Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lissajous Varieties

Francesco Maria Mascarin, Simon Telen

Abstract

This paper studies affine algebraic varieties parametrized by sine and cosine functions, generalizing algebraic Lissajous figures in the plane. We show that, up to a combinatorial factor, the degree of these varieties equals the volume of a polytope. We deduce defining equations from rank constraints on a matrix with polynomial entries. We discuss applications of Lissajous varieties in dynamical systems, in particular the Kuramoto model. This leads us to study connections with convex optimization and Lissajous discriminants.

Lissajous Varieties

Abstract

This paper studies affine algebraic varieties parametrized by sine and cosine functions, generalizing algebraic Lissajous figures in the plane. We show that, up to a combinatorial factor, the degree of these varieties equals the volume of a polytope. We deduce defining equations from rank constraints on a matrix with polynomial entries. We discuss applications of Lissajous varieties in dynamical systems, in particular the Kuramoto model. This leads us to study connections with convex optimization and Lissajous discriminants.

Paper Structure

This paper contains 8 sections, 17 theorems, 69 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related work.
  3. Acknowledgements.
  4. Dimension and degree
  5. Determinantal equations
  6. Kuramoto oscillators
  7. Positive points and convex optimization
  8. Lissajous discriminants

Key Result

Theorem 1.4

Let $A = \in \mathbb{Z}^{d \times n}$ be such that ${\rm rank}(A) = d$ and $\mathbb{Z}A = \mathbb{Z}^d$. Let $P_A \subset \mathbb{R}^d$ be the polytope obtained as the convex hull of the lattice points $\{ \pm a_1, \ldots, \pm a_n \} \subset \mathbb{Z}^d$. For any $b \in \mathbb{C}^n$, the variety

Figures (8)

  • Figure 1: The surfaces ${\cal C}_A$ (left) and ${\cal S}_A$ (right) from Example \ref{['ex:elliptopeintro']}.
  • Figure 2: The spring network of the triangle graph $C_3$.
  • Figure 3: Positive regions for $x \in {\cal S}_A$ (left) and $\omega \in \mathbb{R}^2$ (right).
  • Figure 4: A graph and its reduced incidence matrix.
  • Figure 5: The line $x + y = \omega$ has precisely one intersection point with ${\cal L}_{A,b}^+$ for $\omega \in (-1,1)$.
  • ...and 3 more figures

Theorems & Definitions (54)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • ...and 44 more