Chaotic Dynamics of Conformable Semigroups via Classical Theory
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866911428077682688 |
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| author | Khoulane, Mohamed Ghazouani, Aziz El Elomari, M'hamed |
| author_facet | Khoulane, Mohamed Ghazouani, Aziz El Elomari, M'hamed |
| contents | Conformable derivatives involve a fractional parameter while preserving locality: on smooth functions they reduce to a classical derivative multiplied by an explicit weight. Exploiting this structural feature, we show that conformable time evolution does not give rise to a genuinely new semigroup theory. Rather, it can be fully interpreted as a classical $C_0$--semigroup observed through a nonlinear change of time. For $δ\in(0,1]$, we introduce the conformable clock \[ Ψ(t)=\frac{t^δ}δ, \] and prove that every $C_0$--$δ$--semigroup $\mathcal S_δ$ admits the representation \[ \mathcal S_δ(t)=\mathcal T(Ψ(t)), \] where $\mathcal T$ is a uniquely determined classical $C_0$--semigroup on the same state space. This correspondence is exact at the infinitesimal level: the $δ$--generator of $\mathcal S_δ$ coincides with the generator of $\mathcal T$ on a common domain, and conformable mild solutions are in one-to-one correspondence with classical mild solutions under the reparametrization $s=Ψ(t)$. In particular, orbit sets are unchanged by the conformable clock, so orbit-based linear dynamical properties are invariant; $δ$--hypercyclicity and $δ$--chaos coincide with their classical counterparts. As an application, we derive a conformable version of the Desch--Schappacher--Webb chaos criterion by transporting the classical result. The analysis is carried out in conformable Lebesgue spaces $L^{p,δ}$, which are shown to be isometrically equivalent to standard $L^p$ spaces, allowing a direct transfer of estimates and spectral arguments. Altogether, the results clarify which dynamical features of conformable models are intrinsic and which arise solely from a nonlinear change of time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06782 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Chaotic Dynamics of Conformable Semigroups via Classical Theory Khoulane, Mohamed Ghazouani, Aziz El Elomari, M'hamed Dynamical Systems Mathematical Physics Analysis of PDEs Classical Analysis and ODEs 47D06, 47A10, 37D45, 26A33, 35K57 Conformable derivatives involve a fractional parameter while preserving locality: on smooth functions they reduce to a classical derivative multiplied by an explicit weight. Exploiting this structural feature, we show that conformable time evolution does not give rise to a genuinely new semigroup theory. Rather, it can be fully interpreted as a classical $C_0$--semigroup observed through a nonlinear change of time. For $δ\in(0,1]$, we introduce the conformable clock \[ Ψ(t)=\frac{t^δ}δ, \] and prove that every $C_0$--$δ$--semigroup $\mathcal S_δ$ admits the representation \[ \mathcal S_δ(t)=\mathcal T(Ψ(t)), \] where $\mathcal T$ is a uniquely determined classical $C_0$--semigroup on the same state space. This correspondence is exact at the infinitesimal level: the $δ$--generator of $\mathcal S_δ$ coincides with the generator of $\mathcal T$ on a common domain, and conformable mild solutions are in one-to-one correspondence with classical mild solutions under the reparametrization $s=Ψ(t)$. In particular, orbit sets are unchanged by the conformable clock, so orbit-based linear dynamical properties are invariant; $δ$--hypercyclicity and $δ$--chaos coincide with their classical counterparts. As an application, we derive a conformable version of the Desch--Schappacher--Webb chaos criterion by transporting the classical result. The analysis is carried out in conformable Lebesgue spaces $L^{p,δ}$, which are shown to be isometrically equivalent to standard $L^p$ spaces, allowing a direct transfer of estimates and spectral arguments. Altogether, the results clarify which dynamical features of conformable models are intrinsic and which arise solely from a nonlinear change of time. |
| title | Chaotic Dynamics of Conformable Semigroups via Classical Theory |
| topic | Dynamical Systems Mathematical Physics Analysis of PDEs Classical Analysis and ODEs 47D06, 47A10, 37D45, 26A33, 35K57 |
| url | https://arxiv.org/abs/2602.06782 |