Chaotic Dynamics of Conformable Semigroups via Classical Theory

This paper demonstrates that conformable semigroups do not constitute a genuinely new theory but are mathematically equivalent to classical $C_0$-semigroups under a nonlinear time reparametrization, thereby proving that their chaotic and hypercyclic dynamical properties are identical to those of the corresponding classical systems.

The Gist

Imagine you are watching a movie of a ball rolling down a hill. In the "classical" version of physics, the ball moves at a steady, predictable pace. Now, imagine a special version of this movie where the speed of the ball changes depending on how far it has traveled, but the path itself remains exactly the same. This is the core idea behind Conformable Calculus, a mathematical tool used to describe how things change over time.

For a long time, mathematicians wondered if this "special movie" (conformable dynamics) created entirely new, mysterious behaviors that classical physics couldn't explain. This paper, titled "Chaotic Dynamics of Conformable Semigroups via Classical Theory," answers that question with a surprising "no."

Here is the breakdown of what the authors discovered, using simple analogies:

1. The "Time Dial" Analogy

The authors introduce a concept called the "Conformable Clock."

Think of a standard clock as a ruler where every second is the same length. The conformable clock is like a rubber ruler. When you stretch it, the seconds get longer or shorter depending on where you are on the ruler.

• The Discovery: The authors proved that a "conformable" system isn't a new kind of physics. It is simply a classical system (the standard ball rolling down the hill) being watched through this rubber ruler.
• The Formula: They found a precise mathematical formula, $\Psi(t) = t^\delta / \delta$, that acts as the "dial" to switch between the two views. If you know how the ball moves in the classical world, you can instantly know how it moves in the conformable world just by adjusting the time dial.

2. The "Orbit" is Unchanged

In mathematics, an "orbit" is the path an object takes over time.

• The Metaphor: Imagine a runner on a track. In the classical view, they run at a constant speed. In the conformable view, they might sprint at the start and jog later, or vice versa.
• The Claim: The paper proves that the track itself doesn't change. The runner visits the exact same spots in the exact same order; they just arrive at those spots at different times.
• Why it matters: Because the path (the orbit) is identical, any property that depends on the path—like whether the runner eventually visits every part of the track (hypercyclicity) or loops back to the start (chaos)—is exactly the same in both worlds. If the classical system is chaotic, the conformable one is too. If the classical one is calm, the conformable one is calm.

3. The "Translator" for Chaos

The paper tackles a famous rule for detecting chaos called the Desch–Schappacher–Webb criterion.

• The Analogy: Imagine you have a complex, foreign language (conformable math) and a standard language (classical math). For years, people tried to write a new dictionary for the foreign language to understand chaos.
• The Solution: The authors showed you don't need a new dictionary. You just need a translator. They proved that you can take any rule for chaos from the classical world, "translate" it through their time-dial formula, and it works perfectly for the conformable world.
• The Result: They created a "conformable version" of the chaos rule, but it wasn't a new discovery; it was just the old rule wearing a different hat.

4. Real-World Examples: The "Spatial Clock"

The authors didn't just talk about time; they showed how this works with space, too.

• The Diffusion Example: They looked at a problem involving heat or particles spreading out (diffusion) in a weird, weighted space. By changing the "spatial clock" (stretching the space coordinate just like they stretched the time), they turned a complicated conformable equation into a simple, standard equation.
• The Transport Example: They showed a problem where things move (transport) could be turned into a simple "sliding" motion (translation) just by renaming the coordinates.
• The Takeaway: In both cases, the chaotic behavior of the complex conformable system was proven to be exactly the same as the chaotic behavior of the simple classical system.

Summary: What Does This Mean?

The paper's main message is one of simplification and clarity.

• Before: People thought conformable calculus might be a brand-new, mysterious branch of math with its own unique, unpredictable rules.
• Now: The authors show that conformable calculus is not a new branch. It is a repackaging of classical math.
• The "Fractional" Illusion: The "fractional" nature of these models isn't due to some deep, mysterious memory effect (like a system remembering its past). It is purely a result of re-labeling time and space.

In a nutshell: If you have a conformable model, you don't need to invent new theories to understand it. You just need to look at the corresponding classical model, apply a simple time or space transformation, and the answers are already there. The "chaos" isn't new; it's just the same old chaos seen through a distorted lens.

Technical Summary

Technical Summary: Chaotic Dynamics of Conformable Semigroups via Classical Theory

Problem Statement
The paper addresses a fundamental conceptual question regarding conformable calculus: to what extent do conformable models generate dynamics that are genuinely distinct from those produced by classical evolution families? While fractional-type models (e.g., Riemann–Liouville or Caputo derivatives) are widely used to describe anomalous transport and memory effects via nonlocal operators, conformable derivatives are characterized by their locality. For smooth functions, the conformable derivative reduces to a classical derivative multiplied by an explicit weight. The authors investigate whether this structural feature implies that conformable time evolution constitutes a new semigroup theory or if it can be fully interpreted within the established framework of classical $C_0$-semigroups.

Methodology
The authors employ a structural approach based on nonlinear time reparametrization. The core methodological tool is the introduction of the "conformable clock," defined for a fixed order $\delta \in (0, 1]$ as:
$$\Psi(t) = \frac{t^\delta}{\delta}, \quad t \ge 0.$$
The analysis proceeds through the following steps:

1. Operator-Theoretic Correspondence: The authors define a conformable $C_0$-$\delta$-semigroup $S_\delta$ and construct an associated family of operators $T(s) = S_\delta(\Psi^{-1}(s))$. They prove that $T$ is a classical $C_0$-semigroup on the same Banach space.
2. Generator Identification: They demonstrate that the $\delta$-generator of $S_\delta$ coincides exactly with the classical generator of $T$ on a common domain.
3. Functional Space Equivalence: The paper establishes that conformable Lebesgue ($L^{p,\delta}$) and Sobolev ($W^{m,p}_\delta$) spaces, defined via the weighted measure $d\mu_\delta(t) = t^{\delta-1}dt$, are isometrically equivalent to standard classical $L^p$ and Sobolev spaces via the change of variables $s = \Psi(t)$.
4. Dynamical Invariance: By leveraging the bijection of the conformable clock, the authors analyze orbit-based properties (hypercyclicity, periodic points, chaos) to determine their invariance under the time reparametrization.
5. Application via Conjugacy: The methodology is applied to specific partial differential equations (diffusion-transport and transport models) by introducing spatial changes of variables that reduce conformable operators to classical drift-diffusion or translation operators.

Key Contributions and Results

• Exact Structural Equivalence: The primary result is that every conformable $C_0$-$\delta$-semigroup $S_\delta$ admits the representation $S_\delta(t) = T(\Psi(t))$, where $T$ is a uniquely determined classical $C_0$-semigroup. This correspondence is exact at the infinitesimal level; the generators are identical, and mild solutions are in one-to-one correspondence under the reparametrization $s = \Psi(t)$.
• Invariance of Linear Dynamics: The paper proves that orbit-based dynamical properties are invariant under the conformable clock. Specifically, $S_\delta$ is $\delta$-hypercyclic (resp. $\delta$-chaotic) if and only if the associated classical semigroup $T$ is hypercyclic (resp. chaotic). The sets of periodic points and the asymptotic behaviors (decay to zero, approximation of states) are identical for both systems.
• Transfer of Criteria: As a direct consequence, classical chaos criteria can be transported to the conformable setting without new spectral analysis. The authors derive a conformable version of the Desch–Schappacher–Webb chaos criterion by applying the classical theorem to the generator of $T$.
• Unitary Equivalence in Applications: In the context of conformable diffusion-transport equations on weighted Hilbert spaces, the authors show that a nonlinear change of the spatial variable induces a unitary equivalence between the conformable generator and a classical drift-diffusion operator. Similarly, a conformable transport model is shown to be conjugate to the classical translation semigroup.
• Functional Framework: The study clarifies that conformable function spaces do not introduce a new integrability regime but rather encode the time reparametrization into the measure, allowing for the direct transfer of classical estimates and spectral arguments.

Significance and Claims
The paper claims to settle the conceptual status of conformable evolution within semigroup theory. It asserts that conformable dynamics does not produce a genuinely new fractional behavior characterized by intrinsic nonlocal memory effects, which are the hallmark of integral-type fractional models. Instead, the "fractional" character of conformable calculus is entirely encoded in a deterministic reparametrization of time (and space, in the applications).

The significance of the work lies in providing a precise operator-theoretic mechanism that explains why many conformable results mirror their classical counterparts. It establishes that whenever a conformable model can be reduced to a classical one by an explicit change of variables, qualitative dynamical properties and well-posedness results can be imported directly. This offers a systematic methodology for analyzing conformable models, shifting the focus from ad hoc computations to the application of established classical theory via the conjugacy mechanism.