Gumowski-Mira-Attractor

[josef]

I like Vensim PLE and I am preparing a paper for a teachers’ inservice course using PLE and other tools.
I wonder if it is possible to model the Gumowski-Mira-attractor given by the following system:

There is a function g(x) defined (specially g(x)=-0.9x 2*(1+0.9)*x^2/(1+x^2)

Then

xn = yo + a(1-b*yo^2)*yo + g(xo)
yn = -xo + g(xn)

x0 = y0 = 0.1 (initial values)
xn = x new; xo = x old

yn = y new; yo = y old

Many thanks in advance

Josef Boehm

[Administrator]

I don't see why you cannot do it. To get the old values of x, y and G, use DELAY FIXED with a delay time of one time step.

[tomfid]

Here's an example, which also shows in general how to handle discrete time math.

gumowski mira.mdl

(2.98 KiB) Downloaded 505 times

The behavior is really amazing.

Tom

[josef]

Hi Tom,

many thanks. This helps a lot.
You are right the graphs are very amazing.
Thanks also for the link,

Regards
Josef

[gwr]

Hi Tom,

I do not see any reason for using a DELAY FIXED here. You can handle discrete systems - e.g. difference equations' systems - using the regular System Dynamics notation. What you need to do is:

1. Set the time step to 1.

2. Convert the Stock Equations into difference equations, e.g. Delta X = X(t+1) - X(t) = dx/dt with dt = 1.

3. In the case of the Gumowski-Mira-System a bit of algebra will convince you that no delays are needed.

See the model enclosed.

Cheers,

Guido

[tomfid]

Guido,

You're entirely correct. The INTEG notation that you used is probably the nicest way to do this for formal correspondence with discrete derivative notation. However, as long as TIME STEP=1 and Euler or Diff integration is used, the results will be identical whether INTEG, SMOOTH, or DELAY FIXED is used. (Diff integration is the same as Euler, but the rates and levels are stored differently, which makes it easier to see the initial values of the rates - sometimes useful for discrete systems like this.)

Tom