Four-dimensional Zero-Hopf Bifurcation for a Lorenz-Haken System

Travelling waves and their bifurcations in the Lorenz-96 model

In this paper we study the dynamics of the monoscale Lorenz-96 model using both analytical and numerical means. The bifurcations for positive forcing parameter $F$ are investigated. The main analytical result is the existence of Hopf or Hopf-Hopf bifurcations in any dimension $n\geq4$. Exploiting the circulant structure of the Jacobian matrix enables us to reduce the first Lyapunov coefficient to an explicit formula from which it can be determined when the Hopf bifurcation is sub- or supercritical. The first Hopf bifurcation for $F>0$ is always supercritical and the periodic orbit born at this…

Research on Hopf Bifurcation and Stability of Heterogeneous Lorenz System with Single Time Delay

Time-delay chaotic systems refer to the hyperchaotic systems with multiple positive Lyapunov exponents. It is characterized by more complex dynamics and a wider range of applications as compared to those non-time-delay chaotic systems. In a three-dimensional general Lorenz chaotic system, time delays can be applied at different positions to build multiple heterogeneous Lorenz systems with a single time delay. Despite the same equilibrium point for multiple heterogeneous Lorenz systems with single time delay, their stability and Hopf bifurcation conditions are different due to the difference in…

形状指数、布劳威尔度与庞加莱-霍普夫定理

Shape index, Brouwer degree and Poincaré-Hopf theorem

本文研究向量场布劳威尔度与流动力学的关系，给出庞加莱-霍普夫定理及博苏克、赫希对径定理的新形式，并应用于计算洛伦兹奇异集隔离块中洛伦兹向量场的布劳威尔度。

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.

Lorenz 奇异集演化的 Conley 指标研究

A Conley index study of the evolution of the Lorenz strange set

本文用 Conley 指标理论研究 Lorenz 方程奇异集随参数的演化，分析全局吸引子的 Morse 分解并计算 Morse 方程，同时证明关于任意余维 Poincaré-Andronov-Hopf 分岔、双同宿环分岔及行进排斥子作用的一般定理。

通过持续同调检测霍普夫分岔：从时间序列中的一种函数准则

Topological Detection of Hopf Bifurcations via Persistent Homology: A Functional Criterion from Time Series

本文提出基于持续同调的时间序列霍普夫分岔检测方法，通过一维同调特征的出现或消失构建可计算准则，无需了解底层方程即可识别临界参数。