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

Abstract

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.

Discrete geometric singular perturbation theory

“…are satisfied. The first condition in (25) implies that the map is locally orientation-preserving and that S is normally hyperbolic and attracting near (x, y). The second guarantees that there are no fixed points in the reduced map.…”

“…Thus, alternative methods are needed. A number of works have shown that an adaptation of the well-known geometric blow-up method [10,25,30,32] can be applied to study of discretized fast-slow ODEs with singularities in (i) [1,14,46,44]. The approach adopted in these works relies in an important way on scaling properties of the discretization parameter, but there are significant obstacles to the extension of this approach to the study of general fast-slow maps, i.e.…”

Dynamics of a stochastic excitable system with slowly adapting feedback

“…The equation (32) implies that μ = μ(εt) depends only on the slow time and acts as a parameter in (31). For μ > 1 − I 0 , equation ( 31) has the oscillating solution φ = ϕ μ(t) given by (13). Note that the parameters of this solution can depend on the slow time.…”

“…Investigating the dynamics of such multiscale systems has lead to the development of a number of useful asymptotic and geometric methods, see Refs. [9][10][11][12][13] to name just a few.…”