Fig. 3. Return maps y_i+1(y_i) of the apex height y_APEX of two consecutive flight phases (index i and i+1) for three different retraction speeds _R (A, _R=0 deg s^-1; B, _R=25 deg s^-1; C, _R=50 deg s^-1).The system energy corresponds to a running speed of 5 m s^-1 at an apex height y_APEX=1 m. (A–C) Three characteristic return maps represent the minimum, mean and maximum retraction angle _R (see key in each panel) for stable fixed points (see text, Equation 2). With increasing retraction speed _R, the range of retraction angles _R with stable fixed points increases, and attraction of higher apex heights is observed (max. y₀1.3, 1.9, 2.2 for _R=0, 25, 50 deg s^-1, respectively) as shown by representative tracings (running sequences are indicated by stepped black lines with starting arrows).