Page 531 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

SOME OF THE BASIC METHODS OF NONLINEAR DYNAMICS 50 1

Lyapunov-Schmidt method, since projection operators as defined in (F.84a) and (F.84b)
are also introduced in this case. The purpose of the method is to find a periodic solution
of a nonlinear system in the neighbourhood of a Hopf bifurcation, which is defined by
the occurrence of one or several pairs of complex eigenvalues with zero real parts, *iwO.
Introducing a new time scale, t = %t, the original equation (F.49) takes the form

(F.97)
where @ is a small parameter. The idea is to transform the original differential equation
into two algebraic ones, of the form

y = UY + WI - V)Fty, p), (F.98a)

(F.98b)

where U, V and 3C are some projection operators, and I is the identity operator [see
Bajaj (1982) for their definition]. Note the similarity with equations (F.84a) and (F.84b).
The solution y is split into two components, y = yl + y2, so that the y1 component is
made up of the solutions of the homogeneous part of equation (F.97), whereas the y~
component contains the higher harmonics. It is then possible to solve for y2 explicitly in
equation (F.98a) using the implicit function theorem and, assuming y = y1 + y2(y1, p),
equation (F.98b) becomes the bifurcation equation.