Page 73 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

56 SLENDER STRUCTURES AND AXIAL FLOW

Figure 2.13 Phase-plane trajectories for the system of equation (2.166). Here three stable fixed
points are shown, and two unstable ones (saddle points, denoted by A). Each stable fixed point is
encircled by an unstable limit cycle (---), and farther out by a stable limit cycle (dark oval patch).

to that about the origin: an unstable limit cycle (dashed line) and a stable one farther
out (dark oval). The main difference is that trajectories beyond, e.g. for 1x1 =- 20, cannot
escape to another saddle, as there is none. It could be argued that the structure around
[ 15,O) is qualitatively similar to that about {0, 0} because (i) both fixed points are stable
(hence the two points are statically similar) and (ii) g(i) is invariant to the transformation
y = x - 15, y = i; similarly for the dynamics about {-15, O}. However, such arguments
constitute but pn'nta facie evidence and are not always reliable, as will be demonstrated
for the system of equation (2.167).
For a physical system, the following dynamical behaviour is implied by the results
of Figure 2.13: (i) very small perturbations about the static equilibrium die out, and the
system returns to the origin; (ii) perturbations of amplitude larger than that of the unstable
limit cycle lead the system away from equilibrium and into limit-cycle oscillations (i.e.
to the larger, stable limit cycle); (iii) for still larger perturbations, the system is attracted
by either this same limit cycle or beyond, to the other limit cycles, around xst = f15.
Usually, all the features described in the foregoing do not occur for the same parameter;
as the parameter (U in this case) is varied, some arise, while others disappear. The
apparition of any new feature in the system defines a new bifurcation. Thus, for a certain
U, perhaps the only notable feature may be the stable fixed point along with the saddle
points, which could have arisen earlier via a pitchfork bifurcation. This feature could
remain, or disappear via a merging of these two points. At a higher U, the limit cycle(s)
may emerge via a Hopf bifurcation.