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 | | From: | John Bailey | | Subject: | Re: Neato chaotic equations for analog computers to display? | | Date: | Wed, 22 Dec 2004 12:32:13 GMT |
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 | On 20 Dec 2004 08:04:26 -0800, shoppa@trailing-edge.com wrote:
>I had some fun this past weekend building an analog computer to
>integrate the Lorenz equations. I started with Paul Horowitz's
>design at
>
>http://frank.harvard.edu/~paulh/misc/lorenz.htm
>
>and added some frills like a rotary switch to select the
>integration capacitor sizes and 10-turn pots and knobs for the
>s, r, and b parameters that allow you to turn them and see the
>attractor change in real time as you twist knobs. Lotsa fun.
>
>Are there any other simplistic chaotic systems to try next? Having
>a small number of parameters is good (to keep the number of knobs
>reasonable) and analog multipliers aren't the cheapest thing in the
>world so it's nice to keep the number of analog multipliers
>necessary small too."
(snip)
>Any other ideas you guys might have?
>
>Tim.
>
There was a useful thread on sci.fractals in 1997
In article <33019a72.13686...@nntp.sn.no>,
>Arne.Dehli.Halvor...@computas.no (Arne Dehli Halvorsen) wrote:
>> A new, simple attractor has been discovered which exhibits rotational
>> symmetry around the axis x, y, z
>> (This means that in its definition, x is to y and z
>> as y is to z and x
>> as z is to x and y)
>> The definition:
>> dx/dt = -ax-4y-4z-y*y
>> dy/dt = -ay-4z-4x-z*z
>> dz/dt = -az-4x-4y-x*x
If you convert the three symmetrical differential equations to
an analogous difference equation form, you can get striking
chaotic behaviour with a spreadsheet (I used Microsoft Excel)
ODE form
> dx/dt = -ax-4y-4z-y*y
> dy/dt = -ay-4z-4x-z*z
> dz/dt = -az-4x-4y-x*x
> dz/dt = -az-4x-4y-x*x
Difference equation form
x(new)=Ax(old)+Bsum(old)+Cy*y
y(new)=Ay(old)+Bsum(old)+Cz*z
z(old)=Az(old)+Bsum(old)+Cx*x
sum(old)=x(old)+y(old)+z(old)
Typical values for constants: A= 1.03, B= -0.09, C= 1/1000
http://groups-beta.google.com/group/sci.math/index/browse_frm/thread/c55e66c8b593be2/201768ef4648423d?q=%22difference+equation%22+author:jmb184@servtech.com&_done=%2Fgroups%3Fas_q%3D%26num%3D10%26scoring%3Dr%26hl%3Den%26ie%3DUTF-8%26as_epq%3Ddifference+equation%26as_oq%3D%26as_eq%3D%26as_ugroup%3D%26as_usubject%3D%26as_uauthors%3Djmb184@servtech.com%26lr%3D%26as_drrb%3Dq%26as_qdr%3D%26as_mind%3D1%26as_minm%3D1%26as_miny%3D1981%26as_maxd%3D22%26as_maxm%3D12%26as_maxy%3D2004%26safe%3Doff%26&_doneTitle=Back+to+Search&&d
If that URL doesn't work, a google search for the lead posting might:
Message-ID: <33019a72.13686640@nntp.sn.no>#1/1
(end of quote)
Plotting any two of the three variables against each other you get a
plot not unlike a nonlinear pendulum plot. It was analyzed by
participants in the thread who said:
hendrik richter says:
>I have checked the discrete-time system with the given constants for
>Lyapunov exponents and it's indeed chaotic: The three LE are
>lambda=(0.4848,-0.0072,-0.5115).
And lastly:
An even simpler version based on symmetrical differential equation
model turns out to be:
x(new)= x(old)-(4*sum(old)-y(old)*y(old))/K
y(new)= y(old)-(4*sum(old)-z(old)*z(old))/K
z(new)= z(old)-(4*sum(old)-x(old)*x(old))/K
where sum(old) = x(old)+y(old)+z(old)
The value of K can range from 5.5 to 11.5
The following are Excel formulas which can be
iterated to produce successive values
of the three variables.
D2=4*SUM(A2:C2)
A3=A2-($D2-B2*B2)/$B$1
B3=B2-($D2-C2*C2)/$B$1
C3=C2-($D2-A2*A2)/$B$1
A three lobed scramble of orbits results.
(this quoting myself)
John Bailey
http://home.rochester.rr.com/jbxroads/mailto.html
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