Plot of as a function of , for and three different values of the energy: , and .
Phase space trajectories, for , in the neighborhood of the origin.
“Figure eight” in the plane corresponding to and energies (thick curve) and (dash curve).
Plot of as a function of , for () and three different values of the energy: (small thick curve), (middle curve), and (curve with great oscillations). The curve is very similar to a pure sine or cosine curve but, for higher values of , the plot shows clearly the effects of the nonlinearity.
Phase trajectories corresponding to four different values of the energy (, 0.4, 0.6, and 0.8). The trajectories are closed curves representing periodic motions that for small values of can be considered as rather similar to ellipses; for other values of the curves modify their shape and they lengthen toward the upper side of the phase plane. The motion is asymmetric in the sense that the particle moves from right to left in a slowly way but returns, from left to right, with a much higher velocity that takes its maximum value at the center point .
Closed trajectories (ellipses in the linear case) in the plane corresponding to and four different values of the energy (, 0.4, 0.6 and 0.8).
Nonlinear Lissajous figures in the plane corresponding to “figure eight” trajectories associated to and energies (small eight-looking curve) and (big butterfly-looking curve).
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