Minkowski dimension and slow-fast polynomial Li\'{e}nard equations near infinity

Abstract

In planar slow-fast systems, fractal analysis of (bounded) sequences in $\mathbb R$ has proved important for detection of the first non-zero Lyapunov quantity in singular Hopf bifurcations, determination of the maximum number of limit cycles produced by slow-fast cycles, defined in the finite plane, etc. One uses the notion of Minkowski dimension of sequences generated by slow relation function. Following a similar approach, together with Poincar\'{e}--Lyapunov compactification, in this paper we focus on a fractal analysis near infinity of the slow-fast generalized Li\'{e}nard equations $\dot x=y-\sum_{k=0}^{n+1} B_kx^k,\ \dot y=-\epsilon\sum_{k=0}^{m}A_kx^k$. We extend the definition of the Minkowski dimension to unbounded sequences. This helps us better understand the fractal nature of slow-fast cycles that are detected inside the slow-fast Li\'{e}nard equations and contain a part at infinity.