Fox and Rabbit - part the second

Here the fox and rabbit are free to move in two dimensions [1], thus:

where r is the Euclidean distance between fox and rabbit. This requires all eight integrators (and all four summers) of the computer:

In this patching, the mass of the rabbit is unity, the fox is twice as massive as the rabbit, and b_r = 1 and b_f = 0.1. Also, γ = -1 giving the required inverse square relationship.

To get the reciprocal of the square of the distance (r) squared, I used ln(x) and exp(x) functions. Since these actually give 3ln(x) and exp(x/2)-10 respectively (to make best use of the computer's range), I have to include the -2/3 between them, and then add ten volts, to give the required 1/r².

Fox and rabbit wiring. Analogue computer enclosure nearly complete.

Initial conditions were as per reference [1]. Results below.

Comments: the fox / rabbit trajectories start off similar to those in [1], but decrease in amplitude - I think this is probably due to leakage in the integrators. Also, the trajectories are sensitive to any offsets - particularly in the integrators. Also, some of the integrators / summers are clipping.

Reference

[1] 'Anti-Newtonian dynamics', J. C. Sprott, Am. J. Physics, 77 (9), September 2009, pp. 783 - 787.