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Polynomial slow-fast systems on the Poincaré-Lyapunov sphere

Otavio Henrique Perez, Paulo Ricardo da Silva

Abstract

The main goal of this paper is to study compactifications of polynomial slow-fast systems. More precisely, the aim is to give conditions in order to guarantee normal hyperbolicity at infinity of the Poincaré-Lyapunov sphere for slow-fast systems defined in $\mathbb{R}^{n}$. For the planar case, we prove a global version of the Fenichel Theorem, which assures the persistence of invariant manifolds in the whole Poincaré-Lyapunov disk. We also discuss the appearence of non normally hyperbolic points at infinity, namely: fold, transcritical and pitchfork singularities.

Polynomial slow-fast systems on the Poincaré-Lyapunov sphere

Abstract

The main goal of this paper is to study compactifications of polynomial slow-fast systems. More precisely, the aim is to give conditions in order to guarantee normal hyperbolicity at infinity of the Poincaré-Lyapunov sphere for slow-fast systems defined in . For the planar case, we prove a global version of the Fenichel Theorem, which assures the persistence of invariant manifolds in the whole Poincaré-Lyapunov disk. We also discuss the appearence of non normally hyperbolic points at infinity, namely: fold, transcritical and pitchfork singularities.
Paper Structure (11 sections, 11 theorems, 57 equations, 12 figures)

This paper contains 11 sections, 11 theorems, 57 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on geometric singular perturbation theory and Poincaré--Lyapunov compactification
  3. Geometric singular perturbation theory
  4. Poincaré--Lyapunov compactification of polynomial vector fields
  5. Poincaré--Lyapunov compactification of slow-fast systems
  6. Geometric singular perturbation theory at infinity
  7. Newton polytope of a polynomial vector field
  8. Normal hyperbolicity at infinity
  9. Planar polynomial slow-fast systems
  10. Non normally hyperbolic points at infinity
  11. Acknowledgements

Key Result

Proposition 4

Let $X_{\varepsilon}$ be the polynomial vector field associated to the slow-fast system eq-def-slowfast-2 and denote its PL-compactification by $X^{\infty}_{ \varepsilon}$. Then, in the charts $U_{l}$ for $l = 1,\dots, n$, it follows that:

Figures (12)

  • Figure 1: PL-disk (left) and 3-dimensional PL-ball (right).
  • Figure 2: Critical manifold of slow-fast system \ref{['eq-vdp-r3']} (left) and its phase portrait at infinity (right), which is given by the van der Pol equation \ref{['eq-vdp-infinity']}. The critical manifold is highlithed in green and the canard cycle is highlighted in red (see also DumRou).
  • Figure 3: Planar slow-fast system for $\varepsilon = 0$ (left) and for $\varepsilon > 0$ sufficiently small (right). The Fenichel Theorem assures the existence of a family of invariant manifolds $\mathcal{K}_{\varepsilon}$, and the flow on $\mathcal{K}_{\varepsilon}$ converges to the flow on $\mathcal{K}$. Moreover, Fenichel Theorem also assure the existence of a family of stable manifolds $\mathcal{W}^{s}_{\varepsilon}$ of $\mathcal{K}_{\varepsilon}$.
  • Figure 4: Directional charts that cover the PL-ball $\mathbb{B}^{3}_{\omega}$. Following the terminology of Definition \ref{['def-directions']}, the slow-fast vector field obtained in the chart $U_{1}$ is the compactification in the fast direction, whereas the vector field obtained in the chart $U_{l}$ is the compactification in the slow direction, for $l = 2,\dots,n$.
  • Figure 5: Phase portrait of the slow-fast system \ref{['eq-exe-prop-planar']} in the PL-disk. The critical manifold is highlighted in green.
  • ...and 7 more figures

Theorems & Definitions (33)

  • Example 1
  • Example 2
  • Definition 3
  • Proposition 4
  • proof
  • Example 5
  • Example 6
  • Theorem 7
  • Example 8
  • Example 9
  • ...and 23 more