The basin of attraction of Duffing's equation is infinite -- that is, any initial condition will lead to a solution that eventually settles down to a small amplitude oscillation. On the other hand, our ¶ map approximation in equation (6) will have only a finite basin of attraction. That is, there are initial conditions x(0), y(0) that will lead to a divergent sequence with x, y tending to infinity. The basin of attraction of our ¶ map approximation is defined as the region of the x, y-plane for which this does not happen, and is illustrated in figure 5.
Figure: The basin of attraction -- on and within the thick black curve --
for our 5th-order, 10 circle
approximation to the ¶ map with A = 13. The ¶ section is also
shown and clearly lies well within the basin of attraction.