Chaos of Caputo derivative
Abstract
We study bounded realizations of Caputo differentiation on coefficient Banach spaces of fractional power series. The main contribution is an explicit quantitative analysis of local approximation and periodic reconstruction. We determine the least possible simultaneous error in the initial displacement and the prescribed target. The optimal value is attained, and a minimizing perturbation is given explicitly. We also solve the exact-arrival problem and establish uniqueness of its least-norm solution. These formulas provide computable approximation times and identify the sharp threshold for hypercyclicity, mixing and Devaney chaos. Every periodic point is described by a finite block whose geometric repetition yields exact norm formulas. The resulting approximations attain any prescribed sufficiently large minimal period near any given vector. For distinct positive powers, a simultaneous block construction gives quantitative error estimates and the same sharp threshold for disjoint mixing. Compact embeddings relate the Banach realizations to the Mittag-Leffler-Caputo Fréchet space. They also exhibit dense stable invariant subspaces inside chaotic realizations. The local approximation condition is a norm formulation of classical transitivity. Its quantitative evaluation, together with explicit periodic and simultaneous reconstruction, provides the central results of the study.
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BibTeX
@misc{nafia2026chaos,
title = {{Chaos of Caputo derivative}},
author = {El-Mahdi Nafia and Aziz El Ghazouani and M'hamed El Omari},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31255},
eprint = {2610.31255},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31255},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Chaos of Caputo derivative
AU - El-Mahdi Nafia
AU - Aziz El Ghazouani
AU - M'hamed El Omari
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31255
UR - https://arxiv.org/abs/2610.31255
AB - We study bounded realizations of Caputo differentiation on coefficient Banach spaces of fractional power series. The main contribution is an explicit quantitative analysis of local approximation and periodic reconstruction. We determine the least possible simultaneous error in the initial displacement and the prescribed target. The optimal value is attained, and a minimizing perturbation is given explicitly. We also solve the exact-arrival problem and establish uniqueness of its least-norm solution. These formulas provide computable approximation times and identify the sharp threshold for hypercyclicity, mixing and Devaney chaos. Every periodic point is described by a finite block whose geometric repetition yields exact norm formulas. The resulting approximations attain any prescribed sufficiently large minimal period near any given vector. For distinct positive powers, a simultaneous block construction gives quantitative error estimates and the same sharp threshold for disjoint mixing. Compact embeddings relate the Banach realizations to the Mittag-Leffler-Caputo Fréchet space. They also exhibit dense stable invariant subspaces inside chaotic realizations. The local approximation condition is a norm formulation of classical transitivity. Its quantitative evaluation, together with explicit periodic and simultaneous reconstruction, provides the central results of the study.
ER -