This book - perhaps the only of its kind - gives a comprehensive account of Dynamical Systems in a plain non-technical language which is as rigorous as it can be made at this introductory level. Starting from the first steps of differential equations, on the assumption that readers only have a modest mathematical background, it quickly takes them to nonlinear dynamical systems, linearization theory, limit cycles, Gradient, Lagrangean and Hamiltonian dynamical systems. Apart from a new chapter on Floquet theory, Centre Manifold theorems and Liapunov-Schmidt reduction, this second edition also…mehr
This book - perhaps the only of its kind - gives a comprehensive account of Dynamical Systems in a plain non-technical language which is as rigorous as it can be made at this introductory level. Starting from the first steps of differential equations, on the assumption that readers only have a modest mathematical background, it quickly takes them to nonlinear dynamical systems, linearization theory, limit cycles, Gradient, Lagrangean and Hamiltonian dynamical systems. Apart from a new chapter on Floquet theory, Centre Manifold theorems and Liapunov-Schmidt reduction, this second edition also includes new materials, additional examples, illustrations and applications: almost every chapter has been re-written and enlarged to keep up with rapid advances in this field.Dieses Buch liefert einen umfassenden Überblick über dynamische Systeme und stellt das Gebiet auf leicht verständliche Weise für Studenten dar.
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.
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.
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"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
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