Periodicity of non-expanding piecewise linear maps and effects of random noises

Abstract

We consider non-expanding piecewise linear maps described as Sα, β(x) = αx + β(mod 1), where 0 < α, β < 1. This transformation is known as the Nagumo– Sato (NS) model. The NS model corresponds to a special case of Caianiello's model, and it describes simplified dynamics of a single neuron. In this thesis, we can describe regions explicitly in which Sα, β has a periodic point with period n for an arbitrary integer n, and clarify that these regions are associated with the Farey series, which are already observed experimentally in [5]. We shall explain a mathematical rigorous reason for a complexity of a periodicity of Sα, β that is observed by Aihara-Oku. Furthermore, we introduce an example that noises induce the asymptotic periodicity in the sense of Lasota-Machey, even if an original transformation has no periodicity.