  We consider deterministic fast-slow dynamical systems of the form \[
x_{k+1}^{(n)} = x_k^{(n)} + n^{-1} A(x_k^{(n)}) + n^{-1/\alpha} B(x_k^{(n)})
v(y_k), \quad y_{k+1} = Ty_k, \] where $\alpha\in(1,2)$ and
$x_k^{(n)}\in{\mathbb R}^m$. Here, $T$ is a slowly mixing nonuniformly
hyperbolic dynamical system and the process
$W_n(t)=n^{-1/\alpha}\sum_{k=1}^{[nt]}v(y_k)$ converges weakly to a
$d$-dimensional $\alpha$-stable L\'evy process $L_\alpha$.
  We are interested in convergence of the $m$-dimensional process
$X_n(t)=x_{[nt]}^{(n)}$ to the solution of a stochastic differential equation
(SDE) \[ dX = A(X)\,dt + B(X)\, dL_\alpha. \] In the simplest cases considered
in previous work, the limiting SDE has the Marcus interpretation. In
particular, the SDE is Marcus if the noise coefficient $B$ is exact or if the
excursions for $W_n$ converge to straight lines as $n\to\infty$.
  Outside these simplest situations, it turns out that typically the Marcus
interpretation fails. We develop a general theory that does not rely on
exactness or linearity of excursions. To achieve this, it is necessary to
consider suitable spaces of ``decorated'' c\`adl\`ag paths and to interpret the
limiting decorated SDE. In this way, we are able to cover more complicated
examples such as billiards with flat cusps where the limiting SDE is typically
non-Marcus for $m\ge2$.
