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Mathematical Representation of System Dynamics Models

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Mathematical Representation of System Dynamics Models. Vedat Diker. George Richardson ... The rate of change is represented by a derivative. You can use any ... – PowerPoint PPT presentation

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Title: Mathematical Representation of System Dynamics Models


1
Mathematical Representation of System Dynamics
Models
  • Vedat Diker
  • George Richardson
  • Luis Luna

2
Our Todays Objectives
  • Translate a system dynamics model to a system of
    differential equations
  • Build a system dynamics model from a system of
    differential equations

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
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

5
Introduction
  • System Dynamics describe systems in terms of
    state variables (stocks) and their rates of
    change with respect to time (flows).

State
Rate of change
6
Mathematical Representation
Interest Interest rateMoney in Bank
7
In General
X
8
In General
  • 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
Another Example
(initial 1000)
(0.03)
(65 years)
10
A Two Stock Model
(0.0005)
(0.04)
(3200)
(20)
(0.2)
(0.2)
11
Another Population Model
(0.03)
(0.005)
(1000)
(10000)
(3)
12
How to Describe a Graphical Function?
y (effect of)
x (some ratio)
13
In summary
f (x)gt0 Þ f(x) f (x)lt0 Þ f(x) f (x)gt0 Þ
f(x) f (x)lt0 Þ f(x)
14
Can We Do the Opposite?
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
A Closer Look
f(2)2
f(0)0
f(1)1
17
A Closer Look
Slope is positive f (x) is positive f (x)gt0
18
A Closer Look
The slope is increasing f (x) is increasing
f (x)gt0
19
A Closer Look
The slope is decreasing f (x) is decreasing
f (x)lt0
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