| Abstract: |
| We consider following perturbed systems, $x_{t+1}=S_{\alpha,\beta}(x_t) +\xi_t \ ({\rm mod}\ 1)$ where $S_{\alpha,\beta}(x)=\alpha x+\beta \ ({\rm mod} \ 1)$, $(0<\alpha,\beta<1)$, and $\{\xi_t\}$ are independent random variables each having same density $g$ which is supported on $[0,\theta]$ with $\theta >0$. The non-expanding piecewise linear map $S_{\alpha,\beta}$ is known as the Nagumo-Sato (NS) model, and it describes the simplified dynamics of a single neuron. It is known that the system $S_{\alpha,\beta}$ shows a periodic behavior of the trajectory for almost every $(\alpha, \beta)$. In this talk, we discuss asymptotic properties for the Markov operator corresponding to the perturbed NS model. Especially, we focus on the properties of ``asymptotic periodicity`` and ``asymptotic stability''. Then I will introduce our main result which tells us which the Markov operator corresponding to the perturbed system has either asymptotic periodicity or asymptotic stability for each $(\alpha,\beta)$ and $\theta$. |
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