### View Abstract

Abstract: The non-expanding piecewise linear map, known as the Nagumo-Sato (NS)

model, is described as $S_{alpha,beta}(x)=\alpha x+\beta ({\rm

mod} 1)$, where $0<\alpha,\beta<1$.

The NS model corresponds to a

special case of Caianiello's model, and it describes the simplified

dynamics of a single neuron. It is known that the system shows

periodic behavior of the trajectory for almost every (alpha,

beta). Moreover, the map has one discontinuous point when

$\alpha+\beta>1$, and this leads to a complicated structure for a

periodicity of the NS model. This structure is called a Farey

structure which relates to regions in which $S_{\alpha,\beta}$ has a

periodic point. An important feature of the this structure is that

there exists a region in which $S_{\alpha,\beta}$ has a periodic point

with period (m+n) between the region with period m and n. [1]

In this talk, we will explain above properties for the NS model first,

and next consider a perturbed dynamical system in which noise is

applied to the NS model, $x_{t+1}=S_{\alpha,\beta}(x_t) +\xi_t

({\rm mod}1)$ for $0<\alpha,\beta<1$, where ${\xi_t}$ are

independent random variables each having same density g satisfying

${\rm supp}{g}=[0,\theta]$ with $\theta >0$. We discuss asymptotic

properties for the Markov operator[2] corresponding to this perturbed

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 NS model has asymptotic periodicity or asymptotic

stability for almost all (alpha,beta) and theta. [3]

The topics of this talk are based on following references:

[1] F.Nakamura, Periodicity of non-expanding piecewise linear maps and

effects of random noises, Dynamical Systems, 30(2015), 450-467.

[2] A. Lasota and M. C. Mackey, Chaos, fractals, and noise: stochastic

aspects of dynamics, Springer Science and Business Media, 2013.

[3] F. Nakamura, Asymptotic behavior of non-expanding piecewise linear

maps in the presence of random noise, (accepted to DCDS-B).