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

Hildeberto Jardon-Kojakhmetov; Christian Kuehn

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arXiv · 1901.01402

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

Hildeberto Jardon-Kojakhmetov·Christian Kuehn

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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.

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Hildeberto Jardon-Kojakhmetov, Christian Kuehn. 2019-01-05. A survey on the blow-up method for fast-slow systems. https://arxiv.org/abs/1901.01402

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