I came across an interesting paper [1] which discusses the Lorenz model of general circulation of the atmosphere. The model comprises the following three nonlinear ordinary differential equations,
dX/dt = -Y2 - Z2 -aX +aF
dY/dt = XY - bXZ - Y + G
dZ/dt = bXY + XZ - Z.
Values of a = 1/4, b = 4 and G = 1 are used in [1]; F is given various values.
What's interesting (for me) is that these equations can be implemented on the analogue computer as it stands - just. It makes something of an interesting case to attempt on the computer in its current state. And a good set of equations to try and implement. The important thing is to have big sheets of paper!
Here's a sketch of how I converted the equations into a patching diagram. (At the moment, I am trying to avoid cases which need re-scaling, and so it's a very direct implementation of the set of equations.)
It uses seven of the eight integrator/summers (3 are configured as integrators), plus three of the four available summers, plus two of the three multipliers, and both squarers. Having never programmed an analogue compute before, it's interesting to see what parts of the machine are actually the most useful. For the current problem, the non-inverting summers proved particularly handy. And being able to configure the integrator/summers as either integrator or summer is very useful.
The biggest problem was a rather mundane one of nearly running out of banana leads (I have more on order):
The results (plotted on a Philips PM8043 X-Y recorder) are shown below, for three different values of parameter F, near to the value of 5.198 used for attractor N of reference [1]. Vertical axis is Z(t) , horizontal axis is X(t). a = 1/4, b = 4 and G = 1, as in [1].
An immediate discovery was that the thing is pretty sensitive to the value of F - in particular, the left hand side of the plots. I guess that's chaos for you. Nevertheless, the right hand side in particular, looks pleasingly similar to that given in [1], reproduced below:
Finally, it's incredibly easy to change something - in this case the value of G from 1 to 2 - just by moving one banana plug...and yielding another interesting plot:
This whole area of chaotic dynamics obviously has a very great depth...and as Masoller et al rightly say, is a fascinating subject [1].
[1] 'Characterization of Strange Attractors of Lorenz Model of General Circulation of the Atmosphere', C. Masoller et al, Chaos, Solitons and Fractals, Vol. 6, 1995, pp. 357 - 366.
dX/dt = -Y2 - Z2 -aX +aF
dY/dt = XY - bXZ - Y + G
dZ/dt = bXY + XZ - Z.
Values of a = 1/4, b = 4 and G = 1 are used in [1]; F is given various values.
What's interesting (for me) is that these equations can be implemented on the analogue computer as it stands - just. It makes something of an interesting case to attempt on the computer in its current state. And a good set of equations to try and implement. The important thing is to have big sheets of paper!
Here's a sketch of how I converted the equations into a patching diagram. (At the moment, I am trying to avoid cases which need re-scaling, and so it's a very direct implementation of the set of equations.)
It uses seven of the eight integrator/summers (3 are configured as integrators), plus three of the four available summers, plus two of the three multipliers, and both squarers. Having never programmed an analogue compute before, it's interesting to see what parts of the machine are actually the most useful. For the current problem, the non-inverting summers proved particularly handy. And being able to configure the integrator/summers as either integrator or summer is very useful.
The biggest problem was a rather mundane one of nearly running out of banana leads (I have more on order):
The results (plotted on a Philips PM8043 X-Y recorder) are shown below, for three different values of parameter F, near to the value of 5.198 used for attractor N of reference [1]. Vertical axis is Z(t) , horizontal axis is X(t). a = 1/4, b = 4 and G = 1, as in [1].
 |
| F = 5.1 V |
 |
| F = 5.2 V |
 |
| F = 5.3 V |
 |
| F = 5.2 V; G = 2 V |
[1] 'Characterization of Strange Attractors of Lorenz Model of General Circulation of the Atmosphere', C. Masoller et al, Chaos, Solitons and Fractals, Vol. 6, 1995, pp. 357 - 366.
