Page 519 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

SOME OF THE BASIC METHODS OF NONLINEAR DYNAMICS 489

be computed, usually in terms of power series of x to any desired degree of accuracy.
Examples of how the centre manifold is determined can be found in Rand & Armbruster
(1987) and Wiggins (1990), and one specific example is given in Appendix H.

F.3 NORMAL FORMS

The central idea of the method of normal forms is to use a coordinate transformation to
simplify or eliminate nonlinear terms in a dynamical system (Arnold 1983; Guckenheimer
& Holmes 1983; Wiggins 1990). To demonstrate how the method works, we consider a
differential equation of the form of the first of (F.19), with the nonlinear terms representing
homogeneous polynomials of order k,
X = AX + cfk(X), (F.21)

where E << 1 is a real number used as a book-keeping device. In other words, fk(x) belongs
to the space Hk which is spanned by the vector-valued mononomials

(F.22)

where k = kl + k:! + . . . + k,, is the order of the polynomial fk(x) and ei are unit ortho-
gonal vectors in [w“ (for example, H2 is spanned by the three ‘vectors’ e1,2 = x2el, e2,2 =
xye2, e3.2 = y2e3).
The aim of the method is to find a coordinate transformation,

x = y + chi (y), (F.23)
such that (F.21) takes the ‘simplest possible form’, the so-called normal form,

Y = AY + %l (Y). (F.24)

Substituting (F.23) into (F.21) yields
Y + ~Dhi (YIY = AY + (y> + cfk(~ + chi (Y>>, (F.25)

where Dhl is the Jacobian of hl. Using (F.24) to eliminate y and equating coefficients of
like powers in c leads to
gl(Y> + Dhl(Y)AY - Ahl(Y) = MY) (F.26)

or
%~[hi(J’)l = Dhi(Y)Ay - Ahi(Y) = fk(Y> - gl(Y); (F.27)
%A is known as the Lie or Poisson bracket of the vector fields Ay and hl (y) (Arnold 1988).
Ideally, one would like to remove all nonlinear terms using successive transformations,
in order to reduce the vector field to its linear part, i.e. to transform (F.21) into y = Ay.
This condition means that gl(y) in equation (F.27) is zero, i.e.
fk(y> = zA[hl(y)l. (F.28)

It can be shown easily that %’A is a linear map from Hk -+ Hk (ie. it can be represented
by a matrix), and that the eigenvalues Ak,i of this linear map are related to the eigenvalues