.- 7 Nonlinear Systems.- 7.1 Introduction.- 7.2 Linearization Theory.- 7.2.1 Linearization of Dynamic Systems in the Plane.- 7.2.2 Linearization Theory in Three Dimensions.- 7.2.3 Linearization Theory in Higher Dimensions.- 7.3 Qualitative Solution: Phase Diagrams.- 7.4 Limit Cycles.- Economic Application I: Kaldor’s Trade Cycle Model.- 7.5 The Liénard-Van der Pol Equations and the Uniqueness of Limit Cycles.- Economic Application II: Kaldor’s Model as a Liénard Equation.- 7.6 Linear and Nonlinear Maps.- 7.7 Stability of Dynamical Systems.- 7.7.1 Asymptotic Stability.- 7.7.2 Structural Stability.- 7.8 Conclusion.- 8 Gradient Systems, Lagrangean and Hamiltonian Systems.- 8.1 Introduction.- 8.2 The Gradient Dynamic Systems (GDS).- 8.3 Lagrangean and Hamiltonian Systems.- 8.4 Hamiltonian Dynamics.- 8.4.1 Conservative Hamiltonian Dynamic Systems (CHDS).- 8.4.2 Perturbed Hamiltonian Dynamic Systems (PHDS).- 8.5 Economic Applications.- 8.5.1 Hamiltonian Dynamic Systems (HDS) in Economics.- 8.5.2 Gradient (GDS) vs Hamiltonian (HDS) Systems in Economics.- 8.5.3 Economic Applications: Two-State-Variables Optimal Economic Control Models.- 8.6 Conclusion.- 9 Simplifying Dynamical Systems.- 9.1 Introduction.- 9.2 Poincaré Map.- 9.3 Floquet Theory.- 9.4 Centre Manifold Theorem (CMT).- 9.5 Normal Forms.- 9.6 Elimination of Passive Coordinates.- 9.7 Liapunov-Schmidt Reduction.- 9.8 Economic Applications and Conclusions.- 10 Bifurcation, Chaos and Catastrophes in Dynamical Systems.- 10.1 Introduction.- 10.2 Bifurcation Theory (BT).- 10.2.1 One Dimensional Bifurcations.- 10.2.2 Hopf Bifurcation.- 10.2.3 Some Economic Applications.- 1. The Keynesian IS-LM Model.- 2. Hopf Bifurcation in an Advertising Model.- 3. A Dynamic Demand Supply Model.- 4. Generalized Tobin’s Model of Money and Economic Growth.- 10.2.4 Bifurcations in Discrete Dynamical Systems.- 1. The Fold of Saddle Node Bifurcation.- 2. Transcritical Bifurcation.- 3. Flip Bifurcation’.- 4. Logistic System.- 10.3 Chaotic or Complex Dynamical Systems (DS).- 10.3.1 Chaos in Unimodal Maps in Discrete Systems.- 10.3.2 Chaos in Higher Dimensional Discrete Systems.- 10.3.3 Chaos in Continuous Systems.- 10.3.4 Routes to Chaos.- 1. Period Doubling and Intermittency.- 2. Horseshoe and Homoclinic Orbits.- 10.3.5 Liapunov Characteristic Exponent (LCE) and Attractor’s Dimension.- 10.3.6 Some Economic Applications.- 1. Chaotic Dynamics in a Macroeconomic Model.- 2. Erratic Demand of the Rich.- 3. Structure and Stability of Inventory Cycles.- 4. Chaotic economic Growth with Pollution.- 5. Chaos in Business Cycles.- 10.4 Catastrophe Theory (C.T.).- 10.4.1 Some General Concepts.- 10.4.2 The Morse and Splitting Lemma.- 10.4.3 Codimension and Unfolding.- 10.4.4 Classification of Singularities.- 10.4.5 Some Elementary Catastrophes.- 1. The Fold Catastrophe.- 2. The Cusp Catastrophe.- 10.4.6 Some Economic Applications.- 1. The Shutdown of the Firm (Tu 1982).- 2. Kaldor’s Trade Cycle.- 3. A Catastrophe Theory of Defence Expenditure.- 4. Innovation, Industrial Evolution and Revolution.- 10.4.7 Comparative Statics (CS.), Singularities and Unfolding.- 10.5 Concluding Remarks.- 11 Optimal Dynamical Systems.- 11.1 Introduction.- 11.2 Pontryagin’s Maximum Principle.- 11.2.1 First Variations and Necessary Conditions.- 11.2.2 Second Variations and Sufficient Conditions.- 11.3 Stabilization Control Models.- 11.4 Some Economic Applications.- 1. Intergenerational Distribution of Non-renewable Resources.- 2. Optimal Harvesting of Renewable Resources.- 3. Multiplier-Accelerator Stabilization Model.- 4. Optimal Economic Growth (OEG).- 11.5 Asymptotic Stability of Optimal Dynamical Systems (ODS).- 11.6 Structural Stability of Optimal Dynamical Systems.- 11.6.1 Hopf Bifurcation in Optimal Economic Control Models and Optimal Limit Cycles.- Two-State-Variable Models.- Multisectoral OEG Models.- 11.6.2 Chaos in Optimal Dynamical Systems (ODS).- 11.7 Conclusion.- 12 Some Applications in Economics and Biology.- 12.1 Introduction.- 12.2 Economic Applications of Dynamical Systems.- 12.2.1 Business Cycles Theories.- 1. Linear Multiplier-Accelerator Models.- 2. Nonlinear Models.- 2.1. Flexible Multiplier-Accelerator Models.- 2.2. Kaldor’s Type of Flexible Accelerator Models.- 2.3. Goodwin’s Class Struggle Model.- 3. Optimal Economic Fluctuations and Chaos.- 12.2.2 General Equilibrium Dynamics.- Tâtonnement Adjustment Process.- Non-Tâtonnement Models.- 12.2.3 Economic Growth Theories.- 1. Harrod-Domar’s Models.- 2. Neo-Classical Models.- 2.1. Two Sector Models.- 2.2. Economic Growth with Money.- 2.3. Optimal Economic Growth Models.- 2.4. Endogenous Economic Growth Models.- 12.3 Dynamical Systems in Biology.- 12.3.1 One Species Growth Models.- 12.3.2 Two Species Models.- 1. Predation Models.- 2. Competition Models.- 12.3.3 The Dynamics of a Heartbeat.- 12.4 Bioeconomics and Natural Resources.- 12.4.1 Optimal Management of Renewable and Exhaustible Resources.- 12.4.2 Optimal Control of Prey-Predator Models.- (i) Control by an Ideal Pesticide.- (ii) Biological Control.- 12.5 Conclusion.

Dettagli dell?edizione

• Editore: Springer-Verlag
• Data di pubblicazione: 1994
• ISBN 10: 3540576614
• ISBN 13: 9783540576617
• Rilegatura: Copertina flessibile
• Lingua: Inglese
• Numero di pagine: 336