On the slow passage through a Hopf in a class of real-analytic systems: Exponential asymptotics and maximal delay
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.