The branch processes of vortex filaments and Hopf Invariant Constraint on Scroll Wave

TL;DR

This paper uses topological current theory to analyze vortex filament evolution in excitable media, revealing how the Hopf invariant constrains filament interactions and preserves topological properties during complex branch processes.

Contribution

It introduces a topological framework to describe vortex filament dynamics and demonstrates the preservation of the Hopf invariant during filament interactions in scroll waves.

Findings

Hopf invariant remains conserved during filament branch processes

Vortex filaments generate or annihilate at limit points, and split or merge at bifurcation points

The 'exclusion principle' is a special case of the Hopf invariant constraint

Abstract

In this paper, by making use of Duan's topological current theory, the evolution of the vortex filaments in excitable media is discussed in detail. The vortex filaments are found generating or annihilating at the limit points and encountering, splitting, or merging at the bifurcation points of a complex function $Z(\vec{x},t)$. It is also shown that the Hopf invariant of knotted scroll wave filaments is preserved in the branch processes (splitting, merging, or encountering) during the evolution of these knotted scroll wave filaments. Furthermore, it also revealed that the "exclusion principle" in some chemical media is just the special case of the Hopf invariant constraint, and during the branch processes the "exclusion principle" is also protected by topology.