Chaos and System dynamics

Leon Chang

Edward Lorenz

In the early 1960's using a simple

system of equations to model con

vection in the atmosphere, Edwar

d Lorenz, an MIT meteorologist,

ran headlong into "sensitivity to i

nitial conditions". In the process

he sketched the outlines of one of

the first recognized chaotic attract

ors.

The Butterfly Effect

The "Butterfly Effect" is the propensity of a system to be sensitive to initial conditions.Such systems over time become unpredictable,this idea gave rise to the notion of a butterfly flapping it's wings in one area of the world,causing a tornado or some such weather event to occur in another remote area of the world

Lorenz model

bzxyz

yrxxzy

yxx

H ot

C ool

Lorenz waterwheel

Lorenz model by system dynamicsX

dXdt

Z

Y

dZdt

dYdt

S

B

RminusZ

R

bzxyz

yrxxzy

yxx

Lorenz result

10:13 PM 2004¦~4¤ë3¤é

1.00 13.25 25.50 37.75 50.00

Months

1:

1:

1:

-25.00

5.00

35.00

1: Y

1

1

1

1

Graph 1 (Untitled)

r=28.00

03:41 PM 2004¦~4¤ë4¤é

1.00 13.25 25.50 37.75 50.00

Months

1:

1:

1:

-30.00

0.00

30.00

1: Y 2: Y 3: Y

1

1

1

1

22

2

2

3

3

33

Graph 1 (Untitled)

1: r=28.002: r=28.013: r=28.03

Lorenz result comparison

Strange attractor by X-Y

10:18 PM 2004¦~4¤ë3¤éX

-25.00 0.00 25.00-25.00

5.00

35.00

1: X v. Y

Graph 1 (Untitled)

08:54 PM 2004¦~4¤ë4¤éZ

5.00 30.00 55.00-25.00

5.00

35.00

1: Z v. Y

Graph 1 (Untitled)

Strange attractor by Y-Z

10:25 PM 2004¦~4¤ë3¤éX

-25.00 0.00 25.00-250.00

0.00

250.00

1-2: X v. dXdt

Graph 1 (Untitled)

Strange attractor by X-dX/dt

Lorenz attractor

成長上限

銷售能力和規模

訂單數量

營業收入

欠貨數量

交貨期

銷售困難度

+

++

+

+

-

+

+

++ -

成本與投資不足

成長的行動

需求

績效

認知的投資需求

產能

產能的投資

+ -

-

Using system dynamics to analyse interactions in duopoly competitionPetia Sicea, Erik Mosekildeb, Alfredo Moscardinic, Kevin Lawlerc and Ian Frenchd*System Dynamics Review Vol. 16, No. 2, (Summer 2000): 113–133

Phase planeportrait (FQ against CQ)illustrating the singleperiodlimit cyclebehaviour observed fora = 2 and c = 0.1

Phase planeportrait (FQ against CQ)illustrating the chaoticbehaviour for a = 2 andc = 0.4

Plot of D overthe period 750 to 7000months; the straightline, which has slope0.005, represents the‘best fit’ over theperiod 1500 to 4500

Fractal