On the slow passage through a Hopf in generalized Shishkova systems: Exponential asymptotics and maximal delay

Abstract

In this paper, we consider the slow passage through a Hopf in $\mathbb R^3$ in generalized Shishkova systems. Classically, the Shishkova problem: $\epsilon \frac{dz}{d\mu} = \lambda(\mu) z+\mathcal O(\epsilon)$, $\lambda(\mu) = \mu-{\mathrm i}$, has been used to illustrate the delay associated with slow passage through a Hopf at $\mu=0$. In this paper, we consider general $\lambda$ and add an equation for the complex conjugate. We refer to these systems as generalized Shishkova systems and 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 $\lambda$. In particular, we suppose that $\lambda$ has a simple zero away from the real axis. The remaining assumptions then relate to invariant manifolds of the so-called elliptic system $\dot \mu = -{\mathrm i}\overline{\lambda(\mu)}$ as well as properties of the level set $\operatorname{Re}[{\mathrm i}{\lambda(\mu)}{\lambda(\overline \mu)}]=0$. 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.