Slow-fast systems with stochastic resetting
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915189896511488 |
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| author | Bressloff, Paul C |
| author_facet | Bressloff, Paul C |
| contents | 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 $\dot{y}=ε(x-y)$ with $0<ε\ll 1$. 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 $p(x,t|y)$, which is the solution to a modified Liouville equation. We then show how for $r\gg ε$ the slow dynamics can be approximated by the averaged equation $dy/dτ=\E[x|y]-y$ where $τ=εt$, $\E[x|y]=\int x p^*(x|y)dx$ and $p^*(x|y)=\lim_{t\rightarrow \infty}p(x,t|y)$. 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 $y^*$ that is a solution of the equation $y^*=\E[x|y^*]$. Finally, we numerically explore deviations from averaging theory when $r=O(ε)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_07585 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Slow-fast systems with stochastic resetting Bressloff, Paul C Adaptation and Self-Organizing Systems Statistical Mechanics 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 $\dot{y}=ε(x-y)$ with $0<ε\ll 1$. 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 $p(x,t|y)$, which is the solution to a modified Liouville equation. We then show how for $r\gg ε$ the slow dynamics can be approximated by the averaged equation $dy/dτ=\E[x|y]-y$ where $τ=εt$, $\E[x|y]=\int x p^*(x|y)dx$ and $p^*(x|y)=\lim_{t\rightarrow \infty}p(x,t|y)$. 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 $y^*$ that is a solution of the equation $y^*=\E[x|y^*]$. Finally, we numerically explore deviations from averaging theory when $r=O(ε)$. |
| title | Slow-fast systems with stochastic resetting |
| topic | Adaptation and Self-Organizing Systems Statistical Mechanics |
| url | https://arxiv.org/abs/2503.07585 |