System dynamics math representation

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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

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