IV Chaotic motions
The experience with the driven damped spring according to which the attractor of the driven case is of the type of the undriven, undamped motion, would suggest that adding a temporally periodic driving to the acceleration law of the damped repelling-attracting spring would lead to a long-term motion similar in its form to that of the undriven, undamped case. The attractor would then be periodic, but of a different shape than in the linear case. This is, however, not the case! The seemingly minor change of the appearance of a nonlinear term proportional to $x^3$ in the force leads in the driven version to a qualitative difference: the motion becomes chaotic.
Chaotic systems. To see such a case, study the motion of the damped repelling-attracting spring arising from acceleration $$ a(x,v,t)=x-x^3-0.2v+0.3\cos(t) $$ numerically.
As this worksheet illustrates, there is absolutely no sign of periodicity either in the $x(t)$ or in the $v(t)$ function. This is surprising since the external force is strictly periodic. The term chaos is motivated by the characteristic feature that no simple rules can be derived when observing the motion: it is aperiodic without any repetition, even if observed for a long time (the system is monitored here for more than 11 periods of the drive). This feature is reflected in the $x,v$ representation, too: what we see is a funny ‘coil’ without any clear structure. The long-term motion runs on a chaotic attractor, a much more complicated object than a periodic one, which would be a closed curve.
Butterfly effect. Change the initial condition by a small amount, say add 0.0001 to $x_0$, and check the immediate reaction in the shape of the $x(t), v(t)$ functions and in the $x,v$ view.