Page 528 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

498 SLENDER STRUCTURES AND AXIAL FLOW

Thus the slowly varying variables r and 1c/ are governed by

i- = +br + crr2)r + 6(c2), $J = wo + E(& + cir2) + 6(e2), (F.79)
where 6, and c, are the real parts of b and c, and bi and ci the corresponding imaginary
parts.

F.6.2 The Lyapunov -Schmidt reduction
Similarly to centre manifold reduction, the Lyapunov-Schmidt reduction is a method
which replaces a large and complicated set of equations by a simpler and smaller one
which contains all the essential information concerning a bifurcation. The method is
applicable to the system of equation (F.49), but here the nonlinear functional f(x, p) may
be either finite dimensional (ordinary differential equations) or infinite dimensional (partial
differential or integro-differential equations). However, we shall restrict the analysis to
the case of nondegenerate bifurcations from a stationary solution to another stationary
solution, which excludes the important case of Hopf bifurcations; in principle, this case
can be treated with similar methods (Golubitsky & Schaeffer 1985) - see also the next
section. Thus, we shall be interested in determining steady-state solutions 7 of the system
defined by
f(Y, u) = 0, (F.80)
when the linearized system in the neighbourhood of 7 has a single zero eigenvalue at
the critical parameter, uc, while the other eigenvalues (infinitely many for a PDE) have
negative real parts. Without loss of generality, we shall also assume that the equilibrium
is zero, 7 = 0, since this can be accomplished by a simple coordinate transformation.
As mentioned in Section F.5 for the bifurcation analysis, we would like to find an
equilibrium solution when the parameter u is varied. From the implicit function theorem
(Sattinger 1980), we know that there is no unique solution of (F.80) in the form of
y = y(u) in a small neighbourhood of (0, uc), since we are exactly in the situation where
the linear operator L = D,f(O, u,) is singular, because the Jacobian matrix has a zero
eigenvalue. The basic idea in the Lyapunov-Schmidt method is to decompose the space
in which the solution lies into two subspaces, in order to ‘remove’ this singularity.
Formally, this may be expressed as follows. First, we define vo as the eigenfunction of
L corresponding to the zero eigenvalue, i.e.

=
L(~,)V~ avo = 0, (F.81)
and v;i the eigenfunction of the adjoint system

L*(U,)V;, = 0, (F.82)
where L* is the adjoint operator of L. Furthermore, we assume that the original Banach
space, 72, in which the solution of (F.80) lies, can be decomposed into two subspaces:
72 = kerL + A, where kerL is the kernel of L, and M is a subspace perpendicular to
ker L; in practice, A = range L* [see Kolmogorov & Fomin (1970) for details]. Then, it
is possible to express the solution y as

(F.83)
Y = Yc + YS?