Pages: 340

Trade Paper: ISBN 9788885486522; $59,95

Pdf: ISBN 9788885486539; $41,95

Author: Sandro Salsa, Annamaria Squellati

### Table of Contents

**1 Introduction to Modelling**

1.1 Some Classical Examples

1.1.1 Malthus model

1.1.2 Logistic models

1.1.3 Phillips model

1.1.4 Accelerator model

1.1.5 Evolution of supply

1.1.6 Leslie model

1.1.7 Lotka-Volterra predator-prey model

1.1.8 Time-delay logistic equation

1.2 Continuous Time and Discrete Time Models

1.2.1 Differential and difference equations

1.2.2 Systems of differential and difference equations

**2 First Order Differential Equations**

2.1 Introduction

2.2 Some Solvable Equations

2.2.1 Differential equations with separable variables

2.2.2 Solow-Swan model

2.2.3 Logistic model

2.2.4 Linear equations

2.2.5 Market dynamics

2.2.6 Other types of equations

2.3 The Cauchy Problem

2.3.1 Existence and uniqueness

2.3.2 Maximal interval of existence

2.4 Autonomous Equations

2.4.1 Steady states, stability, phase space

2.4.2 Stability by linearization

2.5 A Neoclassical Growth Model

2.6 Exercises

**3 First Order Difference Equations**

3.1 Introduction

3.2 Linear Equations

3.2.1 Linear homogeneous equations

3.2.2 Nonhomogeneous linear equations

3.2.3 Simple and compounded capitalization

3.2.4 Cobweb model

3.3 Nonlinear Autonomous Equations

3.3.1 Orbits, stairstep diagram, steady states (fixed or equilibrium points)

3.3.2 Steady states and stability

3.3.3 Stability of periodic orbits

3.3.4 Chaotic behavior

3.3.5 Discrete logistic equation

3.4 Exercises

**4 Linear Differential Equations with Constant Coefficients**

4.1 Second Order Equations

4.1.1 Homogeneous equations

4.1.2 Nonhomogeneous equations

4.2 Higher Order Equations

4.2.1 Homogeneous equations

4.2.2 Nonhomogeneous equations

4.2.3 Stability

4.2.4 Phillips model

4.3 Exercises

**5 Linear Difference Equations with Constant Coefficients**

5.1 Second Order Equations

5.1.1 Homogeneous equations

5.1.2 Fibonacci sequence

5.1.3 Nonhomogeneous equations

5.2 Higher Order Equations

5.2.1 Homogeneous equations

5.2.2 Nonhomogeneous equations

5.2.3 Stability

5.2.4 Accelerator model

5.3 Exercises

**6 Systems of Differential Equations**

6.1 The Cauchy Problem

6.2 Linear Systems

6.2.1 Global existence and uniqueness

6.2.2 Homogeneous systems

6.2.3 Nonhomogeneous systems

6.2.4 Equations of order n

6.3 Bidimensional Systems with Constant Coefficients

6.3.1 General integral

6.3.2 Stability of the zero solution

6.4 Systems with Constant Coefficients (higher dimension)

6.4.1 Exponential matrix

6.4.2 Cauchy problem and general integral

6.4.3 Nonhomogeneous systems

6.4.4 Stability of the zero solution

6.5 Exercises

**7 Bidimensional Autonomous Systems**

7.1 Phase Plane Analysis

7.1.1 Orbits

7.1.2 Steady states, cycles and their stability

7.1.3 Phase portrait

7.2 Linear Systems. Classification of steady states

7.3 Non-linear Systems

7.3.1 The linearization method

7.3.2 Outline of the Liapunov method

7.4 Some Models

7.4.1 Lotka-Volterra model

7.4.2 A competitive equilibrium model

7.5 Higher Dimensional Systems

7.6 Exercises

**8 Systems of Difference Equations**

8.1 Linear Systems with Constant Coefficients

8.1.1 Homogeneous systems

8.1.2 Bidimensional homogeneous systems

8.1.3 Nonhomogeneous systems

8.2 Stability

8.2.1 Election polls

8.2.2 A model of students partition

8.2.3 Leslie model

8.3 Autonomous systems

8.3.1 Discrete Lotka-Volterra model

8.3.2 Logistic equation with delay

8.4 Exercises

**9 The Calculus of Variations**

9.1 Introduction

9.2 The Simplest Problem

9.2.1 Fixed boundaries. Euler equation

9.2.2 Special cases of the Euler-Lagrange equation

9.2.3 Free end values. Transversality conditions

9.3 A Sufficient Condition of Optimality

9.4 Infinite Horizon. Unbounded Interval. An Optimal Growth Problem

9.5 The General Variation of a Functional

9.6 Isoperimetric Problems

9.7 Exercises

**10 Optimal Control Problems. Variational Methods**

10.1 Introduction

10.1.1 Structure of a control problem. One-dimensional state and control

10.1.2 Main questions and techniques

10.2 Continuous Time Systems

10.2.1 Free final state. Necessary conditions

10.2.2 Sufficient conditions

10.2.3 Interpretation of the multiplier

10.2.4 Maximum principle. Bounded controls

10.2.5 Discounting. Current values

10.2.6 Applications. Infinite horizon. Comparative analysis

10.2.7 Terminal payoff and various endpoints conditions

10.2.8 Discontinuous and bang-bang control. Singular solutions

10.2.9 An advertising model control

10.3 Discrete Time Problems

10.3.1 The simplest problem

10.3.2 A discrete model for optimal growth

10.3.3 State and control constraints

10.3.4 Interpretation of the multiplier

10.4 Exercises

**11 Dynamic Programming**

11.1 Introduction

11.2 Continuous Time System. The Bellman Equation

11.3 Infinite Horizon. Discounting

11.4 Discrete Time Systems

11.4.1 The value function. The Bellman equation

11.4.2 Optimal resource allocation

11.4.3 Infinite horizon. Autonomous problems

11.4.4 Renewable resource management

**A Appendix**

A.1 Eigenvalues and Eigenvectors

A.2 Functional Spaces

A.3 Static Optimization

A.3.1 Free optimization

A.3.2 Constrained optimization. Equality constraints

A.3.3 Constrained optimization. Inequality constraints

**References**

**Subject index**