Slow-fast systems with stochastic resetting

In this paper we explore the effects of instantaneous stochastic resetting on a planar slow-fast dynamical system of the form $\dot{x}=f(x)-y$ and $\dot{y}=\epsilon (x-y)$ with $0<\epsilon \ll 1$. We assume that only the fast variable $x(t)$ resets to its initial state $x_0$ at a random sequence of times generated from a Poisson process of rate $r$. Fixing the slow variable, we determine the parameterized probability density $p(x,t\mathrm{\mid }y)$, which is the solution to a modified Liouville equation. We then show how for $r\gg \epsilon$ the slow dynamics can be approximated by the averaged equation $dy\mathrm{/}d\tau =\text{E}[x\mathrm{\mid }y]-y$ where $\tau =\epsilon t$, $\text{E}[x\mathrm{\mid }y]=\int x{p}^{\ast }(x\mathrm{\mid }y)dx$ and $p^{\ast }(x\mathrm{\mid }y)={\lim }_{t\to \mathrm{\infty }}p(x,t\mathrm{\mid }y)$. We illustrate the theory for $f(x)$ given by the cubic function of the FitzHugh-Nagumo equation. We find that the slow variable typically converges to an $r$-dependent fixed point $y^{\ast }$ that is a solution of the equation $y^{\ast }=\text{E}[x\mathrm{\mid }{y}^{\ast }]$. Finally, we numerically explore deviations from averaging theory when $r=O(\epsilon )$.