Page 513 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

Appendix F

Some of the Basic Methods of

Nonlinear Dynamics

The purpose of this appendix is to outline some of the methods utilized in modern
nonlinear dynamics. It is intended to help those not already familiar with them.
A common feature of analytical methods in nonlinear dynamics is the transformation of
complicated dynamical systems into simpler ones. Another aspect is that nonlinear analysis
often emphasizes qualitative features of system dynamics, frequently in the neighbourhood
of critical parameter values. After introducing the concept of stability, we briefly go over
several of the most commonly used methods.

F. 1 LYAPUNOV METHOD

F.l.l The concept of Lyapunov stability

Consider a system of differential equations of the form

x = f(x, t), x E R”. (F. 1 )

It is assumed that there exists a unique solution X(t) of (F.l) that is determined by the
initial condition xo at to. This solution is said to be stable if, starting close to X(t) at a
given time, it remains close to X(t) for all later times. More precisely, %(t) is stable if
for any other solution y(t) of (F.l) and for every (arbitrarily small) E > 0 there exists a
S(E) > 0, such that

The norm here may refer to the Euclidian or any other norm. Within this definition it
makes no sense to use terms such as ‘stable system’ or ‘stable differential equation’, since
one and the same differential equation may have stable as well as unstable solutions.
Unfortunately, with this definition of stability, periodic solutions of equation (F. 1) are
not stable! This is because a small change in the initial conditions may produce a slight
change in the period of oscillations and for a reasonably large time, two solutions starting
from nearby points will not remain nearby. It is therefore necessary to enlarge the concept
of stability to cover also the case in which the phase trajectories remain close to each
other. This is the purpose of the concept of a stable trajectory or of orbital stability.

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