Series Preface * Preface to the Third Edition * 1 Linear Systems * 2 Nonlinear Systems: Local Theory * 3 Nonlinear Systems: Global Theory * 4 Nonlinear Systems: Bifurcation Theory * References * Index

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Differential Equations and Dynamical Systems

Introduction to Ordinary Differential Equations * Linear Systems and Stability of Nonlinear Systems * Applications * Hyperbolic Theory * Continuation of Periodic Solutions * Homoclinic Orbits, Melnikov?s Method, and Chaos * Averaging * Local Bifurcation * References * Index.

http://www.math.ecnu.edu.cn/~zmwang/teaching/Advanced_ODE-2014-2015/1-Textbook_and_Reference_book/2-Ordinary%20Differential%20Equations%20with%20Applications.pdf

Ordinary Differential Equations with Applications

Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made thus far to classify different patterns of MMOs, in contrast to the classification of the related phenomena of bursting oscillations. This paper gives a survey of different types of MMOs, concentrating its analysis on MMOs whose small-amplitude oscillations are produced by a local, multiple-time-scale “mechanism.” Recent work gives substantially improved insight into the mathematical properties of these mechanisms. In this survey, we unify diverse observations about MMOs and establish a systematic framework for studying their properties. Numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper.

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Mixed-Mode Oscillations with Multiple Time Scales

The geometric approach to singular perturbation problems is based on powerful methods from dynamical systems theory. These techniques have been very successful in the case of normally hyperbolic critical manifolds. However, at points where normal hyperbolicity fails, the well-developed geometric theory does not apply. We present a method based on blow-up techniques, which leads to a rigorous geometric analysis of these problems. A detailed analysis of the extension of slow manifolds past fold points and canard points in planar systems is given. The efficient use of various charts is emphasized.

Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions ∗

Relaxation Oscillation and Canard Explosion

Canard Cycles and Center Manifolds

A cusp type germ of vector fields is a C∞ germ at 0∈ℝ2, whose 2-jet is C∞ conjugate toWe define a submanifold of codimension 5 in the space of germs consisting of germs of cusp type whose 4-jet is C0 equivalent toOur main result can be stated as follows: any local 3-parameter family in (0, 0) ∈ ℝ2 × ℝ3 cutting transversally in (0, 0) is fibre-C0 equivalent to

/pdf/generic-3-parameter-families-of-vector-fields-on-the-plane-51p0goeqyd.pdf

Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3

In his well known paper `Singularities of vector fields', Takens made a topological classification of vector fields up to codimension 2 and introduced a semialgebraic stratification to distinguish the different cases; from dimensions he had to use the notion of `weak--equivalence'. In this paper we show how to classify singularities of vector fields on up to codimension 4 for the notion of equivalence. To separate the different cases we use a semianalytic stratification and show that a semialgebraic one is not possible, even for the notion of weak--equivalence. Up to codimension 3 the stratification is semialgebraic. We will always suppose that the vector fields are , although it will be clear that the results are valid for , with r sufficiently big. We provide a complete, but short, survey of the different techniques to be used, referring to the existing literature for precise calculations and pictures. We put much emphasis on the new results.

https://iopscience.iop.org/article/10.1088/0951-7715/11/4/015/pdf

Singularities of vector fields on

Hilbert′s 16th Problem for Quadratic Vector Fields

Definitions and notations.- Transformation into normal form.- Bifurcations of codimension 1 and 2.- Elementary properties.- The central rescaling.- Conclusions and discussion of remaining problems.- Abelian integrals in unfoldings of codimension 3 singular planar vector fields.

Freddy Dumortier

Papers

Canard Cycles and Center Manifolds

Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3

Singularities of vector fields on

Hilbert′s 16th Problem for Quadratic Vector Fields

Bifurcations of planar vector fields : nilpotent singularities and Abelian integrals