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Families of Kuramoto models and bounded symmetric domains

M. Olshanetsky

Abstract

We define the families of Kuramoto models (KM) related to bounded symmetric domains. The families include the Lohe unitary model and the spherical models as special cases. Our approach is based on the construction proposed by Watanabe and Strogats WS. We replace the Poincare disc and its $S^1$ boundary in the WS construction on the bounded symmetric domains and on the its Bergman-Shilov (BS) boundaries. In Cartan classifications there are four classical domains of types I-IV. Here we consider the domains of types I,II and III. For a fixed domain there is a decreasing chain of the BS boundaries components. This leads to the KM families we described here.

Families of Kuramoto models and bounded symmetric domains

Abstract

We define the families of Kuramoto models (KM) related to bounded symmetric domains. The families include the Lohe unitary model and the spherical models as special cases. Our approach is based on the construction proposed by Watanabe and Strogats WS. We replace the Poincare disc and its boundary in the WS construction on the bounded symmetric domains and on the its Bergman-Shilov (BS) boundaries. In Cartan classifications there are four classical domains of types I-IV. Here we consider the domains of types I,II and III. For a fixed domain there is a decreasing chain of the BS boundaries components. This leads to the KM families we described here.

Paper Structure

This paper contains 23 sections, 134 equations.

Table of Contents

  1. Intruduction
  2. Preliminaries about SU(1,1) and Poincaré disk
  3. Kuramoto model
  4. Bounded symmetric domains of type I
  5. Hermitian symmetric spaces $X^I$
  6. Complexification
  7. Boundary structure
  8. Kuramoto families
  9. Examples
  10. ${\mathbb C} P^m$ model
  11. Gr$(2,4)$ model
  12. Classical domains of type II
  13. Symmetric space $X^{II}$
  14. The boundary structure
  15. Kuramoto families
  16. ...and 8 more sections