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Some remarkable autonomous systems

Oumar Wone

Abstract

We study the links of the Darboux-Halphen-Ramanujan system, with contact geometry, canonical coordinates of some $3$-dimensional Frobenius manifolds and projective connections on Riemann surfaces. One of our important goals is to highlight the role of contact geometry in this setting. We also study autonomous systems "derived" from the potential given by the Group-determinant, of any cyclic group $\Z/n\Z$, $n\geqslant3$, and the Klein group $\Z/2\Z\times\Z/2\Z$.

Some remarkable autonomous systems

Abstract

We study the links of the Darboux-Halphen-Ramanujan system, with contact geometry, canonical coordinates of some -dimensional Frobenius manifolds and projective connections on Riemann surfaces. One of our important goals is to highlight the role of contact geometry in this setting. We also study autonomous systems "derived" from the potential given by the Group-determinant, of any cyclic group , , and the Klein group .
Paper Structure (7 sections, 11 theorems, 145 equations)

This paper contains 7 sections, 11 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Darboux-Halphen-Ramanujan systems and autonomous systems
  3. Contact geometric interpretation of the Darboux-Ramanujan system
  4. Elementary symmetric functions and the Darboux-Halphen system
  5. WDVV equations and Chazy equation: canonical coordinates
  6. Group determinants and autonomous systems
  7. Wirtinger connections

Key Result

Theorem 2.4

Consider the symplectic $(M=\mathbb C^3\times\mathbb C^\times,\Omega)$ with coordinates $(q_1,q_2,p_1,p_2)$. Let $\mathscr F$ be the following Hamiltonian on $M$ given by Set $q_1=\mathscr{X}_1$, $q_2=\mathscr{Z}_1$, $p_1=\lambda \mathscr{Y}_1$ and $p_2=\lambda$. Then the Hamiltonian equations associated to $\mathscr{F}$, derived from Definition d8 are Moreover the projection of the system d5 on

Theorems & Definitions (29)

  • Definition 2.1: bryant
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Corollary 2.7: Dubrovin
  • proof
  • ...and 19 more