The Lyapunov exponents of 
are defined for discrete systems by
whenever this limit exists. Here 
are the eigenvalues of
where J is the Jacobian matrix. One can show (see, for example, Parker & Chua parker) that a perturbation grows as
Taking the norm of both sides,
Now the perturbation 
and
so
We can see from Eq.(73) that contractivity is asking that
. From properties of the matrix norm, we know that if
then
and
so contractivity is equivalent to 
, or
. Thus from Eq.(107), contractivity
is sufficient to give nonpositive Lyapunov exponents and thence regular
motion. (Note that the reverse is not necessarily true.)
