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

Abstract

We consider non-expanding piecewise linear maps described as S_{α, 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 S_{α, 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 S_{α, 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.

Keywords:

• non-expanding piecewise linear maps
• complexity of periodicity
• random noise

AMS Classification:

• 37C25
• 26A18

Acknowledgements

I would like to express my deepest gratitude to Prof. Michiko Yuri, who gave me a lot of instructions and themes for study, and people who supported me about many things.

Disclosure statement

No potential conflict of interest was reported by the author.