Anti-Newtonian Dynamics Animations

Vladimir Zhdankin and J. C. Sprott
Department of Physics, University of Wisconsin, Madison, WI 53706, USA
April 28, 2009

This animation shows a trivial example of a fox encircling a rabbit with both in synchronous concentric circular orbits using the anti-Newtonian approximation (forces are equal in magnitude and direction). The fox has half the mass of the rabbit, and the motion is frictionless The force law is 1/R in two spatial dimensions. (Actually, the force law doesn't matter for this case since the separation is constant.) The dynamics are periodic with an invariant circle in the 8-dimensional phase space.

This animation shows an example of a fox encircling a rabbit with both in off-centered nested elliptical orbits using the anti-Newtonian approximation (forces are equal in magnitude and direction). The fox has half the mass of the rabbit, and the motion is frictionless The force law is 1/R2 in two spatial dimensions. The dynamics are periodic with an invariant circle in the 8-dimensional phase space.

This animation shows the motion of a fox chasing a rabbit using the anti-Newtonian approximation (forces are equal in magnitude and direction). The fox has half the mass of the rabbit, and the motion is frictionless The force law is 1/R in two spatial dimensions. The dynamics are quasi-periodic with an invariant torus in the 8-dimensional phase space.

This animation shows the motion of a fox chasing a rabbit using the anti-Newtonian approximation (forces are equal in magnitude and direction). The fox has twice the mass of the rabbit and 1/10 of the friction. The force law is 1/R in two spatial dimensions. The dynamics are quasi-periodic with an attracting torus in the 8-dimensional phase space. The distance of closest approach is 1.637298.

This animation shows the motion of a fox chasing a rabbit using the anti-Newtonian approximation (forces are equal in magnitude and direction). The fox has twice the mass of the rabbit and half the friction. The force law is 1/R in two spatial dimensions. The dynamics are chaotic with a strange attractor in the 8-dimensional phase space. The distance of closest approach is ~0.1.

This animation shows the motion of two foxes chasing a rabbit using the anti-Newtonian approximation (forces are equal in magnitude and direction). The foxes have twice the mass of the rabbit and 1/3 of the friction. The force law is 1/R in two spatial dimensions, and the foxes do not interact. The dynamics are chaotic with a largest Lyapunov exponent of 0.1346 and a strange attractor in the 12-dimensional phase space.