Pierre N. V. Tu

Dynamical Systems (eBook, PDF)

An Introduction with Applications in Economics and Biology

• Format: PDF

Produktbeschreibung

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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Produktdetails

• Verlag: Springer Berlin Heidelberg
• Seitenzahl: 314
• Erscheinungstermin: 6. Dezember 2012
• Englisch
• ISBN-13: 9783642787935
• Artikelnr.: 53134919

Inhaltsangabe

1 Introduction.- 2 Review of Ordinary Differential Equations.- 2.1 First Order Linear Differential Equations.- 2.2 Second and Higher Order Linear Differential Equations.- 2.3 Higher Order Linear Differential Equations with Constant Coefficients.- 2.6 Conclusion.- 3 Review of Difference Equations.- 3.1 Introduction.- 3.2 First Order Difference Equations.- 3.3 Second Order Linear Difference Equations.- 3.4 Higher Order Difference Equations.- 3.5 Stability Conditions.- 3.6 Economic Applications.- 3.7 Concluding Remarks.- 4 Review of Some Linear Algebra.- 4.1 Vector and Vector Spaces.- 4.2 Matrices.- 4.3 Determinant Functions.- 4.4 Matrix Inversion and Applications.- 4.5 Eigenvalues and Eigenvectors.- 4.6 Quadratic Forms.- 4.7 Diagonalization of Matrices.- 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.- 5.3 Jordan Canonical Form of ODE Systems.- 5.4 Alternative Methods for Solving ? = Ax.- 5.5 Reduction to First Order of ODE Systems.- 5.6 Fundamental Matrix.- 5.7 Stability Conditions of ODE Systems.- 5.8 Qualitative Solution: Phase Portrait Diagrams.- 5.9 Some Economic Applications.- 6 First Order Difference Equations Systems.- 6.1 First Order Linear Systems.- 6.2 Jordan Canonical Form.- 6.3 Reduction to First Order Systems.- 6.4 Stability Conditions.- 6.5 Qualitative Solutions: Phase Diagrams.- 6.6 Some Economic Applications.- 7 Nonlinear Systems.- 7.1 Introduction.- 7.2 Linearization Theory.- 7.3 Qualitative Solution: Phase Diagrams.- 7.4 Limit Cycles.- 7.5 The Liénard-Van der Pol Equations and the Uniqueness of Limit Cycles.- 7.6 Linear and Nonlinear Maps.- 7.7 Stability of Dynamical Systems.-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.- 8.5 Economic Applications.- 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).- 10.3 Chaotic or Complex Dynamical Systems (DS).- 10.4 Catastrophe Theory (C.T.).- 10.5 Concluding Remarks.- 11 Optimal Dynamical Systems.- 11.1 Introduction.- 11.2 Pontryagin's Maximum Principle.- 11.3 Stabilization Control Models.- 11.4 Some Economic Applications.- 11.5 Asymptotic Stability of Optimal Dynamical Systems (ODS).- 11.6 Structural Stability of Optimal Dynamical Systems.- 11.7 Conclusion.- 12 Some Applications in Economics and Biology.- 12.1 Introduction.- 12.2 Economic Applications of Dynamical Systems.- 2.1. Flexible Multiplier-Accelerator Models.- 2.2. Kaldor's Type of Flexible Accelerator Models.- 2.3. Goodwin's Class Struggle Model.- 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.- 12.4 Bioeconomics and Natural Resources.- 12.5 Conclusion.

Rezensionen

"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