Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A survey on the blow-up method for fast-slow systems   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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The categories of this paper are: hep-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-th
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The categories of this paper are: dg-ga
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The categories of this paper are: eess
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The categories of this paper are: solv-int
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The categories of this paper are: cs
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The categories of this paper are: atom-ph
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The categories of this paper are: adap-org
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The categories of this paper are: physics
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The categories of this paper are: funct-an
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The categories of this paper are: bayes-an
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The categories of this paper are: nucl-ex
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The categories of this paper are: patt-sol
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-ex