Chaotic Dynamics of Conformable Semigroups via Classical Theory

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 $\hat{I}´\in (0,1]$, we introduce the conformable clock \[ Ψ(t)=\frac{t^δ}δ, \] and prove that every $C_0$--$\hat{I}´$--semigroup ${\mathcal{S}}_{\hat{I}}´$ 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 $\hat{I}´$--generator of ${\mathcal{S}}_{\hat{I}}´$ 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 $s=\hat{I}¨(t)$. In particular, orbit sets are unchanged by the conformable clock, so orbit-based linear dynamical properties are invariant; $\hat{I}´$--hypercyclicity and $\hat{I}´$--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 $L^{p,\hat{I}´}$, 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.

23 pages