Page 419 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

(a)

0.45
0.4

0.35 4
0.3
P
0.25

0.2
0.15

ni
". 1
0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 u - u,
(b) 65 (C)
Figure 5.56 (a) The sequence of states associated with Hopf-pitchfork heteroclinic cycles in 3-D motions of the pipe-spring system [from Steindl
& Troger (1996)l. (b) Map of stable Hopf-Hopf heteroclinic cycles (+) of the same system; ---, marks the boundary for the onset of the secondary
bifurcation. (c) Qualitative bifurcation diagram, showing the two principal limit-cycle branches for oscillations in one plane (SWl) or the other
perpendicular to it (SW2): -, stable solutions; ---, unstable solutions; HC marks the range of existence of heteroclinic cycles (Steindl 1996).