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PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 387

as u = [u - uint]'/*. Plots of T and u versus u for this problem show excellent agreement
with this scaling (Semler & Pdidoussis 1995).
One worry concerning this work with p < 0 is that the robustness of the results for
N > 4 has not been checked. In view of the high values of u necessary for the post-Hopf
bifurcations, especially for very small Ipl, N = 4 may well be insufficient.
To conclude, it is clear that the system with p = 0 is truly singular and relatively
dull, capable of limit-cycle oscillations 'only'! In contrast, both for p < 0 and p > 0, a
succession of interesting dynamical states and chaos generally follow the emergence of
limit-cycle oscillations.

5.8.4 Chaos near double degeneracies
As discussed in Section 5.7.3, a number of systems have been studied in the neigh-
bourhood of double degeneracies, in the process determining conditions, e.g. heteroclinic
orbits, which when perturbed could lead to chaos; indeed, in several cases, finding chaos
was the principal aim.
The first case to be discussed here is that of the so-called up-standing cantilever, in
which the double degeneracy involves coincident Hopf and pitchfork bifurcations in the
(u, p, y}-space, see Section 5.7.3(d). Keeping p fixed at 0.2, this double degeneracy occurs
at u, = 2.2458 and y, = -46.0014 for N = 2; the work that follows is for this particular
set of parameters. Furthermore, it is recalled that heterociinic orbits for this system arise
on a line in Figure 5.29 defined by equation (5.130), in which the constants are b = 1.518
and c = 3.954 for fi = 0.2. Thus, the system is studied at

u = u, + p and y = y, +x, (5.142)

where p and x are determined via (5.126) for the system to both be doubly degenerate
and to have heteroclinic orbits: u = 2.2466, y = -46.0200.
The system is perturbed by varying the nonlinear coefficient aijk/ in equation (5.1 16a),
+
a;,kt = (u, + ~)~ai,k/ (y + x)bijkt + cijkl, and then varying p or x. It is stressed that
to keep the characteristics of heteroclinic cycles in the unperturbed system, u and y in
the linear part of the system are kept constant at the values given in the last paragraph.
Simulations have been conducted by using the full nonlinear equations of motion.
Variations in u do not lead to chaos, contrary to expectations, but variations in y do. Results
for x E (13, 14) are summarized in the bifurcation diagram of Figure 5.54(a). Note that,
although y is significantly far from yc for x = 14, still y/yc 2 0.3 only. It is clear from
Figure 5.54 and other calculations given in Li & Pdidoussis (1994) that a period-doubling
bifurcation occurs for x 2 13.4, and then chaos develops for x 2 13.55. The phase-
plane diagram of Figure 5.54(c) is reminiscent of some depicting chaotic oscillation of
a two-well-potential oscillator (Moon 1992), which is associated with a homoclinic orbit
(two loops connected by a saddle), whereas the analytical subsystem in this case exhibits
a heteroclinic orbit. Nevertheless, physically, the existence of homoclinic orbits of the
doubly degenerate up-standing cantilever does make sense. Thus, decreasing y means
that two attractors (buckled states) on either side of the straight equilibrium are created,
and the oscillator jumps back and forth between the two attracting domains in a stochastic
manner.