Minkowski dimension and slow-fast polynomial Liénard 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 Poincaré--Lyapunov compactification, in this paper we focus on a fractal analysis near infinity of the slow-fast generalized Liénard equations $\dot{x}=y-{\sum }_{k=0}^{n+1}{B}_k{x}^k,\dot{y}=-\epsilon {\sum }_{k=0}^m{A}_k{x}^k$. 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 Liénard equations and contain a part at infinity.