Computer-assisted proofs of Hopf bubbles and degenerate Hopf bifurcations

TL;DR

This paper introduces a computer-assisted methodology to rigorously prove the existence of complex bifurcation structures like Hopf bubbles and degenerate Hopf bifurcations in various differential equations, including delay differential equations.

Contribution

The paper develops a novel computer-assisted approach for proving nonlocal bifurcation structures in differential equations, applied to multiple models.

Findings

Successfully proved Hopf bubbles in FitzHugh-Nagumo model

Confirmed degenerate Hopf bifurcations in Lorenz-84 model

Validated bifurcation structures in a delay SI model

Abstract

We present a computer-assisted approach to prove the existence of Hopf bubbles and degenerate Hopf bifurcations in ordinary and delay differential equations. We apply the method to rigorously investigate these nonlocal bifurcation structures in the FitzHugh- Nagumo equation, the extended Lorenz-84 model and a time-delay SI model.