Minkowski Dimension and Slow–Fast Polynomial Liénard Equations Near Infinity

Abstract

In planar slow–fast systems, fractal analysis of (bounded) sequences in 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. 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.