Duffing equation (strange attractor)

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).
Since the equation is not autonomous, the full phase space is that of triplets (t, x, v) or, since t only appears in the cosinus, (ωt mod 2π, x, v). The simulation will display a projection of the orbit in a space in which the phase ωt mod 2π coils around an axis.
The simulation will also display 3 stroboscopic Poincaré sections defined by the conditions ωt mod 2π = φ, φ+2/3π, φ+4/3π.
The point of view of the three-dimensional projections can be changed with the cursors or the mouse.
Each whole image can be moved with the mouse while pressing Ctrl.
To change the zoom in the projections, press Shift when moving up or down the mouse pointer.
Put the mouse pointer over an element to get information about it.

Activities

Uncheck the options Sections and Axes.
Start the simulation by pressing Start. It is difficult to understand the three dimensional orbit.
Check the options Sections and Axes.
After a couple of seconds press Clear to erase the initial transitory, i.e., to wait until the system is on the attractor (well, very very close to it).
You will see three stroboscopic sections. (You may uncheck Orbit to see them better.)
To explore (faster and in greater detail) one section use the simulations Duffing_Poincare and Duffing_baker.

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.