Duffing equation

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' for the middle point of the set of N solutions that are simultaneously computed. Initial conditions for that point can also be selected by moving with the mouse the point on the display Phase space. You can also select the form (filled square, hollow square or circle) of the set of initial conditions, as well as its diameter d.
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.

Activities

Set γ = 0 and f = 0 and solve the system with the default initial conditions. May you describe and explain what is happening?
Try different initial conditions to see non-linear periodic oscillations.Try also initial conditions near (x,x') = (0,0).
What happens with the phase surface: is it conserved?
Repeat the previous analysis with γ = 0.1
Put now f = 0.1. What happens?
Set γ = 0.1 and f = 0.3. Describe what is happening.

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.