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On the singularity structure of Kahan discretizations of a class of quadratic vector fields

René Zander

Abstract

We discuss the singularity structure of Kahan discretizations of a class of quadratric vector fields and provide a classification of the parameter values such that the corresponding Kahan map is integrable, in particular, admits an invariant pencil of elliptic curves.

On the singularity structure of Kahan discretizations of a class of quadratic vector fields

Abstract

We discuss the singularity structure of Kahan discretizations of a class of quadratric vector fields and provide a classification of the parameter values such that the corresponding Kahan map is integrable, in particular, admits an invariant pencil of elliptic curves.

Paper Structure

This paper contains 15 sections, 13 theorems, 119 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Birational maps of surfaces
  4. Birational quadratic maps of $\mathbb {CP}^2$
  5. The $(\gamma_1,\gamma_2,\gamma_3)$-class
  6. The case $(\gamma_1,\gamma_2,\gamma_3)=(1,1,1)$
  7. Lifting the map to a surface automorphism
  8. The case $(\gamma_1,\gamma_2,\gamma_3)=(1,1,2)$
  9. Lifting the map to a surface automorphism
  10. The case $(\gamma_1,\gamma_2,\gamma_3)=(1,2,3)$
  11. Lifting the map to a surface automorphism
  12. The case $(\gamma_1,\gamma_2,\gamma_3)=(1,1,0)$
  13. Lifting the map to an analytically stable map
  14. The case $(\gamma_1,\gamma_2,\gamma_3)=(n,1,-1)$
  15. Acknowledgements

Key Result

Theorem 1.1

Let $\phi\colon\mathbb {CP}^2\rightarrow\mathbb {CP}^2$ be the Kahan map of (nahm_intro). The sequence of degrees $d(m)$ of iterates $\phi^m$ grows exponentially, so that the map $\phi$ is non-integrable, except for the following cases: Here, $(\gamma_1,\gamma_2,\gamma_3)$ are fixed up to permutation and multiplication by $\lambda\in\mathbb R\setminus\{0\}$.

Figures (4)

  • Figure 1: The curves $\overline{\mathcal{E}}_0$, $\overline{\mathcal{E}}_{\infty}$, $\overline{\mathcal{E}}_{0.01}$ in resp. red, blue and green for $H(x,y)=H_1(x,y)$, $\varepsilon=1$.
  • Figure 2: The curves $\overline{\mathcal{E}}_0$, $\overline{\mathcal{E}}_{\infty}$, $\overline{\mathcal{E}}_{0.001}$ in resp. red, blue and green for $H(x,y)=H_2(x,y)$, $\varepsilon=1$.
  • Figure 3: The curves $\overline{\mathcal{E}}_0$, $\overline{\mathcal{E}}_{\infty}$, $\overline{\mathcal{E}}_{-0.002}$ in resp. red, blue and green for $H(x,y)=H_3(x,y)$, $\varepsilon=1$.
  • Figure 4: The curves $\overline{\mathcal{E}}_0$, $\overline{\mathcal{E}}_{\infty}$, $\overline{\mathcal{E}}_{0.1}$ in resp. red, blue and green for $\ell_1(x,y)=x+y$, $\ell_2(x,y)=x-y$, $\ell_3(x,y)=x$ and $\varepsilon=1$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 2.2: Diller, Favre DF01, Theorem 0.2
  • Theorem 2.3: Recurrence relations
  • proof
  • Corollary 2.4: Generating functions
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 14 more