Anisotropic oscillator

A mass can move without friction in a horizontal table while tied by equal springs from two fixed points. It is assumed that both springs are always straight.

The unit length is equal to half the distance between the fixed points and the springs have proper length L and negligible mass.
The unit time is (m/k)^1/2, where m is the mass and k is the spring constant.

Activities

Write down the equations of motion.
Show that when L goes through 1 a bifurcation happens: the number and stability of fixed points change.
Check the previous result in the simulation.
Since the oscillations are anisotropic and anharmonic, we expect the orbit to be different from Lissajous curves. Check this around each stable fixed point in the different cases.
When L = 1 and the initial horizontal displacement is much greater than the vertical one, which is the oscillation center? Why?

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.