Duffing equation (Baker transformation)

This simulation explores the Duffing equation, which reads (in dimensionless variables) as follows:
x'' + 2 γ x' - x (1-x²) = f cos ωt
where each ' denotes a time derivative.

You can select below the parameters γ and f, as well as the initial conditions for the elongation x and the velocity v = x'.
The unit time is 1/ ω (so that ω = 1).
For information on other elements, put over them the mouse pointer to get a tooltip.
The simulation will display N² stroboscopic Poincaré sections defined by a series of conditions in the form
ωt mod 2π = φ + 2 n π/N²
for n = 0, 1, …, N²-1 (from left to right and then from top to bottom)

Activities

Start the simulation by pressing Start. According to the default settings, the first ten cycles will be erased to wait until the system is on the attractor (well, very very close to it).
Use different initial values for x and v. Is there any difference in the displayed section? Discuss your answer.
You can see the stretch-and-fold mechanism of the baker transformation at work.
Discuss how the stretching generates deterministic chaos, i.e., sensible dependence on initial conditions.
Discuss how the folding makes the attractor bounded while generating its fractal (Cantor-like) structure.
You can run the examples Duffing2, Duffing3 and Duffing4 of Dynamics Solver for faster simulations of the baker transformation. The sensible dependence on initial conditions is shown in Duffing5.

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.