Modeling of Economic Dynamics https://ek.biem.sumdu.edu.ua/

1. LECTION 5
LINEAR DISCRETE DYNAMIC
MODELS OF ECONOMIC SYSTEMS
Lecturer: PhD Iryna Didenko

2. PLAN
 Algorithm for constructing a phase portrait of a
linear dynamic system.
 The main isoclines.
 Special phase trajectories of the node and saddle.
 Direction of movement on phase trajectories.
 Examples.
2

3. ALGORITHM FOR CONSTRUCTING A PHASE
PORTRAIT OF LDS
3
1. Calculation of the determinant ( ) and trace ( ) of
the corresponding matrix.
2. Finding the eigenvalues of the matrix of the previous
system from Equation .
3. If , determining the equilibrium position by solving
the system.
4. Transition to p. 4. If , - p. 7.
5. Determination of the type of equilibrium point and its
stability according to the table.
6. Finding the equation of the main isoclines and
plotting them on the phase plane.
 








d
c
b
a
0
)
(
2




 bc
ad
d
a 

0









0
0
2
1
2
1
dx
cx
bx
ax
0



4. АЛГОРИТМ ПОБУДОВИ ФАЗОВОГО ПОРТРЕТУ
ЛІНІЙНОЇ ДИНАМІЧНОЇ СИСТЕМИ (ПРОДОВЖЕННЯ)
4
6. If the equilibrium position is a saddle or a node, it is
necessary to find those phase trajectories that lie on
straight lines passing through the equilibrium point.
Go to item 9.
7. Determination of the type of equilibrium points and
their equations (in the case of direct equilibrium
points) according to the table.
8. Finding other phase lines as a solution of the integral
.
9. Construction of phase trajectories.
10. Determining the direction of movement along phase
trajectories and depicting it with arrows on phase
lines.
 dx
dy

5. VERTICAL ISOCLINE
5
.
)
,
(
)
,
(
2
1
2
1
2
'
2
2
1
2
1
1
'
1











dx
cx
x
x
f
x
bx
ax
x
x
f
x
A vertical isocline – a set of
points in the phase plane at
which a tangent to the
phase trajectory is parallel to
the vertical axis. Since at
these points ,
for
system (2.12) the equation of
the vertical isoline has the
form
(2.12)
0
)
(
'
1 
t
x
0
2
1 
bx
ax

6. HORIZONTAL ISOCLINE
6
.
)
,
(
)
,
(
2
1
2
1
2
'
2
2
1
2
1
1
'
1











dx
cx
x
x
f
x
bx
ax
x
x
f
x
Horizontal isocline – a set of
points in the phase plane, in
which the tangent to
the phase trajectory is parallel
to the horizontal axis. Since
at these points,
for system (2.12) the equation
of the vertical isocline (ВІ) has
the form
(2.12)
0
)
(
'
2 
t
x
0
2
1 
 dx
cx
The equilibrium point on the phase plane is the point of intersection of
the main isoclines.

7. THE EQUILIBRIUM POINT ON THE PHASE PLANE
IS THE POINT OF INTERSECTION OF THE MAIN
ISOCLINES.
For the equilibrium position of the saddle or node, there are
phase trajectories that lie on straight lines passing through
the origin.
7

8. THE EQUILIBRIUM POINT ON THE PHASE PLANE IS THE
POINT OF INTERSECTION OF THE MAIN ISOCLINES.
The equations of such lines can be found in the form:
(1)
If we substitute this expression in:
(2)
then to determine k we get:
or (3)
1
2 kx
x 
2
1
2
1
1
2
bx
ax
dx
cx
dx
dx



bk
a
dk
c
k



.
0
)
(
2



 c
k
d
a
bk
8

9. 9

10. EXAMPLE
For a given matrix
find the equation of the line corresponding to the
smallest eigenvalue by the modulus through the
solution of the matrix equation :
.
3
1
2
4








0
)
( 
 X
E
A 
10
10
1
2
3
4 





7
3
4 


 0
10
7
2


 
 5
,
2 2
1 
 

0
2
0
0
2
3
1
2
4
































y
x
,
0
1
1
2
2

















y
x
0
2
2 
 y
x
0

 y
x .
x
y 


11. THE TYPES OF EQUILIBRIUM POINTS AND LINES
USED TO REPRESENT THE CORRESPONDING
PHASE PORTRAIT
The types of
equilibrium
Lines used to represent the corresponding phase
portrait
Saddle Hyperbolas and lines (separatrices)
Node Parabolas and the line to which they touch
Focus Spirals
Center Closed circles or ellipses
Others Lines
11

12. DIRECTION OF MOVEMENT ON PHASE TRAJECTORIES
It is indicated by arrows
If the position of equilibrium is a node or focus, the
direction of movement along the phase trajectories is
uniquely determined by its stability (before the
coordinates) or instability (from the origin).
12

13. DIRECTION OF MOVEMENT ON PHASE
TRAJECTORIES
In the case of focus, it is necessary to set the direction of
"twisting" ("untwisting") of the spiral - clockwise or
counterclockwise.
Determined by the sign of the derivative )
(
'
2 t
x at the axis points х. 13

14. DIRECTION OF MOVEMENT ON PHASE
TRAJECTORIES
If
of a point moving along the phase trajectory when crossing
the "positive" beam of the axis х1 increases.
The "twisting" ("untwisting") of the trajectories occurs
counterclockwise.
0
1
0
'
2
2



cx
x
x
when 0
1 
x , is the ordinate
14
If
("untwisting") of the trajectory occurs clockwise.
0
1
0
'
2
2



cx
x
x
when 0
1 
x , then "twisting"

15. DIRECTION OF MOVEMENT ON PHASE
TRAJECTORIES
If the equilibrium position is the center, then the
"twisting" ("untwisting") of the movement along the phase
trajectories is established in the same way as in the case
of the focus.
In the case of a saddle, movement along one of its
separatrixes occurs in the direction of the origin of
coordinates, along the other - away from the origin of
coordinates. For all other phase trajectories, the
movement occurs in accordance with the movement along
the separatrixes.
15

16. DIRECTION OF MOVEMENT ON PHASE
TRAJECTORIES
It is enough to set the
direction of movement along
any one trajectory, and then
you can unambiguously set
the direction of movement
along all others.
The movement along the separatrix corresponding to
the negative eigenvalue occurs to the equilibrium point.
16

17. EXAMPLES
,
2
2
2
)
1 '
'









y
x
y
y
x
x
,
3
2
4
)
2 '
'







y
x
y
y
x
x
,
2
4
4
)
3 '
'








y
x
y
y
x
x
,
4
)
4 '
'







y
x
y
y
x
x
,
3
)
5 '
'








y
x
y
y
x
x
,
2
2
4
)
6 '
'








y
x
y
y
x
x
.
2
4
2
)
7 '
'







y
x
y
y
x
x
17

18. 18

19. 19

20. 20

21. 21

22. 22

23. 23

24. 24

25. 25