IN the past few years numerical analysts have come to realize that linear stability theory cannot be applied to nonlinear systems. One cannot say that the Jacobian represents the local behaviour of the solutions except at a fixed point. This had not previously been appreciated in numerical analysis, and there was a tendency to believe that looking at the Jacobian at one point as a constant, the solutions nearby would behave like the linearized system produced from this `frozen' Jacobian. Numerical analysts have now recognized the failings of linear stability theory when applied to nonlinear systems, and have constructed a new theory of nonlinear stability.
The theory looks at systems that have a property termed
contractivity;
if 
and 
are any two solutions of the system 
satisfying different initial conditions, then if
for all 
, 
where 
, the system is said to
be contractive.
An analogous definition may be framed for a discrete system; if
the discrete system is said to be contractive. Now we must define another
property of the system which numerical analysts have termed
dissipativity. Since we have already used this word in dynamics, we shall
call this new property NA-dissipativity
(numerico-analytic dissipativity). The system 
is said
to be NA-dissipative in 
if
where 
is an inner product,
holds for all y, 
in 
and for 
, where 
is the domain of 
regarded as a function of y.
NA-dissipativity can be shown to imply contractivity.
We needed to define NA-dissipativity because we can obtain a usable test
for it; a system is NA-dissipative if
where 
is the logarithmic norm of
the Jacobian 
.
If 
, 
are the eigenvalues of
, Eq.(75)
can be shown to be equivalent to
We now have a practical sufficient condition for contractivity.
We further define: if a Runge--Kutta method applied with any step length to a
NA-dissipative autonomous system is contractive, then the method is said to
be B-stable.
If we have the same situation with a nonautonomous system,
then the method is said to be BN-stable.
A sufficient condition for both
of these properties is given by
algebraic stability. A Runge--Kutta
method is said to be algebraically stable if
and
are both nonnegative definite. 
and 
here come from
Eq.(11), the Butcher array
of the Runge--Kutta method.
Algebraic stability implies A-stability, but the reverse is not true.
Let us look at this theory from a dynamics viewpoint. The problem is that
contractivity
is a very severe requirement to impose; in fact it precludes the
possibility of chaos occurring in the system. Is is easy to see that this must
be so, since chaos demands that neighbouring trajectories be divergent, whereas
contractivity demands that they be convergent. We show in
Appendix A.2 that contractivity is sufficient to give nonpositive
Lyapunov exponents;
positive Lyapunov exponents are a necessary condition for
chaos to occur. Unfortunately then,
one can only investigate the stability of contractive, nonchaotic systems
with the nonlinear stability theory as it stands. However, as we have remarked
previously, practical numerical codes do not use any stability theory to
evaluate their accuracy, so this is a theoretical, rather than a practical,
problem.
It should be pointed out that what numerical analysts call dissipativity,
what we have termed NA-dissipativity, is a much stronger requirement even than
strict dissipativity (
)
in dynamics. The latter merely requires
that 
, whereas the former insists that
(
being the Lyapunov exponents of the system).
