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Beatrice Venturi1 Economic Faculty STABILITY AND DINAMICAL SYSTEMS prof. Beatrice Venturi

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mathematics for economics Beatrice Venturi 2 1.STABILITY AND DINAMICAL SYSTEMS §We consider a differential equation: with f a function independent of time t, represents a dynamical system.

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mathematics for economics Beatrice Venturi 3 a = is an equilibrium point of our system x(t) = a is a constant value. such that f(a)=0 The equilibrium points of our system are the solutions of the equation f(x) = 0 1.STABILITY AND DINAMICAL SYSTEMS

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mathematics for economics Beatrice Venturi 4 Market Price

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mathematics for economics Beatrice Venturi 5 Dynamics Market Price §The equilibrium Point

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mathematics for economics Beatrice Venturi 6 Dynamics Market Price The general solution with k>0 (k<0) converges to (diverges from) equilibrium asintotically stable (unstable)

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mathematics for economics Beatrice Venturi 7 The Time Path of the Market Price

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mathematics for economics Beatrice Venturi 8 1.STABILITY AND DINAMICAL SYSTEMS Given

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mathematics for economics Beatrice Venturi 9 1.STABILITY AND DINAMICAL SYSTEMS Let B be an open set and a Є B, §a = is a stable equilibrium point if for any x(t) starting in B result:

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Mathematics for Economics Beatrice Venturi 10 A Market Model with Time Expectation : Let the demand and supply functions be:

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A Market Model with Time Expectation mathematics for economics Beatrice Venturi 11 In equilibrium we have

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Mathematics for Economics Beatrice Venturi 12 A Market Model with Time Expectation We adopt the trial solution: In the first we find the solution of the homogenous equation

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Mathematics for Economics Beatrice Venturi 13 A Market Model with Time Expectation We get: The characteristic equation

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Mathematics for Economics Beatrice Venturi 14 A Market Model with Time Expectation We have two different roots the general solution of its reduced homogeneous equation is

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A Market Model with Time Expectation mathematics for economics Beatrice Venturi 15 The intertemporal equilibrium is given by the particular integral

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A Market Model with Time Expectation §With the following initial conditions mathematics for economics Beatrice Venturi 16 The solution became

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mathematics for economics Beatrice Venturi 17 The equilibrium points of the system STABILITY AND DINAMICAL SYSTEMS

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mathematics for economics Beatrice Venturi 18 STABILITY AND DINAMICAL SYSTEMS §Are the solutions :

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mathematics for economics Beatrice Venturi 19 The linear case

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mathematics for economics Beatrice Venturi 20 We remember that x'' = ax' + bcx + bdy §by = x' ax §x'' = (a + d)x' + (bc ad)x x(t) is the solution (we assume z=x) z'' (a + d)z' + (ad bc)z = 0. (*)

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mathematics for economics Beatrice Venturi 21 The Characteristic Equation If x(t), y(t) are solution of the linear system then x(t) and y(t) are solutions of the equations (*). The characteristic equation of (*) is p(λ) = λ 2 (a + d)λ + (ad bc) = 0

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mathematics for economics Beatrice Venturi 22 Knot and Focus The stable case

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mathematics for economics Beatrice Venturi 23 Knot and Focus The unstable case

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mathematics for economics Beatrice Venturi 24 Some Examples Case a) λ 1 = 1 e λ 2 = 3

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mathematics for economics Beatrice Venturi 25 Case b) λ 1 = -3 e λ 2 = -1

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mathematics for economics Beatrice Venturi 26 Case c) Complex roots λ 1 = 2+i and λ 2 = 2-i,

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mathematics for economics Beatrice Venturi 27 System of LINEAR Ordinary Differential Equations §Where A is the matrix associeted to the coefficients of the system:

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mathematics for economics Beatrice Venturi 28 STABILITY AND DINAMICAL SYSTEMS §Definition of Matrix §A matrix is a collection of numbers arranged into a fixed number of rows and columns. Usually the numbers are real numbers. Here is an example of a matrix with two rows and two columns:

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mathematics for economics Beatrice Venturi 29 STABILITY AND DINAMICAL SYSTEMS

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mathematics for economics Beatrice Venturi 30

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mathematics for economics Beatrice Venturi 31 STABILITY AND DINAMICAL SYSTEMS §Examples

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mathematics for economics Beatrice Venturi 32 STABILITY AND DINAMICAL SYSTEMS

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mathematics for economics Beatrice Venturi 33

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Eigenvectors and Eigenvalues of a Matrix The eigenvectors of a square matrix are the non-zero vectors that after being multiplied by the matrix, remain parellel to the original vector.

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mathematics for economist Beatrice Venturi 35 Eigenvectors and Eigenvalues of a Matrix §Matrix A acts by stretching the vector x, not changing its direction, so x is an eigenvector of A. The vector x is an eigenvector of the matrix A with eigenvalue λ (lambda) if the following equation holds:

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Eigenvectors and Eigenvalues of a Matrix §This equation is called the eigenvalues equation. mathematics for economist Beatrice Venturi 36

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Eigenvectors and Eigenvalues of a Matrix §The eigenvalues of A are precisely the solutions λ to the equation: §Here det is the determinant of matrix formed by A - λI ( where I is the 2×2 identity matrix). §This equation is called the characteristic equation (or, less often, the secular equation) of A. For example, if A is the following matrix (a so-called diagonal matrix): mathematics for economist Beatrice Venturi 37

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mathematics for economist Beatrice Venturi 38 Eigenvectors and Eigenvalues of a Matrix §Example

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mathematics for economics Beatrice Venturi 39 §We consider STABILITY AND DINAMICAL SYSTEMS

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mathematics for economics Beatrice Venturi 40 §We get the system: STABILITY AND DINAMICAL SYSTEMS

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mathematics for economics Beatrice Venturi 41 STABILITY AND DINAMICAL SYSTEMS

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mathematics for economics Beatrice Venturi 42 The Characteristic Equation

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mathematics for economics Beatrice Venturi 43 STABILITY AND DINAMICAL SYSTEMS The Characteristic Equation of the matrix A is the same of the equation (1)

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mathematics for economics Beatrice Venturi 44 STABILITY AND DINAMICAL SYSTEMS its equivalent to : EXAMPLE

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mathematics for economics Beatrice Venturi 45 STABILITY AND DINAMICAL SYSTEMS

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mathematics for economics Beatrice Venturi 46 Eigenvalues §p( λ) = λ 2 (a + d) λ + (ad bc) = 0 The solutions are the eigenvalues of the matrix A.

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mathematics for economics Beatrice Venturi 47 STABILITY AND DINAMICAL SYSTEMS

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mathematics for economics Beatrice Venturi 48 STABILITY AND DINAMICAL SYSTEMS Solving this system we find the equilibrium point of the non-linear system (3): :

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mathematics for economics Beatrice Venturi 49 STABILITY AND DINAMICAL SYSTEMS

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mathematics for economics Beatrice Venturi 50 STABILITY AND DINAMICAL SYSTEMS

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mathematics for economics Beatrice Venturi 51 Jacobian Matrix

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mathematics for economics Beatrice Venturi 52 Jacobian Matrix

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mathematics for economics Beatrice Venturi 53 Jacobian Matrix

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mathematics for economics Beatrice Venturi 54 Stability and Dynamical Systems.

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mathematics for economics Beatrice Venturi 55 Stability and Dynamical Systems §Given the non linear system:

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mathematics for economics Beatrice Venturi 56 Stability and Dynamical Systems

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mathematics for economics Beatrice Venturi 57 Stability and Dynamical Systems

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mathematics for economics Beatrice Venturi 58 Stability and Dynamical Systems

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mathematics for economics Beatrice Venturi 59 Stability and Dynamical Systems

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mathematics for economics Beatrice Venturi 60 Stability and Dynamical Systems

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61 LOTKA-VOLTERRA Prey – Predator Model

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The Lotka-Volterra Equations,

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63 We shall consider an ecologic system PREyPREDATOR

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mathematics for economics Beatrice Venturi 64 The Model

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Steady State Solutions a x 1 -bx 1 x 2 =0 c x 1 x 2 – d x 2 =0 a prey growth rate; d mortality rate

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The Jacobian Matrix J= a 11 a 12 a 21 a 22

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mathematics for economics Beatrice Venturi 67 Eigenvalues §p( λ) = λ 2 (a + d) λ + (ad bc) = 0 The solutions are the eigenvalues of the matrix A.

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68 TrJ = a 11 + a 22 a 11 a 12 a 21 a 22 J = THE TRACE

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69 THE DETERMINANT Det J = a 11 a 22 – a 12 a 21

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70 The equilibrium solutions x = 0 y = 0 Unstable x = d/g y = a/b Stable center

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72 Cycles

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