发表机构
Technical University of Denmark（丹麦技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究一类实解析系统通过Hopf点的慢穿越，在全局假设下给出吸引与排斥慢流形指数级小分裂及最大延迟的渐近公式，方法基于几何与椭圆路径。
AI 中文摘要
本文考虑在$\mathbb R^3$中一类实解析系统通过Hopf点的慢穿越问题。经典地，（简化的）Shishkova问题：$\epsilon \frac{dz}{d\mu} = \lambda(\mu) z+\mathcal O(\epsilon)$，其中$\lambda(\mu) = \mu-ı$，已被用于说明在$\mu=0$处通过Hopf点所伴随的延迟。在本文中，我们考虑一般的$\lambda(\mu)$，满足$\lambda(0)\in i\mathbb R\setminus \{0\}$，$\real[\lambda'(0)]\ne 0$，并为复共轭添加一个方程。基本事实是，这些系统是$\mathbb R^3$中任何经历通过Hopf点慢穿越的实解析$(1,2)$慢快系统的局部正规形。在本文中，我们在对$\lambda$的额外（全局）假设下研究这些系统。特别地，我们假设$\lambda$在远离实轴处有一个简单零点。其余假设则涉及所谓的椭圆系统$\dot \mu = -ı\overline{\lambda(\mu)}$的不变流形以及水平集$\operatorname{Re}[ı{\lambda(\mu)}{\lambda(\overline \mu)}]=0$的性质。在这些假设下，我们提供了吸引和排斥慢流形指数级小分裂的渐近公式。作为推论，我们还获得了最大延迟的渐近公式。我们的方法是几何的，受到Hayes等人（2016）的工作（使用吹胀）和Neishtadt（1987,1988）的工作（使用所谓的椭圆路径）以及第二作者最近关于零-Hopf分岔开折中指数级小分裂的工作的启发。
英文摘要
In this paper, we consider the slow passage through a Hopf in $\mathbb R^3$ in a certain class of real-analytic systems. Classically, the (simplified) Shishkova problem: $ε\frac{dz}{dμ} = λ(μ) z+\mathcal O(ε)$, $λ(μ) = μ-ı$, has been used to illustrate the delay associated with slow passage through a Hopf at $μ=0$. In this paper, we consider a general $λ(μ)$ with $λ(0)\in i\mathbb R\setminus \{0\}$, $\real[λ'(0)]\ne 0$, and add an equation for the complex conjugate. It is a basic fact that these systems are local normal forms for any real-analytic $(1,2)$ slow-fast system in $\mathbb R^3$ experiencing slow passage through a Hopf. In this paper, we study these systems under additional (global) assumptions on $λ$. In particular, we suppose that $λ$ has a simple zero away from the real axis. The remaining assumptions then relate to invariant manifolds of the so-called elliptic system $\dot μ= -ı\overline{λ(μ)}$ as well as properties of the level set $\operatorname{Re}[ı{λ(μ)}{λ(\overline μ)}]=0$. Under these assumptions, we then provide an asymptotic formula for the exponentially small splitting of attracting and repelling slow manifolds. As a corollary, we also obtain an asymptotic formula for the maximal delay. Our approach is geometric and is inspired by the work of Hayes et al (2016) (using blowup) and Neishtadt (1987,1988) (using so-called elliptic paths) but also by recent work of the second author on exponentially small splitting in unfoldings of the zero-Hopf bifurcation.
CommentsFixed a mistake in Lemma 3.5. The main result has been refined accordingly
