A spring in a smooth table

A particle of mass m moves without friction in a horizontal table while tied to a fixed point through an elastic spring of constant k and length l. The equations of motion are as follows (each prime is a time derivative):
m x''+k [1-l (x²+y²)^-1/2] x=0
m y''+k [1-l (x²+y²)^-1/2] y=0
On the right the mass orbit is displayed, along with the return circles (i.e., the maximum and minimum distances in the orbit).

Activities

Select Graph to view the evolution in the equivalent one-dimensional problem.
Select different initial condition and parameters and observe that, in general, orbits are open, i.e., non periodic. This is what one would expect due to Bertrand's theorem (which states that in a central forces field all bounded orbits are closed, the force is Newtonian, F∝-r^-2, or harmonic, F∝-r.
However, nothing prevents the existence of isolated periodic orbits. Try, for instance,
x = -0.854775390624998, y = 1.155, vx = 0, vy = -0.1, k/m = 1.
How small must be a change in the above initial conditions to get an orbit which is quickly recognized as non periodic?
With a bit of patience, you may ba able to find (good approximations to) other periodic orbits.

This is an English translation of the Basque original for a course on mechanics, oscillations and waves.
It requires Java 1.5 or newer and was created by Juan M. Aguirregabiria with Easy Java Simulations (Ejs) by Francisco Esquembre. I thank Wolfgang Christian and Francisco Esquembre for their help.