Formal normal form of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:math> slow–fast systems

Jardón-Kojakhmetov, Hildeberto

doi:10.1016/j.crma.2015.06.009

Cited by 3 publications

References 19 publications

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Stabilization of a class of slow–fast control systems at non-hyperbolic points

2019

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In this paper we study the stabilization problem of a general class of slow-fast systems with one fast and arbitrarily many slow states. Moreover, the class of systems under study is slowly actuated, meaning that only the slow states are subject to the action of a controller. Furthermore, we are particularly interested in the case where normal hyperbolicity is lost. We show that by using the Geometric Desingularization method, it is possible to design controllers to locally stabilize non-hyperbolic points of any finite degeneracy. The main novelty of this paper is that, unlike previous research on the topic, we make use of more than one chart of the blow up space to enhance the region of attraction of the operating point. A couple of numerical examples highlight our contribution.

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Stabilization of a class of slow–fast control systems at non-hyperbolic points

2019

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Analysis of a slow–fast system near a cusp singularity

Jardón-Kojakhmetov

Broer

Roussarie

2016

Journal of Differential Equations

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International audienceThis 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. (C) 2015 Elsevier Inc. All rights reserved

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

Jardón-Kojakhmetov¹,

Kuehn²

2021

Mexican Mathematicians in the World

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In this document we review a geometric technique, called 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 to the study of fast-slow systems in the seminal work by Dumortier and Roussarie in 1996, whose aim was to give a geometric approach and interpretation of canards in the van der Pol oscillator. Following Dumortier and Roussarie, 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. Due to their great importance in the theory of fast-slow systems, we cover fold points 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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Formal normal form of Ak slow–fast systems

Cited by 3 publications

References 19 publications

Stabilization of a class of slow–fast control systems at non-hyperbolic points

Stabilization of a class of slow–fast control systems at non-hyperbolic points

Analysis of a slow–fast system near a cusp singularity

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

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