Fig. 6.29 Euler loops of a buckled elastica with a periodical imperfection. The figures show
solutions to (6.65) for a = 0.01, xl = 200, u(0) = 3.1, and _
uð0Þ ¼ 0. (a) Undeformed elastica,
definition of variables; (b) ^ k
2 ¼ 0:001; (c) ^ k
2 ¼ 0:01; (d) ^ k
2 ¼ 0:1; (e) ^ k
2 ¼ 1. In (e) loops appear
seemingly at random (elastostatical chaos)
Fig. 6.30 (a) Poincaré map for (6.65). (b) Enlargement of the boxed region in (a). (Parameters as
for Fig. 6.29(e), except that xl = 50,000)
380
6 Chaotic Vibrations