Chaotic Dynamics of Conformable Semigroups via Classical Theory
Mohamed Khoulane, Aziz El Ghazouani, M'hamed Elomari
TL;DR
The paper establishes a precise structural equivalence between conformable evolution (with order $\delta\in(0,1]$) and classical $C_0$-semigroups by introducing the conformable clock $\Psi(t)=\frac{t^{\delta}}{\delta}$, yielding $\mathcal{S}_\delta(t)=\mathcal{T}(\Psi(t))$ where $\mathcal{T}$ is classical. It proves generator coincidence $\mathcal{A}_\delta=\mathcal{B}$ on a common domain, and shows conformable mild solutions correspond to classical ones under $s=\Psi(t)$; the conformable Lebesgue/Sobolev scales $L^{p,\delta}$ and $W^{m,p}_\delta$ are isometrically equivalent to their classical counterparts through the same clock. The paper then demonstrates that orbit-based dynamics, including $\delta$-hypercyclicity and $\delta$-chaos, are invariant under the clock, enabling transport of the Desch--Schappacher--Webb chaos criterion to conformable time. In applications, conformable problems on weighted spaces are reduced to classical problems via unitary conjugacies, yielding immediate chaoticity results and clarifying which dynamical features are intrinsic to the model versus clock-induced effects.
Abstract
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.