The following interesting system is given by Sprott in [1]:
Interesting because the system is either dissipative (exhibiting chaos) or conservative, dependent on the initial conditions.
Here's my patching for Sprott's system:
Results
The plotter results shown below were obtained with 0.2 µF integration capacitors throughout (with, as always, ×1 integrator inputs equating to 1 MΩ input resistances). The initial conditions were (2,0,0), corresponding to Sprott's first (red) solution [1].
 |
| XY-plane (analogue computer left (570 seconds of trajectory), MATLAB right (100000 steps of 0.005 seconds)) |
 |
| XZ-plane (240 seconds of trajectory) |
 |
| YZ-plane (600 seconds of trajectory) |
Interestingly, after a period of time (typically between 5 and 10 minutes ), the analogue computer result ceases being chaotic and settles into a stable(ish) periodic orbit - pretty much circular in the XY-plane, and corresponding (at least approximately) to the toroidal solution in Sprott's paper [1]. This is more apparent when the computer is run in FAST mode, with the output to the oscilloscope. The following results were obtained with 2 second exposure times:
 |
| Chaotic regime with periodic orbit. XY-plane. 0.5 V/div. |
 |
| Chaotic regime with periodic orbit. XZ-plane. 0.5 V/div. |
 |
| Chaotic regime with periodic orbit. YZ-plane. 0.5 V/div. |
The trajectory starts in the (dissipative) chaotic regime (expected with the initial conditions assumed, according to [1]) but, after a period of time, collapses into the conservative regime, ending in a periodic (quasi?) orbit. The latter appears similar in location / size to the (outer dimension) of the invariant torus of [1], reached with initial conditions (1,0,0).
It's interesting that, the obvious circular orbit visible in the XY-plane is offset by an amount along the positive x-axis which is very close to 1/√2, which seems connected with the comment in [1] regarding the boundary between conservative and dissipative regions.
I am thinking that this eventual failure of the analogue computer's solution (and move from dissipative to conservative regime) may be either due to (a) offset nulls in various parts of the analogue computer effectively acting as sources of energy, or (b) due to losses in the integration capacitors.
Reference
[1] J. C. Sprott, 'A dynamical system with a strange attractor and invariant tori', Physics Letters A, 378, 2014, pp. 1361 - 1363.
Interesting because the system is either dissipative (exhibiting chaos) or conservative, dependent on the initial conditions.
