ARNOLD V.
Equations Differentielles Ordinaires, Editions MIR, 1974
Equazioni Differenziali Ordinarie, Edizioni MIR, 1979
Ordinary Differential Equations, Springer, 1992
CHIANG A. C.
Elements of Dynamic Optimization, Waveland Press Inc., 1992
GIUSTI E.
Analisi Matematica 1, seconda edizione, Edizioni Bollati-Boringhieri, 1984
GIUSTI E.
Analisi Matematica 2, seconda edizione, Edizioni Bollati-Boringhieri, 1989
GUCKENHEIMER J. , HOLMES P.
Non Linear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,
Applied Mathematical Sciences (Vol. 42), Springer, 1983
HIRSCH M. , SMALE S.
Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, 1974
LANG S.
Linear Algebra, Addison Wesley, 1966
LORENZ H.W.
Nonlinear Dynamical Economics and Chaotic Motion, Springer, 1993
SALSA S. , SQUELLATI A.
Modelli Dinamici e Controllo Ottimo, Egea, 2010
SYDSAETER K. , HAMMOND P. , SEIERSTAD A. , STROERN A.
Further Mathematics for Economic Analisys, Prentice Hall, 2008
Learning Objectives
adequate knowledge of basic mathematical techniques in economic dynamics theory
Prerequisites
topics in elementary linear algebra and real calculus in one and several variables
Teaching Methods
lectures and seminars
Type of Assessment
written and oral final exam
Course program
MATHEMATICS for ECONOMICS
Academic Year 2014 - 2015
Prof. Franco Gori
PROGRAM (abridged version)
0. Complements in Topology, Linear Algebra and Calculus
0.1. Cardinality of a Set
0.2. Topology of Metric Spaces
0.3. The Field of Complex Numbers
0.4. Eigenvectors and Eigenvalues
0.5. Diagonalization of Matrices
0.6. Linear Normed Spaces
0.7. Banach Spaces
0.8. Scalar Products and Orthogonality
0.9. Hilbert Spaces
0.10. Series of Real Numbers
0.11. Sequence and Series in Banach Spaces
0.12. The Banach-Caccioppoli Contraction Theorem
1. Dynamical Systems
1.1. Introduction to Dynamical Modeling
1.2. Models in Continuous and Discrete Time
1.3. Finite and Infinite Dimensional Models
1.4. Deterministic, Semi-Deterministic and Stochastic Models
1.5. Classical Models in Economic Dynamics
1.6. The Flow of a Dynamical System
2. Ordinary Differential Equations
2.1. Generalities and Ordinary Differential Equations in Normal Form
2.2. First Order Differential Equations
2.3. The Cauchy Problem
2.4. Existence and Uniqueness
2.5. On Extending Solutions
2.6. Separation of Variables Integrating Method
2.7. First Order Linear Differential Equation
2.8. Second Order Linear Differential Equation with Constant Coefficients
2.9. Higher Order Linear Differential Equation with Constant Coefficients
3. Systems of Ordinary Differential Equations
3.1. Generalities and Systems in Normal Form
3.2. Vector Fields
3.3. The Cauchy Problem
3.4. The Fundamental Theorem of Existence and Local Uniqueness
3.5. On Extending Solutions
3.6. Maximal Intervals
3.7. Unbounded Maximal Intervals
3.8. Gronwall’s Inequality
3.9. Continuous Dependence of Solutions on Initial Conditions
3.10. Maximal Intervals Dependence on Initial Conditions
4. Systems of Linear Differential Equations
4.1. Generalities on Systems of Linear Differential Equations
4.2. Autonomous and Non-Autonomous Systems
4.3. Homogeneous and Non-Homogeneous Systems
4.4. Subspace and Linear Manifolds of Solutions
4.5. Fundamental Systems of Solutions and their Wronskian
4.6. Wronskian Differential Equation and Liouville’s Theorem
4.7. Constant Coefficient Homogeneous Systems of Linear Differential Equations
4.8. Exponential of a Real Matrix and its Properties
4.9. Closed Form of Solutions and Unbounded Maximal Intervals
4.10 Closed Form of Solutions in the Constant Coefficient Non- Homogeneous Case
4.11. Computing the Exponential of a Real Matrix : Jordan Canonical Form
4.12. The 2-Dimensional Constant Coefficients Linear Case
4.13. Attracting , Repelling and Saddle Equilibria in the 2-Dimensional Case
4.14. Trajectories, Closed Orbits and Phase Portraits in the 2-Dimensional Case
4.15. The n-Dimensional Constant Coefficients Linear Case
4.16. Hyperbolic Equilibria
4.17. Attracting and Repelling Hyperbolic Equilibria
4.18. Generic Properties of Hyperbolic Equilibria
5. Non Linear Dynamical Systems : Local Analysis
5.1. Stable and Asymptotically Stable Equilibria
5.2. Liapunov’s Functions
5.3. Liapunov’s Theorems
5.4. Systems Linearization in the Neighborhood of an Equilibrium Point
5.5. Hyperbolic Equilibria
5.6. Attracting and Repelling Hyperbolic Equilibria
5.7. The Hartman and Grobman Theorem
6. Non Linear Dynamical Systems : Global Analysis
6.1. Generalities on Equilibria, Closed Orbits, Invariant Regions and Attractors
6.2. and Limit Sets
6.3. Elementary Attractors : Equilibria and Limit Cycles
6.4. Complex Attractors
6.5. Example 1 : The Lorenz’s Attractor
6.6. Example 2 : The Roesller’s Attractor
6.7. Example 3 : Hyperbolic Dynamical Systems and Attracting Tori
6.8. Recurrent and Non-Wandering Points
6.9. Non Linear Dynamical Systems in the Plane
6.10. Limit Cycles of Non Linear Dynamical Systems in the Plane
6.11. Local Cross-Sections and the Flow-Box Technique
6.12. The Poincarè-Bendixon’s Theorem and its Corollaries
7. Elements of Variational Calculus and Optimal Control
7.1. Generalities on Calculus of Variations
7.2. Functionals
7.3. Fixed Boundary Problems
7.4. Euler’s Equation
7.5. Special Cases
7.6. Free Boundary Problems
7.7. Euler’s Equation and Transversality Conditions
7.8. Generalities on Optimal Control Theory
7.9. The “Simplest” Optimal Control Problem
7.10. Maximun Principle
7.11. Heuristic Assessments on the Maximun Principle
7.12. Alternative Boundary Conditions
7.13. Finite and Infinite Horizon Problems
7.14. Optimal Control Theory and Economic Modelling
CHIANG A. C.
Elements of Dynamic Optimization, Waveland Press Inc., 1992
GIUSTI E.
Analisi Matematica 1, seconda edizione, Edizioni Bollati-Boringhieri, 1984
GIUSTI E.
Analisi Matematica 2, seconda edizione, Edizioni Bollati-Boringhieri, 1989
GUCKENHEIMER J. , HOLMES P.
Non Linear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,
Applied Mathematical Sciences (Vol. 42), Springer, 1983
HIRSCH M. , SMALE S.
Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, 1974
LANG S.
Linear Algebra, Addison Wesley, 1966
LORENZ H.W.
Nonlinear Dynamical Economics and Chaotic Motion, Springer, 1993
SALSA S. , SQUELLATI A.
Modelli Dinamici e Controllo Ottimo, Egea, 2010
SYDSAETER K. , HAMMOND P. , SEIERSTAD A. , STROERN A.
Further Mathematics for Economic Analisys, Prentice Hall, 2008