Duffing equation

• 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 either display. 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).
• The upper graphics will display the simultaneous positions and velocities of the N solutions at every time t = 2 n π / ω, for (n = 0, 1, 2,...), i.e., with the frequency of the external force.
• In the lower graphics the Poincaré section corresponding to the middle initial conditions will be computed step by step at times t = 2 n π / ω, for (n = n₀, n₀+1, n₀+2,...), where n₀ is the numbers of initial cycles to be skipped (this is done to make sure the solution is, approximately, on the attractor).
• To change the ranges displayed in each window you may right click on it and use the corresponding menu under the entry called Zoom
• For information on other elements, put over them the mouse pointer to get a tooltip.

Activities

1. Solve the system with the default settings. Discuss the similarities and differences between the two graphics.
   
2. Select a high value for N and look at the screen from a couple of meters: comment what you see.
3. Try different values of N (and R) to get a good compromise between display quality and computing time.
4. What happens in non-chaotic (conservative or dissipative) cases.
5. Set γ = 0.1 and f = 0.3. Describe what is happening.