  We study the local dynamics of $L^{2}\left(\mathbb{R}\right)$-perturbations
to the zero solution of spatially $2\pi$-periodic coefficient
reaction-diffusion systems. In this case the spectrum of the linearization
about the zero solution is purely essential and may be described via the point
spectrum of a one-parameter family of Bloch operators. When this essential
spectrum is unstable, we characterize a large class of initial perturbations
which lead to nonlinear instability of the trivial solution. This is
accomplished by using the Bloch transform to construct an appropriate
projection to capture the maximum amount of linear exponential growth
associated to the initial perturbation arising from the unstable eigenvalues of
the Bloch operators. This result is also extended to dissipative systems of
conservation laws.
