Stability and Dynamical Systems - people.unica.it€¦ · STABILITY AND DINAMICAL SYSTEMS Lesson #...
Transcript of Stability and Dynamical Systems - people.unica.it€¦ · STABILITY AND DINAMICAL SYSTEMS Lesson #...
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Beatrice Venturi 1
STABILITY AND
DINAMICAL SYSTEMS
Lesson # 4
prof. Beatrice Venturi
PhD in Economics and Business
Course:
Quantitative Methods
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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
(*)
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Market Price
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)]()([ padt
dp
)()( apadt
dp
( )d s
dpa Q Q
dt
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Dynamics Market Price
The Equilibrium Point
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costante)(tp
)( pfdt
dp0)( pf
0)]()([ pa
)(
)(p
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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)
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The Time Path of the Market Price
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1.STABILITY AND DINAMICAL SYSTEMS
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)(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:
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atxt
)(lim
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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
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A Market Model with Time
Expectation
45)(522
2
tPdt
dP
dt
Pd
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In equilibrium we have
sD QQ
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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
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A Market Model with Time
Expectation
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We get:
0)52( 2teC
The characteristic equation
0522
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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
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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
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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
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The equilibrium points of the system
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))(),((
))(),((
)1(
2122
2111
xyxyfdx
dy
xyxyfdx
dy
STABILITY AND DINAMICAL
SYSTEMS
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STABILITY AND DINAMICAL SYSTEMS
• Are the solutions :
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0))(),((
0))(),(()2(
212
211
xyxyf
xyxyf
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The Linear Case
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)()(
)()(
(*)
tdytcxdt
dy
tbytaxdt
dx
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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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Knot and Focus The stable case
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Knot and Focus The Unstable Case’
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Some Examples Case a)λ1= 1 e λ2 = 3
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)(2)(
)()(2
)1(
212
211
txtxdt
dx
txtxdt
dx
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Case b) λ1= -3 e λ2 = -1
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)(2)(
)()(2
)2(
212
211
txtxdt
dx
txtxdt
dx
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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
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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
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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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STABILITY AND DINAMICAL SYSTEMS
20
01A
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STABILITY AND DINAMICAL SYSTEMS
• Examples
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)(2
)(
)1(
22
11
txdt
dx
txdt
dx
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STABILITY AND DINAMICAL SYSTEMS
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t
t
ectx
ectx
2
22
11
)(
)(
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STABILITY AND DINAMICAL SYSTEMS
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)(2
)(
)2(
22
11
txdt
dx
txdt
dx
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STABILITY AND DINAMICAL SYSTEMS
20
01A
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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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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
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Eigenvectors and Eigenvalues of a Matrix
• This equation is called the eigenvalues equation.
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xAx
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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):
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Eigenvectors and Eigenvalues of a Matrix
• Example
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020
01det)det( IA
0)2)(1(
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• We consider
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)()()1( 212
2
xfxyadx
yda
dx
yd
STABILITY AND DINAMICAL SYSTEMS
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• We get the system:
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)sin(
)2(
pydt
dp
ydt
dp
STABILITY AND DINAMICAL SYSTEMS
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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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)()(
10
12 xaxaA
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The Characteristic Equation
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0)()(
1det
)det(
12 xaxa
IA
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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
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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
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STABILITY AND DINAMICAL SYSTEMS
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Eigenvalues
• p( λ) = λ2 − (a + d) λ + (ad − bc) = 0
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The solutions
are the eigenvalues of the matrix A.
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STABILITY AND DINAMICAL SYSTEMS
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)(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):
:
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0)(3
1)()(
0)()(2)(
)4(221
211
txtxtx
txtxtx
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STABILITY AND DINAMICAL SYSTEMS
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),()(3
1)()(
),()()(2)(
)3(
212212
212111
xxgtxtxtxdt
dx
xxftxtxtxdt
dx
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STABILITY AND DINAMICAL SYSTEMS
Beatrice Venturi 51
)0,0(),( 21 xx
)2
1,
3
1(),( 21 xx
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Jacobian Matrix
Beatrice Venturi 52
21
2
1
1
21 ),(
x
g
x
g
x
f
x
f
xxJ
3
1
221
),(12
12
21xx
xx
xxJ
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Jacobian Matrix
Beatrice Venturi 53
3
10
01
)0,0(J
3
10
01
)det( AI
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Jacobian Matrix
??
??)2/1,3/1(J
Beatrice Venturi 54
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Stability and Dynamical Systems
.
Beatrice Venturi 55
01
dt
dx02
dt
dx
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Stability and Dynamical Systems
• Given the non linear system:
Beatrice Venturi 56
1)()(
)()(
)4(
2
2
12
211
txtxdt
dx
txtxdt
dx
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Stability and Dynamical Systems
Beatrice Venturi 57
01
dt
dx
)()(
0)()(
12
21
txtx
txtx
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Stability and Dynamical Systems
Beatrice Venturi 58
02
dt
dx
1)()(
01)()(
2
2
2
2
1
1
txtx
txtx
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Stability and Dynamical Systems
Beatrice Venturi 59
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
Beatrice Venturi 60
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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61
LOTKA-VOLTERRA
Prey – Predator Model
Beatrice Venturi
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The Lotka-Volterra Equations,
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63
We shall consider an ecologic system
PREy PREDATOR
Beatrice Venturi
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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
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Beatrice Venturi 65
),()()()(
),()()()(
(*)
SFgtStdFtcSdt
dS
SFftStbFtaFdt
dF
The Model
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Steady State Solutions
a F –b F S=0
d F S– c S=0
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The Jacobian Matrix
J =
∂f/∂F ∂f/∂S
∂g/∂F ∂g/∂S
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Eigenvalues
p( λ) = λ2 − Tr J λ + det J = 0
Beatrice Venturi 68
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69
TrJ = a11+ a22
Det J = a11 a22 – a12 a21
a11 a12
a21 a22 J =
THE TRACE & THE DETERMINANT
Beatrice Venturi
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70
The equilibrium solutions
F = 0 S = 0 Unstable
F = c/d S = a/b Stable Center
Beatrice Venturi
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Fish Cycles
Beatrice Venturi 71
0 200 400 600 8001000
900
1000
1100
Time
Fis
h
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Sharks Cycles
Beatrice Venturi 72
0 200 400 600 800100098
99
100
101
102
Time
Shar
ks
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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
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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
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75
bFSaF
cSdFS
dt
dF
dt
dS/
cFcdFbSSa ||ln||ln
||ln||ln),( 21 FcdFbSSaxxHBeatrice Venturi
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76
1 2 3 4 5 6 7
1
2
3
4
Cycles
Beatrice Venturi
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77 Beatrice Venturi