  In this paper we introduce the notion of fractal codimension of a nilpotent
contact point $p$, for $\lambda=\lambda_0$, in smooth planar slow$-$fast
systems $X_{\epsilon,\lambda}$ when the contact order $n_{\lambda_0}(p)$ of $p$
is even, the singularity order $s_{\lambda_0}(p)$ of $p$ is odd and $p$ has
finite slow divergence, i.e., $s_{\lambda_0}(p)\leq 2(n_{\lambda_0}(p)-1)$. The
fractal codimension of $p$ is a generalization of the traditional codimension
of a slow-fast Hopf point of Li\'{e}nard type, introduced in (Dumortier and
Roussarie (2009)), and it is intrinsically defined, i.e., it can be directly
computed without the need to first bring the system into its normal form. The
intrinsic nature of the notion of fractal codimension stems from the Minkowski
dimension of fractal sequences of points, defined near $p$ using the
so$-$called entry$-$exit relation, and slow divergence integral. We apply our
method to a slow$-$fast Hopf point and read its degeneracy (i.e., the first
nonzero Lyapunov quantity) as well as the number of limit cycles near such a
Hopf point directly from its fractal codimension. We demonstrate our results
numerically on some interesting examples by using a simple formula for
computation of the fractal codimension. We demonstrate our results numerically
on some interesting examples by using a simple formula for computation of the
fractal codimension.
