return trajectories are plotted in Figure \[fig:sao\]. The production of small-amplitude oscillations is dominated by the saddle-focus mechanism: in the example shown, the red orbit exhibits four oscillations before the (relatively) large-amplitude return, whereas the blue orbit exhibits seven oscillations. We can select trajectories with increasing numbers of small-amplitude oscillations by picking points closer to $W^s \cap \{z=0\}$. A complication in this analysis is that jumps at intermediate heights, which are clearly shown to occur in these examples, blur the distinction between ‘large’ and ‘small’ oscillations in a mixed-mode cycle.

\(a) ![\[fig:2dmap\] (a) Partition of a compact subset of the cross-section $\Sigma_0 = \{z = 0\}$. Black dashed line is the tangency of the vector field $\{\dot{z} = 0\}$, separating the subset $\{\dot{z} > 0\}$ (black points) from the other subsets. Yellow (resp. green): points above (resp. below) the line $\{y = 0\}$ with winding number less than three. Red (resp. blue): points whose forward trajectories reach a maximal height greater than (resp. less than) 0.18 and have winding number three or greater. (b) Overlay of red and blue subsets of domain (points) with images of yellow, green, and black subsets (crosses). (c) Overlay of red and blue subsets of 