Fractal analysis of slow-fast and regular systems: A survey of recent results and future perspectives

Abstract

We survey recent developments in fractal analysis of regular and slow-fast dynamical systems using Minkowski dimension. Our focus is on spiral trajectories near monodromic limit periodic sets in regular systems and entry-exit sequences in slow-fast systems with degenerate singularities. For regular systems, we recall connections between Minkowski dimension and cyclicity. Key results include fractal classifications of weak foci, degenerate foci, and polycycles, where dimensional relationships predict limit cycle birth. For slow-fast systems, we survey the coordinate-free fractal methodology analyzing slow-fast Hopf points and canard cycles through slow divergence integrals and entry-exit sequences. The Minkowski dimension takes discrete values yielding upper bounds for the number of limit cycles without normal form transformations. The fractal approach provides computational advantages, works with original coordinates, and reveals geometric structures underlying bifurcation phenomena. Applications span neuroscience, chemistry, population dynamics, and climate modeling. We also discuss extensions to piecewise smooth and three-dimensional systems.