  The leading-order approximation to a Filippov system $f$ about a generic
boundary equilibrium $x^*$ is a system $F$ that is affine one side of the
boundary and constant on the other side. We prove $x^*$ is exponentially stable
for $f$ if and only if it is exponentially stable for $F$ when the constant
component of $F$ is not tangent to the boundary. We then show exponential
stability and asymptotic stability are in fact equivalent for $F$. We also show
exponential stability is preserved under small perturbations to the pieces of
$F$. Such results are well known for homogeneous systems. To prove the results
here additional techniques are required because the two components of $F$ have
different degrees of homogeneity. The primary function of the results is to
reduce the problem of the stability of $x^*$ from the general Filippov system
$f$ to the simpler system $F$. Yet in general this problem remains difficult.
We provide a four-dimensional example of $F$ for which orbits appear to
converge to $x^*$ in a chaotic fashion. By utilising the presence of both
homogeneity and sliding motion the dynamics of $F$ can in this case be reduced
to the combination of a one-dimensional return map and a scalar function.
