Analysis of a slow-fast system near a cusp singularity

TL;DR

This paper analyzes the dynamics of a slow-fast system near a cusp singularity using normal form theory and blow-up techniques, providing refined insights into the transition behavior around the cusp.

Contribution

It introduces a formal normal form for such systems and applies blow-up methods to study the cusp transition, refining previous results in the field.

Findings

Refined understanding of the cusp transition dynamics.

Application of normal form theory to slow-fast systems.

Enhanced analysis using blow-up techniques.

Abstract

This paper studies a slow-fast system whose principal characteristic is that the slow manifold is given by the critical set of the cusp catastrophe. Our analysis consists of two main parts: first, we recall a formal normal form suitable for systems as the one studied here; afterwards, taking advantage of this normal form, we investigate the transition near the cusp singularity by means of the blow up technique. Our contribution relies heavily in the usage of normal form theory, allowing us to refine previous results.