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

Fumihiko Nakamura

doi:10.3934/dcdsb.2018055

Article Contents

2018, Volume 23, Issue 6: 2457-2473. Doi: 10.3934/dcdsb.2018055

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

Fumihiko Nakamura

Department of Mathematics, Hokkaido university, Sapporo, Hokkaido, 060-0808, Japan

Received: March 31, 2017

Revised: June 30, 2017

Early access: February 2018

Published: June 2018

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Abstract

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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Mathematics Subject Classification: Primary: 37G15, 37A30; Secondary: 37E05.

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Figure 1. The region of the parameter space $(\alpha,\beta)$ in which $S_{\alpha,\beta}$ has a periodic point with period $n = 2,3,4,5$.

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Figure 2. Asymptotic periodicity illustrated. Here we show histograms obtain after iterating 5,000,000 initial points uniformly distributed on $[0,1]$ with $\alpha = 1/2, \beta = 4/7$, and $\theta = 1/14$ in Equation (2) for (a) $t = 200$; (b) $t = 201$; (c) $t = 202$; and (d) $t = 203$. A correspondence of the histograms for $t = 200$ and $t = 203$ indicates that the sequence of densities has period 3.

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Figure 3. Asymptotic stability illustrated. Here we show histograms obtain after 200 iterating 5,000,000 initial points uniformly distributed on $[0,1]$ with $\alpha = 1/2, \beta = 4/7$, and $\theta = 1/14+0.02$ in Equation (2).

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Figure 4. Asymptotic stability illustrated. Here we show histograms obtain after (a)200; (b)1,000; (c)10,000; (d)100,000 iterating 5,000,000 initial points uniformly distributed on $[0,1]$ with $\alpha = 1/2, \beta = 17/30$, and $\theta = 1/15$ in Equation (2).

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