Page 523 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

SOME OF THE BASIC METHODS OF NONLINEAR DYNAMICS 493

in which fl and f2 represent the cubic nonlinearities in (F.30). The algebra involved
may be carried out easily using, for example, MATHEMATICA. This leads finally to the
same results as obtained by the normal form, equation (F.41).
A derivation of the averaging method, more adapted to the case of PDEs is given in
Section F.6.1; but, before this, bifurcation theory is discussed briefly, together with the
calculation of the unfolding parameters.

F.5 BIFURCATION THEORY AND UNFOLDING PARAMETERS

As mentioned already, systems of physical interest typically have parameters which appear
in their defining equations; in fluid-structure interaction systems, for example, one of the
major concerns is the study of the effect of increasing flow velocity on the dynamics. The
original problem (F. 13) may therefore be rewritten as

x = f(x, p), x E R", p E Rk, (F.49)

where ,LL represents all the parameters. As these parameters are varied, changes may
occur in the qualitative structure of the solutions for certain parameter values po (Chow
& Hale 1982). These changes are called bifurcations and the parameter values are called
bi&rcution values.
In light of the stability theory that was introduced in Section F.l, it is clear that in
the case of a fixed point X of (F.49), a necessary condition for a bifurcation to occur
is that the Jacobian D,f(Si, KO) have at least one eigenvalue with a zero real part, in
which case E is referred to as being nonhyperbolic. This of course is the interesting
case from the nonlinear dynamics point of view. In the simplest cases, these bifurcations
have been classified and are now well known. For example, if the linearized system
contains a pair of purely imaginary eigenvalues at the critical parameter, uc, it is called
a Hopf bifurcation; if it contains a single zero eigenvalue, it may be a saddle-node, a
transcritical or a pitchfork bifurcation, depending on the nonlinear terms, or there might
not even exist a bifurcation (e.g. for the system defined by X = p -x3, x E R, p E [w).
In the case of higher degeneracy (e.g. when zero and purely imaginary eigenvalues occur
simultaneously, as discussed in Section 5.7.3), the situation is even more complicated.
Here, we show how to take into account the variation of system parameters in the
neighbourhood of critical values for cases where the linearized system has a single pair
of imaginary eigenvalues or a single zero eigenvalue. To this end, the ordinary differential
equation (F.49) is replaced by

x = L(u)x + f(x, u), x E R", (F.50)

where u E [w represents the system parameter in question (the dimensionless flow
velocity), L is an n x n matrix, and f contains all the nonlinear terms. At u = uc, suppose
that L(u,) contains a pair of purely imaginary eigenvalues, h = Aiwo. Then, when u is
varied by a small amount, u = u, + p, p << I, and the new eigenvalues corresponding to
fiwo can be expressed as h = u f iw, with cr = yl and o = wo + 11.2. Here, p~ and ,LL~ are
called the unfolding parameters and can be determined from the following characteristic
equation:
det[L(u) - AI] = 0 %(a, w, u) + i9(a, w, u) = 0. (F.51)