Mathematical Representation
of System Dynamics Models
Vedat Diker
George Richardson
Luis Luna
Our Today’s Objectives




Translate a system dynamics model to
a system of differential equations
Build a system dynami...
Introduction


Many phenomena
can be expressed
by equations which
involve the rates of
change of quantities
(position,
po...
Introduction




The state of the
system is
characterized by
state variables,
which describe the
system.
The rates of ch...
Introduction


System Dynamics describe systems in terms
of state variables (stocks) and their rates of
change with respe...
Mathematical Representation
I
n
t
e
r
s

M
o
n
e
y
i
B
a
n
k

Interest= Interest rate*Money in Bank

I
n
t
e
r
s
a

x
d
x
...
In General
S
t
o
c
k
O
u
t
f
l
o
w

I
n
f
l
o
w

X

dx 
= x = net flow = inflow - outflow
dt
 dx
∆x changein x 

 dt ...
In General
dx 
= x = net flow = inflow - outflow
dt






This equation that describes a rate
of change is a different...
Another Example
(initial = 1000)
P
o
p
u
ln
a
t
i
o
B
i(
r
tB
h
s
)

B
if(
rtf
ta
h
c
i)
o
n

(0.03)

(
P
)

D
e
a
tD
h
s
...
A Two Stock Model

(0.0005)

(0.04)
Rso
aIw
briu
iett
ttch
N
n
e
a
Pc
riF
ern
dt(
aia
ta
o)
n
o

P
ri
e
d
a
t
o
n
F
rn
a
c...
Another Population Model
(0.03)

(0.005)

(1000)

C
u
r
e
n
t

B
if(
rtf
ta
h
c
i)
o
n

E
P
D
f

D
e(
a)
tc
h
fr
r
t
i
o
n...
How to Describe a Graphical
Function?
C
u
r
e
n
t

E
f

E
f

y (effect of…)

C
u
r
e
n
t

2

2
1
.
5

1
.
5

1

1
0
.
5

0...
In summary
f ’(x)>0 ⇒ f(x)
f ’(x)<0 ⇒ f(x)
f ’’(x)>0 ⇒ f(x)
f ’’(x)<0 ⇒ f(x)
Can We Do the Opposite?
dx
=y
dt
dy
k
c
= − x− y
dt
m
m
where :
k / m = 64
c / m = 0.2
xo = 4.5
y o = −0.45
Final ideas






Any System Dynamics model can be
expressed as a system of differential
equations
The differential equ...
C A Closer Look
u
r
e
n
t
E
f
2
f(2)=2

1
.
5

f(0)=0

1

f(1)=1

0
.
5
0

0

1

2
C A Closer Look
u
r
e
n
t
E
f
2

Slope is
positive

1
.
5

f ’(x) is
positive

1
0
.
5

f ’(x)>0

0

0

1

2
1
.
5

A Closer Look

1

0
.
5
0

0

The slope is increasing
f ‘(x) is increasing

1
X

f ’’(x)>0
A Closer Look

The slope is
decreasing
f ‘(x) is decreasing

f ’’(x)<0
Upcoming SlideShare
Loading in...5
×

System dynamics math representation

192

Published on

System Dynamics Math Representation

Published in: Education
0 Comments
1 Like
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total Views
192
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
Downloads
9
Comments
0
Likes
1
Embeds 0
No embeds

No notes for slide

System dynamics math representation

  1. 1. Mathematical Representation of System Dynamics Models Vedat Diker George Richardson Luis Luna
  2. 2. Our Today’s Objectives   Translate a system dynamics model to a system of differential equations Build a system dynamics model from a system of differential equations
  3. 3. Introduction  Many phenomena can be expressed by equations which involve the rates of change of quantities (position, population, principal, quality…) that describe the state of the phenomena.
  4. 4. Introduction   The state of the system is characterized by state variables, which describe the system. The rates of change are expressed with respect to time G rra aln p h fu o P p t i o 1 1 1 1 1 1 1 1 1 2 1 3 1 4 1 5 1 0 . 7 5 1 0 . 5 1 0 . 2 5 1 1 0 1 0 1 1 1 1 1 2 3 4 5 6 7 8 9 1 0 T ie m (d P r i) o Pu or pe un li:t a tn o C 1 1 1 1 1 1 1 1 1 1 Iu n d ia v l s Gc rran aet pp hr fro od A gi tu o 1 1 , 0 1 1 9 0 1 1 8 0 1 1 1 1 7 0 1 1 1 1 1 6 0 0 2 4 6 8 1 1 0 1 2 1 4 1 6 1 8 2 0 2 T i() m e Y e a r Anp gdsr rpB eue gt:ro aca triap eo om 1 1 1 1 1 1 1 D o ls a r 2 4
  5. 5. Introduction  System Dynamics describe systems in terms of state variables (stocks) and their rates of change with respect to time (flows). I n t e r s M o n e y i B a n k State I n t e r s Rate of change P e r c n t a g e
  6. 6. Mathematical Representation I n t e r s M o n e y i B a n k Interest= Interest rate*Money in Bank I n t e r s a x d x / t r  dx = r x or x = r x dt where : r = 0.15 x o = 100
  7. 7. In General S t o c k O u t f l o w I n f l o w X dx  = x = net flow = inflow - outflow dt  dx ∆x changein x    dt comes from ∆t = changein t    
  8. 8. In General dx  = x = net flow = inflow - outflow dt    This equation that describes a rate of change is a differential equation. The rate of change is represented by a derivative. You can use any letter, not just “x.”
  9. 9. Another Example (initial = 1000) P o p u ln a t i o B i( r tB h s ) B if( rtf ta h c i) o n (0.03) ( P ) D e a tD h s ( ) A v( en re as g lp ia f) s (65 years)
  10. 10. A Two Stock Model (0.0005) (0.04) Rso aIw briu iett ttch N n e a Pc riF ern dt( aia ta o) n o P ri e d a t o n F rn a c t( ib o ) (3200) R a b i( tR s ) R ah b( irI tt) B s R) at( bD ie ta D h s C o n ts a c ( N ) F o x e s ( F ) F) o x B is rO t( h Er fng ici ctn e y o f u pb ren eit dto as tr d a f( o x e s ) (0.2) (20) F) oT x D e a t( h s No atrn taa uc rai lh d ei f t af( bo sd ec n) c o (0.2)
  11. 11. Another Population Model (0.03) (0.005) (1000) C u r e n t B if( rtf ta h c i) o n E P D f D e( a) tc h fr r t i o n 8 6 4 2 0 P o p u ln a t i o ( P ) B i( r) tB h s (10000) D e a tD h s ( ) E fp et co tu o fa ln i dte ea n( sh iv te y) o r d s P o p u ln a t i o d e n s iE t) y ( A r e a ( A ) (3) N o rd m a l i z e d e n s i) t y ( N P oy p u ln a td ie o s i t n o r) m a l ( n E P D f 0 2 X 4
  12. 12. How to Describe a Graphical Function? C u r e n t E f E f y (effect of…) C u r e n t 2 2 1 . 5 1 . 5 1 1 0 . 5 0 . 5 0 0 1 X 2 0 0 x (some ratio) 1 X 2
  13. 13. In summary f ’(x)>0 ⇒ f(x) f ’(x)<0 ⇒ f(x) f ’’(x)>0 ⇒ f(x) f ’’(x)<0 ⇒ f(x)
  14. 14. Can We Do the Opposite? dx =y dt dy k c = − x− y dt m m where : k / m = 64 c / m = 0.2 xo = 4.5 y o = −0.45
  15. 15. Final ideas    Any System Dynamics model can be expressed as a system of differential equations The differential equations can be linear or non-linear (linear and non-linear systems) We can have 1 or more differential equations (order of the system)
  16. 16. C A Closer Look u r e n t E f 2 f(2)=2 1 . 5 f(0)=0 1 f(1)=1 0 . 5 0 0 1 2
  17. 17. C A Closer Look u r e n t E f 2 Slope is positive 1 . 5 f ’(x) is positive 1 0 . 5 f ’(x)>0 0 0 1 2
  18. 18. 1 . 5 A Closer Look 1 0 . 5 0 0 The slope is increasing f ‘(x) is increasing 1 X f ’’(x)>0
  19. 19. A Closer Look The slope is decreasing f ‘(x) is decreasing f ’’(x)<0
  1. A particular slide catching your eye?

    Clipping is a handy way to collect important slides you want to go back to later.

×