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

## TL;DR

This survey reviews recent advances in fractal analysis of dynamical systems using Minkowski dimension, highlighting classifications, bifurcation insights, and applications across various scientific fields.

## Contribution

It provides a comprehensive overview of fractal methods applied to regular and slow-fast systems, emphasizing coordinate-free techniques and future research directions.

## Key findings

- Minkowski dimension predicts limit cycle birth in regular systems.
- Fractal methods offer computational advantages and geometric insights.
- Applications include neuroscience, chemistry, ecology, and climate science.

## 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.