Torus Knot in a Dissipative Fifth-Order System

Abstract

In order to show that some periodic orbits of a fifth-order system of magnetoconvection are embedded in a three-dimensional subspace, main projections onto a three-dimensional subspace from the five-dimensional space are numerically investigated. It is found that the periodic orbits are topologically equivalent to a (p, q)-torus knot, where its curve closes after rotating q times in the meridional direction and p times in the longitudinal direction. In terms of a braid word for the torus knot, a (2, 7)-torus knot is finally obtained in the fifth-order system through the complicated bifurcations under parameter variation. This suggests that topological invariants embedded in a three-manifold can be extracted from realistic dissipative higher dimensional dynamical systems.