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Economic Faculty. STABILITY AND DINAMICAL SYSTEMS. prof. Beatrice Venturi. 1.STABILITY AND DINAMICAL SYSTEMS. We consider a differential equation:. with f a function independent of time t , represents a dynamical system. 1.STABILITY AND DINAMICAL SYSTEMS.

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Economic Faculty

STABILITY AND

DINAMICAL SYSTEMS

prof. Beatrice Venturi

Beatrice Venturi

1.STABILITY AND DINAMICAL SYSTEMS
• We consider a differential equation:

with f a function independent of time t , represents a dynamical system .

mathematics for economics Beatrice Venturi

1.STABILITY AND DINAMICAL SYSTEMS

a= is an equilibrium point of our system

x(t) = a is a constant value.

such that

f(a)=0

The equilibrium points of oursystem are the solutions of the equation

f(x) = 0

mathematics for economics Beatrice Venturi

Market Price

mathematics for economics Beatrice Venturi

Dynamics Market Price
• The equilibrium Point

mathematics for economics Beatrice Venturi

Dynamics Market Price

The general solution withk>0 (k<0) converges to (diverges from) equilibrium asintotically stable (unstable)

mathematics for economics Beatrice Venturi

The TimePath of the Market Price

mathematics for economics Beatrice Venturi

1.STABILITY AND DINAMICAL SYSTEMS

Given

mathematics for economics Beatrice Venturi

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:

mathematics for economics Beatrice Venturi

A Market Model with Time Expectation

:

Let the demand and supply functions be:

Mathematics for Economics Beatrice Venturi

A Market Model with Time Expectation

In equilibrium we have

mathematics for economics Beatrice Venturi

A Market Model with Time Expectation

In the first we find the solution of the homogenous equation

Mathematics for Economics Beatrice Venturi

A Market Model with Time Expectation

We get:

The characteristic equation

Mathematics for Economics Beatrice Venturi

A Market Model with Time Expectation

We have two different roots

the general solution of its reduced

homogeneous equation is

Mathematics for Economics Beatrice Venturi

A Market Model with Time Expectation

The intertemporal equilibrium is given by the particular integral

mathematics for economics Beatrice Venturi

A Market Model with Time Expectation
• With the following initial conditions

The solution became

mathematics for economics Beatrice Venturi

STABILITY AND DINAMICAL SYSTEMS
• Are the solutions :

mathematics for economics Beatrice Venturi

The linear case

mathematics for economics Beatrice Venturi

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. (*)

mathematics for economics Beatrice Venturi

The Characteristic Equation

If x(t), y(t) are solution of the linear system thenx(t) and y(t) are solutions of the equations (*).

The characteristicequation of (*) is

p(λ) = λ2 − (a + d)λ + (ad − bc) = 0

mathematics for economics Beatrice Venturi

Knot and Focus The stable case

mathematics for economics Beatrice Venturi

Knot and Focus The unstable case’

mathematics for economics Beatrice Venturi

Some ExamplesCase a)λ1=1 e λ2= 3

mathematics for economics Beatrice Venturi

Case b) λ1= -3 e λ2= -1

mathematics for economics Beatrice Venturi

Case c) Complex roots λ1 =2+i and λ2 = 2-i,

mathematics for economics Beatrice Venturi

System of LINEAR Ordinary Differential Equations
• Where A is the matrix associeted to the coefficients of the system:

mathematics for economics Beatrice Venturi

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:

mathematics for economics Beatrice Venturi

STABILITY AND DINAMICAL SYSTEMS

mathematics for economics Beatrice Venturi

STABILITY AND DINAMICAL SYSTEMS

mathematics for economics Beatrice Venturi

STABILITY AND DINAMICAL SYSTEMS
• Examples

mathematics for economics Beatrice Venturi

STABILITY AND DINAMICAL SYSTEMS

mathematics for economics Beatrice Venturi

STABILITY AND DINAMICAL SYSTEMS

mathematics for economics Beatrice Venturi

### 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.

Eigenvectors and Eigenvalues of a Matrix
• Matrix A acts by stretching the vectorx, not changing its direction, so x is an eigenvector of A. The vector x is an eigenvector of the matrixA with eigenvalueλ (lambda) if the following equation holds:

mathematics for economist Beatrice Venturi

Eigenvectors and Eigenvalues of a Matrix
• This equation is called the eigenvalues equation.

mathematics for economist Beatrice Venturi

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

Eigenvectors and Eigenvalues of a Matrix
• Example

mathematics for economist Beatrice Venturi

STABILITY AND DINAMICAL SYSTEMS

• We consider

mathematics for economics Beatrice Venturi

STABILITY AND DINAMICAL SYSTEMS

• We get the system:

mathematics for economics Beatrice Venturi

STABILITY AND DINAMICAL SYSTEMS

mathematics for economics Beatrice Venturi

The CharacteristicEquation

mathematics for economics Beatrice Venturi

STABILITY AND DINAMICAL SYSTEMS

The CharacteristicEquation of the matrix A is the same of the equation (1)

mathematics for economics Beatrice Venturi

STABILITY AND DINAMICAL SYSTEMS

EXAMPLE

it’s equivalent to :

mathematics for economics Beatrice Venturi

STABILITY AND DINAMICAL SYSTEMS

mathematics for economics Beatrice Venturi

Eigenvalues
• p( λ) = λ2 − (a + d) λ + (ad − bc) = 0

The solutions

are the eigenvalues of the matrix A.

mathematics for economics Beatrice Venturi

STABILITY AND DINAMICAL SYSTEMS

mathematics for economics Beatrice Venturi

STABILITY AND DINAMICAL SYSTEMS

Solving this system we find the equilibrium point of the non-linear system (3):

:

mathematics for economics Beatrice Venturi

STABILITY AND DINAMICAL SYSTEMS

mathematics for economics Beatrice Venturi

STABILITY AND DINAMICAL SYSTEMS

mathematics for economics Beatrice Venturi

Jacobian Matrix

mathematics for economics Beatrice Venturi

Jacobian Matrix

mathematics for economics Beatrice Venturi

Jacobian Matrix

mathematics for economics Beatrice Venturi

Stability and Dynamical Systems

.

mathematics for economics Beatrice Venturi

Stability and Dynamical Systems
• Given the non linear system:

mathematics for economics Beatrice Venturi

Stability and Dynamical Systems

mathematics for economics Beatrice Venturi

Stability and Dynamical Systems

mathematics for economics Beatrice Venturi

Stability and Dynamical Systems

mathematics for economics Beatrice Venturi

Stability and Dynamical Systems

mathematics for economics Beatrice Venturi

Stability and Dynamical Systems

mathematics for economics Beatrice Venturi

LOTKA-VOLTERRA

Prey – Predator Model

The Model

mathematics for economics Beatrice Venturi

a x1-bx1x2=0

c x1x2– d x2=0

a prey growth rate;

d mortality rate

a11 a12

a21 a22

### The Jacobian Matrix

J=

Eigenvalues
• p( λ) = λ2 − (a + d) λ + (ad − bc) = 0

The solutions

are the eigenvalues of the matrix A.

mathematics for economics Beatrice Venturi

THE TRACE

a11 a12

a21 a22

J =

TrJ = a11+ a22

THE DETERMINANT

Det J = a11 a22 – a12 a21

The equilibrium solutions

x = 0 y = 0

Unstable

Stable center

x = d/g y = a/b