The parameters used to get this plot are:
initial position = 1.0
initial velocity = 0.0
forcing amplitude = 0.22
forcing frequency = 1.0
dampening constant = 0.25
It is seen that the double well oscillator can produce stable limit cycles. Any perturbation from this stable cycle will eventually fall back into the cycle, so long as the perturbation doesn't push the oscillator over to the other side. This solution predicts that the oscillator will rock back and forth over top of one magnet, never gaining enough energy to escape it's grasp. This is the kind of solution that everyone is familiar with, periodic and stable. So far the equation has not produced anything too interesting.
Here are position and velocity plots for this same parameters . We note that there is another symmetric stable cycle on the other side of the zero line.
Position vs. Time Plot
Velocity vs. Time Plot
This is certainly not the only behavior that this equation can produce, so let's proceed and see what else can be discovered in this equation.