Fractal codimension of nilpotent contact points in two-dimensional slow-fast systems

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.