Instability of slowly varying systems

Instability criteria are obtained for systems described by
\dot{x} = A(t)x
when the parameters are slowly varying. In particular it is shown that, when
A(t)
has eigenvalues in the right-half plane and all eigenvalues are bounded away from the imaginary axis, then if
\sup_{t \geq 0} \parallel \dot{A}(t)\parallel
is sufficiently small, the system has unbounded solutions. Results are also given for systems of the form
\dot{x} = A(t)x + f(x, t)
, and the dichotomy of solutions is studied in both the linear and nonlinear cases.