beatrice venturi1 economic faculty stability and dinamical systems prof. beatrice venturi
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Beatrice Venturi 1
Economic Faculty
STABILITY AND DINAMICAL SYSTEMS
prof. Beatrice Venturi
mathematics for economics Beatrice Venturi
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1.STABILITY AND DINAMICAL
SYSTEMS We consider a differential equation:
)((*) xfx
dt
d
with f a function independent of time t , represents a dynamical system .(*)
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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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Market Price
)]()([ padt
dp
)()( apadt
dp
( )d s
dpa Q Q
dt
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Dynamics Market Price
The equilibrium Point
)(
)(
p
costante)( tp
)( pfdt
dp 0)( pf
0)]()([ pa
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Dynamics Market Price
)(
,))0(()(
akdove
pepptp kt
The general solution with k>0 (k<0) converges to (diverges from) equilibrium asintotically stable
(unstable)
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The Time Path of the Market Price
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1.STABILITY AND DINAMICAL SYSTEMS
Given
)(xdt
df
x
)(xfx
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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:
atxt
)(lim
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A Market Model with Time Expectation:
Let the demand and supply functions be:
40)(222
2
tPdt
dP
dt
PdQd
5)(3 tPQs
A Market Model with Time Expectation
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45)(522
2
tPdt
dP
dt
Pd
In equilibrium we have
sD QQ
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A Market Model with Time Expectation
tCetP )(
tt eCdt
PdandeC
dt
dP 22
2
We adopt the trial solution:
In the first we find the solution of the homogenous equation
tt eCdt
PdandeC
dt
dP 22
2
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A Market Model with Time Expectation
We get:
0)52( 2 teC
The characteristic equation
0522
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A Market Model with Time Expectation
We have two different rootsiandi 2121 21
the general solution of its reduced homogeneous equation is
tectectP tt 2sin2cos)( 21
A Market Model with Time Expectation
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95/45)( tP
The intertemporal equilibrium is given by the particular integral
92sin2cos)( 21 tectectP tt
A Market Model with Time ExpectationWith the following initial conditions
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12)0( P
1)0(' PThe solution became
92sin22cos3)( tetetP tt
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The equilibrium points of the system
))(),((
))(),(()1(
2122
2111
xyxyfdx
dy
xyxyfdx
dy
STABILITY AND DINAMICAL SYSTEMS
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STABILITY AND DINAMICAL SYSTEMS
Are the solutions :
0))(),((
0))(),(()2(
212
211
xyxyf
xyxyf
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)()(
)()((*)
tdytcxdt
dy
tbytaxdt
dx
The linear case
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We remember that
x'' = ax' + bcx + bdyby = 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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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) = λ2 − (a + d)λ + (ad − bc) = 0
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Knot and Focus The stable case
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Knot and Focus The unstable case’
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Some ExamplesCase a)λ1= 1 e λ2 = 3
)(2)(
)()(2)1(
212
211
txtxdt
dx
txtxdt
dx
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Case b) λ1= -3 e λ2 = -1
)(2)(
)()(2)2(
212
211
txtxdt
dx
txtxdt
dx
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Case c) Complex roots λ1 =2+i and λ2 = 2-i,
)(2)(
)()(2)3(
212
211
txtxdt
dx
txtxdt
dx
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System of LINEAR Ordinary Differential Equations
Where A is the matrix associeted to the coefficients of the system:
)()(
)()(
2221
1211
xaxa
xaxaA
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STABILITY AND DINAMICAL SYSTEMS
Definition of MatrixA 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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STABILITY AND DINAMICAL SYSTEMS
t
t
ectx
ectx2
22
11
)(
)(
STABILITY AND DINAMICAL SYSTEMS
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20
01A
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STABILITY AND DINAMICAL SYSTEMS
Examples
)(2
)()1(
22
11
txdt
dx
txdt
dx
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STABILITY AND DINAMICAL SYSTEMS
)(2
)()2(
22
11
txdt
dx
txdt
dx
STABILITY AND DINAMICAL SYSTEMS
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20
01A
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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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:
xAx
Eigenvectors and Eigenvalues of a Matrix
This equation is called the eigenvalues equation.
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xAx
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):
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Eigenvectors and Eigenvalues of a Matrix
Example
020
01det)det(
IA
0)2)(1(
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We consider
)()()1( 212
2
xfxyadx
yda
dx
yd
STABILITY AND DINAMICAL SYSTEMS
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We get the system:
)()()()(
)()2(
21122
21
xfxyxaxyadx
dy
xydx
dy
STABILITY AND DINAMICAL SYSTEMS
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STABILITY AND DINAMICAL SYSTEMS
)()(
10
12 xaxaA
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0)()(
1det
)det(
12
xaxa
IA
The Characteristic Equation
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STABILITY AND DINAMICAL SYSTEMS
The Characteristic Equation of the matrix A is the same of the equation (1)
0)()1( 212
2
xyadx
yda
dx
yd
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STABILITY AND DINAMICAL SYSTEMS
0)(23)3(2
2
txdt
xd
dt
xd
)(3)(2
)()4(
212
21
txtxdt
dx
txdt
dx
it’s equivalent to :
EXAMPLE
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STABILITY AND DINAMICAL SYSTEMS
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Eigenvalues
p( λ) = λ2 − (a + d) λ + (ad − bc) = 0
The solutions
are the eigenvalues of the matrix A.
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STABILITY AND DINAMICAL SYSTEMS
)(3
1)()(
)()(2)()3(
2212
2111
txtxtxdt
dx
txtxtxdt
dx
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STABILITY AND DINAMICAL SYSTEMS
Solving this system we find the equilibrium point of the non-linear system (3):
:
0)(3
1)()(
0)()(2)()4(
221
211
txtxtx
txtxtx
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STABILITY AND DINAMICAL SYSTEMS
),()(3
1)()(
),()()(2)()3(
212212
212111
xxgtxtxtxdt
dx
xxftxtxtxdt
dx
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STABILITY AND DINAMICAL SYSTEMS
)0,0(),( 21 xx
)2
1,
3
1(),( 21 xx
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Jacobian Matrix
21
2
1
121 ),(
x
g
x
g
x
f
x
f
xxJ
3
1
221),(
12
12
21xx
xxxxJ
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Jacobian Matrix
3
10
01)0,0(J
3
10
01)det( AI
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Jacobian Matrix
??
??)2/1,3/1(J
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Stability and Dynamical Systems
.01
dt
dx02
dt
dx
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Stability and Dynamical Systems
Given the non linear system:
1)()(
)()()4(
22
12
211
txtxdt
dx
txtxdt
dx
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Stability and Dynamical Systems
01 dt
dx
)()(
0)()(
12
21
txtx
txtx
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Stability and Dynamical Systems
02 dt
dx
1)()(
01)()(
22
22
1
1
txtx
txtx
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Stability and Dynamical Systems
f(x)=(x^2)-1
f(x)=x
-4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5
-4
-3
-2
-1
1
2
3
4
x
f(x)
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Stability and Dynamical Systems
20
01A
11 22
t
t
ectx
ectx2
22
11
)(
)(
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Stability and Dynamical Systems
f(x)=e^x
f(x)=e^(-2x)
-4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5
-4
-3
-2
-1
1
2
3
4
x
f(x)
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LOTKA-VOLTERRAPrey – Predator Model
The Lotka-Volterra Equations,
63
We shall consider an ecologic
system
PREy PREDATOR
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)()()(
)()((*)
2212
2111
tdxtxtcxdt
dx
txbxtaxdt
dx
The Model
Steady State Solutions
a x1-bx1x2=0c x1x2– d x2=0
a prey growth rate; d mortality rate
The Jacobian Matrix
J= a11 a12
a21 a22
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Eigenvalues
p( λ) = λ2 − (a + d) λ + (ad − bc) = 0
The solutions
are the eigenvalues of the matrix A.
68
TrJ = a11+ a22
a11 a12
a21 a22 J =
THE TRACE
69
THE DETERMINANT
Det J = a11 a22 – a12 a21
70
The equilibrium solutions x = 0 y = 0 Unstable
x = d/g y = a/b Stable center
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211
22112 /xxx
xxx
dt
dx
dt
dx
22
)( dxx
1
1
)( dxx
cxxxx ||ln||ln 1122
||ln||ln),( 112221 xxxxxxF
72
1 2 3 4 5 6 7
1
2
3
4
Cycles
73