Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise

We consider the perturbed dynamical system applied to non expanding piecewise linear maps on \begin{document}$[0, 1]$\end{document} which describe simplified dynamics of a single neuron. It is known that the Markov operator generated by this perturbed system has asymptotic periodicity with period \begin{document}$n≥1$\end{document} . In this paper, we give a sufficient condition for \begin{document}$n>1$\end{document} , asymptotic periodicity, and for \begin{document}$n = 1$\end{document} , asymptotic stability. That is, we show that there exists a threshold of noises \begin{document}$θ_{*}$\end{document} such that the Markov operator generated by this perturbed system displays asymptotic periodicity (asymptotic stability) if a maximum value of noises is less (greater) than \begin{document}$θ_{*}$\end{document} . This result indicates that an existence of phenomenon called mode-locking is mathematically clarified for this perturbed system.