Chaotic Dynamics of Conformable Semigroups via Classical Theory

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.

💡 Research Summary

The paper investigates the dynamical consequences of using the conformable derivative, a “fractional‑type” operator that retains locality: for smooth functions the conformable derivative of order δ∈(0,1] is simply a classical derivative multiplied by the weight t^{1‑δ}. The authors exploit this structural property to show that conformable time evolution does not generate a genuinely new semigroup theory; instead, it can be completely understood as a classical C₀‑semigroup observed through a nonlinear re‑parametrization of time.

The central construction is the conformable clock
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