stability and dynamical systems -...
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Beatrice Venturi 1
STABILITY AND
DINAMICAL SYSTEMS
Lesson # 4
prof. Beatrice Venturi
PhD in Economics and Business
Course:
Quantitative Methods
1.STABILITY AND DINAMICAL SYSTEMS
• We consider a differential equation:
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)((*) 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
(*)
Dynamics Market Price
The Equilibrium Point
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costante)(tp
)( pfdt
dp0)( pf
0)]()([ pa
)(
)(p
Dynamics Market Price
)(
,))0(()(
akdove
pepptp kt
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The general solution with k>0 (k<0) converges to
(diverges from) equilibrium asintotically stable
(unstable)
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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atxt
)(lim
A Market Model with Time
Expectation
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:
Let the demand and supply functions be:
40)(222
2
tPdt
dP
dt
PdQd
5)(3 tPQs
A Market Model with Time
Expectation
45)(522
2
tPdt
dP
dt
Pd
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In equilibrium we have
sD QQ
A Market Model with Time
Expectation
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tCetP )(
tt eCdt
PdandeC
dt
dP 2
2
2
We adopt the trial solution:
In the first we find the solution of the homogenous equation
tt eCdt
PdandeC
dt
dP 2
2
2
A Market Model with Time
Expectation
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We get:
0)52( 2teC
The characteristic equation
0522
A Market Model with Time
Expectation
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We have two different roots
iandi 2121 21
the general solution of its reduced
homogeneous equation is
tectectP tt 2sin2cos)( 21
A Market Model with Time
Expectation
95/45)(tP
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The intertemporal equilibrium is given by the
particular integral
92sin2cos)( 21 tectectP tt
A Market Model with Time
Expectation
• With the following initial conditions
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12)0(P
1)0('P
The solution became
92sin22cos3)( tetetP tt
The equilibrium points of the system
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))(),((
))(),((
)1(
2122
2111
xyxyfdx
dy
xyxyfdx
dy
STABILITY AND DINAMICAL
SYSTEMS
STABILITY AND DINAMICAL SYSTEMS
• Are the solutions :
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0))(),((
0))(),(()2(
212
211
xyxyf
xyxyf
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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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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Case c) Complex roots λ1 =2+i and λ2 = 2-i,
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)(2)(
)()(2
)3(
212
211
txtxdt
dx
txtxdt
dx
System of LINEAR Ordinary Differential Equations
• Where A is the matrix associeted to the coefficients of the linear system of ODE ‘s:
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)()(
)()(
2221
1211
xaxa
xaxaA
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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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 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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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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STABILITY AND DINAMICAL SYSTEMS
• The equilibrium solutions are
• P1(0,0) and P2(0, ).
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STABILITY AND DINAMICAL SYSTEMS
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The Characteristic Equation of the matrix A is the
same of the equation (1)
0)()1( 212
2
xyadx
yda
dx
yd
STABILITY AND DINAMICAL SYSTEMS
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0)(23)3(2
2
txdt
xd
dt
xd
)(3)(2
)(
)4(
212
21
txtxdt
dx
txdt
dx
it’s equivalent to :
EXAMPLE
Eigenvalues
• p( λ) = λ2 − (a + d) λ + (ad − bc) = 0
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The solutions
are the eigenvalues of the matrix A.
STABILITY AND DINAMICAL SYSTEMS
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)(3
1)()(
)()(2)(
)3(
2212
2111
txtxtxdt
dx
txtxtxdt
dx
STABILITY AND DINAMICAL SYSTEMS
Solving this system we find the equilibrium point of the non-linear system (3):
:
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0)(3
1)()(
0)()(2)(
)4(221
211
txtxtx
txtxtx
STABILITY AND DINAMICAL SYSTEMS
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),()(3
1)()(
),()()(2)(
)3(
212212
212111
xxgtxtxtxdt
dx
xxftxtxtxdt
dx
Jacobian Matrix
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21
2
1
1
21 ),(
x
g
x
g
x
f
x
f
xxJ
3
1
221
),(12
12
21xx
xx
xxJ
Stability and Dynamical Systems
• Given the non linear system:
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1)()(
)()(
)4(
2
2
12
211
txtxdt
dx
txtxdt
dx
Stability and Dynamical Systems
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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)
Stability and Dynamical Systems
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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)
Predator-Prey cycles
1
1
dFa b S
F dtdS
c d FS dt
Rate of growth of Fish
Food supply Interactions with Sharks
Rate of growth of Sharks
Rate of death in absence of Fish to eat
Interactions with Fish
• Generates a cycle: – Lots of fish—>lots of interactions with Sharks—>rapid growth of Sharks—>Fall in Fish numbers—>less interactions with Sharks —>Fall in Shark numbers—>Lots of fish again...
dFa F b S F
dtdS
c S d F Sdt
Linear bits unstable
near equilibrium
Nonlinear bits stabilise far from
equilibrium
69
TrJ = a11+ a22
Det J = a11 a22 – a12 a21
a11 a12
a21 a22 J =
THE TRACE & THE DETERMINANT
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Predator-Prey cycles
98 99 100 101 102
900
950
1000
1050
1100
Sharks
Fis
h
Equilibrium here, but
system will never reach it
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Predator-Prey cycles
0 200 400 600 8001000
900
1000
1100
Time
Fis
h
0 200 400 600 800100098
99
100
101
102
Time
Shar
ks
98 99 100 101 102
900
950
1000
1050
1100
Sharks
Fis
h
Equilibrium here, but system will never reach it
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely