User guide on Hopf bifurcation and time periodic orbits with pde2path

Computer-assisted proofs of Hopf bubbles and degenerate Hopf bifurcations

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.

Rigorous verification of Hopf bifurcations via desingularization and continuation

In this paper we present a general approach to rigorously validate Hopf bifurcations as well as saddle-node bifurcations of periodic orbits in systems of ODEs. By a combination of analytic estimates and computer-assisted calculations, we follow solution curves of cycles through folds, checking along the way that a single nondegenerate saddle-node bifurcation occurs. Similarly, we rigorously continue solution curves of cycles starting from their onset at a Hopf bifurcation. We use a blowup analysis to regularize the continuation problem near the Hopf bifurcation point. This extends the applicab…

Rigorous validation of a Hopf bifurcation in the Kuramoto-Sivashinsky PDE

We use computer-assisted proof techniques to prove that a branch of non-trivial equilibrium solutions in the Kuramoto-Sivashinsky partial differential equation undergoes a Hopf bifurcation. Furthermore, we obtain an essentially constructive proof of the family of time-periodic solutions near the Hopf bifurcation. To this end, near the Hopf point we rewrite the time periodic problem for the Kuramoto-Sivashinsky equation in a desingularized formulation. We then apply a parametrized Newton-Kantorovich approach to validate a solution branch of time-periodic orbits. By construction, this solution b…

Hopf 分岔点的优化

Optimization of Hopf bifurcation points

本文提出一种基于扩展系统约束优化的数值方法，用于控制动力系统中 Hopf 分岔的位置、稳定性及振荡频率，并在生物与流体动力学模型中验证其有效性。

从Hopf到Neimark-Sacker分岔：一种计算算法

From Hopf to Neimark-Sacker bifurcation: a computational algorithm

本文提出一种基于傅里叶谱方法的算法，用于逼近Neimark-Sacker分岔点处产生的不变环面，通过参数化方法实现低阶和高阶近似。