Chaotic Dynamics of Conformable Semigroups via Classical Theory

TL;DR

This paper shows that conformable semigroups are essentially classical semigroups viewed through a nonlinear time change, clarifying which dynamical features are intrinsic versus time-reparametrization effects.

Contribution

It establishes a precise correspondence between conformable and classical $C_0$--semigroups via a nonlinear time change, clarifying their structural relationship.

Findings

Conformable semigroups are equivalent to classical semigroups under a nonlinear time reparametrization.

Orbit-based properties like hypercyclicity and chaos are invariant under the conformable clock.

A conformable chaos criterion is derived by transferring classical results through the time change.

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 $\delta\in(0,1]$, we introduce the conformable clock \[ \Psi(t)=\frac{t^\delta}{\delta}, \] and prove that every $C_0$--$\delta$--semigroup $\mathcal S_\delta$ admits the representation \[ \mathcal S_\delta(t)=\mathcal T(\Psi(t)), \] 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 $\delta$--generator of $\mathcal S_\delta$ 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 $s=\Psi(t)$. In particular, orbit sets are unchanged by the conformable clock, so orbit-based linear dynamical properties are invariant; $\delta$--hypercyclicity and $\delta$--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 $L^{p,\delta}$, 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.