Page 515 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

SOME OF THE BASIC METHODS OF NONLINEAR DYNAMICS 485

Autonomous case
If A is a constant real-valued matrix, then the solution of (F.7) is asymptotically stable
if all the eigenvalues of A have negative real parts. On the other hand, if at least one
eigenvalue of A has a positive real part, then the solution is unstable.
If there exist real numbers B > 1, a ? 0, such that the condition

MY, t)l 5 4YlB F.8)

is satisfied in a neighbourhood of y = 0, then the stability of trivial solutions of the
nonlinear system (F.6) can be obtained from the eigenvalues of A in the following form:
(i) if all of the eigenvalues of A have negative real parts, then the equilibrium solution
of (F.6) is asymptotically stable;
(ii) if at least one eigenvalue of A has a positive real part, then the trivial solution of
(F.6) is unstable.
These statements are valid, independently of the higher order terms; h(y, t) need only
satisfy the inequality (F.8). In cases where A has at least one eigenvalue with vanishing
real part, then the effect of nonlinear terms must be taken into account in the stability
analysis.
Periodic case

In the case where A(t) is a periodic function of time, A(t + T) = A(t), the stability of the
trivial solution of (F.7) is obtained using Floquet theory: for the system (F.71, it can be
shown that a&ndamental solution matrix can be found, in the form

Y(t) = Z(t) exp(tR), (F.9)

where Z(t) is also periodic of period T, Z(t + T) = Z(t), and R is a (nonunique) constant
matrix (Nayfeh & Mook 1979). Furthermore, if Z(0) is equal to the identity matrix, then
Y(T) = eTR. It thus becomes obvious that the stability of the trivial solution is related to
the eigenvalues of the matrix eTR, since after n periods the trivial solution will be related to
en TR . These eigenvalues are called the characteristic or Floquet multipliers. Consequently,
the trivial solution of (F.7) is asymptotically stable if and only if all of the eigenvalues
of the matrix eTR have absolute values (modulus) less than unity, while it is unstable if
one of the eigenvalues has a modulus greater than 1. In the case where one or several
eigenvalues have modulus equal to 1, then the trivial solution of (F.7) may be stable or
unstable, depending on the structure of the Jordan normal form' corresponding to eTR.
Furthermore, linearization theorems as in the case of systems with constant coefficients
can be proved, which means that it is possible to relate the stability of the nonlinear
system to the stability of the linearized one.
In practice, an analytical determination of the fundamental matrix is very difficult,
except in some special cases. Nevertheless, it can be found using perturbation methods,
or using numerical schemes.

+The Jordan form is the simplest form a matrix can take, when transformed in the appropriate vector space
(Hirsch & Smale 1974).