We consider the perturbed dynamical system applied to non expanding piecewise linear maps on $[0, 1]$ 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 $n≥1$. In this paper, we give a sufficient condition for $n>1$, asymptotic periodicity, and for $n = 1$, asymptotic stability. That is, we show that there exists a threshold of noises $θ_{*}$ 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 $θ_{*}$. This result indicates that an existence of phenomenon called mode-locking is mathematically clarified for this perturbed system.
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Figure 2.
Asymptotic periodicity illustrated. Here we show histograms obtain after iterating 5,000,000 initial points uniformly distributed on
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