Duffing equation (Poincaré section)

• 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 a stroboscopic Poincaré section defined by a condition in the form
  ωt mod 2π = φ.

Activities

1. Start the simulation by pressing Start. By default the section will be erased after the first 10 cycles to skip the transitory until the system is on the attractor (well, very very close to it).
2. Use different initial values for x and v. Is there any difference in the displayed section? Discuss your answer.
3. Use the mouse to select smaller and smaller parts of the attractor section. The increased resolution will show that the attractor has structure at all magnification levels: it is a fractal, much like a Cantor set.
4. Notice that at higher magnifications you will have to wait longer to get a number of solution points high enough to see the attractor.
5. Press Window to recover the original magnification.
6. Change φ by steps from 0 to 2π to get an idea of the full attractor in the three-dimensional space (t,x,v). You can see at work one of the mechanisms of chaos: the so-called baker transformation, which is better shown in Duffing_baker.
7. You can run the examples Duffing1, Duffing2, Duffing3 and Duffing4 of Dynamics Solver for faster simulations, which allow exploring the Cantor set structure of the attractor, as well as the baker transformation. The sensible dependence on initial conditions is shown in Duffing5.