Kato's chaos created by quadratic mappings associated with spherical orthotomic curves

Takashi Nishimura

Journal of Singularities
volume 21 (2020), 205-211

Received: 13 April 2018. Received in revised form: 12 May 2019.

DOI: 10.5427/jsing.2020.21l

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Abstract:

In this paper, we first show that for a given generic spherical curve γ: I -> S^n and a generic point P in S^n, the spherical orthotomic curve relative to γ and P naturally yield a simple quadratic mapping Φ_P: R^{n+1}-> R^{n+1}. Since S^n is compact and Φ_P|_{S^n}: S^n -> S^n is the spherical counterpart of the trivial expanding mapping x -> 2x, it is natural to expect a chaotic behavior for the iteration of Φ_P|_{S^n}. Accordingly, we show that Φ_P|_{S^n} (and incidentally Φ_P|_{D^{n+1}} as well) actually creates Kato's chaos. Therefore, by investigating spherical orthotomic curves, %it turns out that an example of singular quadratic mapping creating Kato's chaos is naturally obtained.

2010 Mathematical Subject Classification:

37D45, 54H20, 26A18, 39B12

Key words and phrases:

Kato's chaos, Sensitivity, Accessibility, Quadratic mapping, Spherical orthotomic curve

Author(s) information:

Takashi Nishimura
Research Institute of Environment
and Information Sciences
Yokohama National University
Yokohama 240-8501, Japan
email: nishimura-takashi-yx@ynu.ac.jp