# A survey on the blow-up method for fast-slow systems

## TL;DR

This survey reviews the blow-up method, a geometric technique from algebraic geometry, used to analyze and understand the dynamics of fast-slow systems near non-hyperbolic points, including various singularities.

## Contribution

It consolidates and summarizes the development and applications of the blow-up method in the geometric theory of fast-slow systems with non-hyperbolic singularities.

## Key findings

- Effective analysis of fold points and other singularities using the blow-up method.
- Extension of the method to various types of singularities like Hopf, pitchfork, and cusp.
- Successful application of the blow-up method in specific models and scenarios.

## Abstract

In this document we review a geometric technique, called \emph{the blow-up method}, as it has been used to analyze and understand the dynamics of fast-slow systems around non-hyperbolic points. The blow-up method, having its origins in algebraic geometry, was introduced in 1996 to the study of fast-slow systems in the seminal work by Dumortier and Roussarie \cite{dumortier1996canard}, whose aim was to give a geometric approach and interpretation of canards in the van der Pol oscillator. Following \cite{dumortier1996canard}, many efforts have been performed to expand the capabilities of the method and to use it in a wide range of scenarios. Our goal is to present in a concise and compact form those results that, based on the blow-up method, are now the foundation of the geometric theory of fast-slow systems with non-hyperbolic singularities. We cover fold points due to their great importance in the theory of fast-slow systems as one of the main topics. Furthermore, we also present several other singularities such as Hopf, pitchfork, transcritical, cusp, and Bogdanov-Takens, in which the blow-up method has been proved to be extremely useful. Finally, we survey further directions as well as examples of specific applied models, where the blow-up method has been used successfully.