Chaos of Caputo derivative

Technical Summary: Caputo Dynamics: Optimal Approximation and Chaos

Problem Statement
The paper investigates the dynamical properties of the Caputo fractional differentiation operator when realized as a bounded linear operator on specific coefficient Banach spaces of fractional power series. While the qualitative dynamics (such as hypercyclicity and chaos) of Caputo operators on Fréchet spaces have been previously studied, this work focuses on a quantitative analysis within a Banach space setting. The central problem is to determine the optimal cost of approximating a target vector from an initial state under the action of the Caputo operator. Specifically, the authors seek to solve two optimization problems: the "balanced approximation" (minimizing the maximum of initial displacement and terminal error) and the "exact-arrival" problem (finding the least-norm perturbation to reach a target exactly at a prescribed time). Furthermore, the study aims to establish sharp thresholds for hypercyclicity, mixing, Devaney chaos, and disjoint mixing of powers, and to characterize the structure of periodic points.

Methodology
The authors construct a specific class of Banach spaces, denoted $X^p_{\alpha, \rho}$, consisting of fractional power series $f(t) = \sum a_n \frac{\rho^n t^{\alpha n}}{\Gamma(\alpha n + 1)}$ where the coefficient sequence $(a_n)$ belongs to $\ell^p$. The norm is defined directly on these coefficients.

1. Shift Conjugacy: The core methodological insight is the identification of the Caputo operator $C_\alpha$ on $X^p_{\alpha, \rho}$ as being conjugate (via an isometric isomorphism) to a scaled unilateral backward shift $\rho B$ on $\ell^p$. Specifically, $J^{-1}_{\alpha, \rho} C_\alpha J_{\alpha, \rho} = \rho B$. This conjugacy reduces the complex fractional calculus problem to the well-understood dynamics of the backward shift.
2. Explicit Optimization: Leveraging the shift structure, the authors derive explicit formulas for the optimal perturbations. They utilize the properties of the backward shift $B$ and its right inverse (the forward shift) to solve the minimization problems for initial displacement and terminal error.
3. Periodic Reconstruction: Periodic points are constructed by repeating finite blocks of coefficients. The authors analyze the geometric repetition of these blocks to derive exact norm formulas and prove that these periodic points are dense.
4. Disjoint Mixing: For distinct positive powers of the operator, a "separated-block" construction is employed. By placing target coefficients in disjoint intervals of the sequence space, the authors control multiple targets simultaneously to establish disjoint mixing properties.
5. Embedding: The Banach spaces $X^p_{\alpha, \rho}$ are embedded into the Mittag–Leffler–Caputo Fréchet space studied in prior literature. The authors demonstrate that these embeddings are compact and dense, allowing the transfer of dynamical properties from the Banach realizations to the Fréchet setting.

Key Contributions and Results

• Optimal Approximation Formulas: The paper provides explicit, attained values for the local defect $\Delta_{A,n}(f, g)$, which measures the minimal simultaneous error in initial displacement and terminal error for an operator $A = \lambda C_\alpha$. The formula is given by:
  $$\Delta_{A,n}(f, g) = \frac{\|g - A^n f\|}{1 + (|\lambda|\rho)^n}$$
  A unique minimizing perturbation $h$ is explicitly constructed, attaining equal initial and terminal errors.
• Exact-Arrival Solution: For the problem of reaching a target $g$ exactly at time $n$ ($A^n(f+h)=g$), the paper establishes the existence of a unique least-norm solution. The norm of this perturbation is explicitly calculated as $(|\lambda|\rho)^{-n} \|g - A^n f\|$.
• Sharp Dynamical Thresholds: The study identifies a precise threshold for chaotic behavior. The operator $\lambda C_\alpha$ is hypercyclic, topologically mixing, and Devaney chaotic if and only if $|\lambda|\rho > 1$. Below or at this threshold ($|\lambda|\rho \le 1$), the operator is stable (orbits converge to zero) and possesses no non-trivial periodic points.
• Periodic Structure: Every periodic point is shown to correspond to a finite block of coefficients whose geometric repetition yields the full vector. The paper proves that every nonempty open set contains points of every sufficiently large minimal period, confirming the density of periodic points.
• Disjoint Mixing: The authors prove that for distinct positive powers $r_1, \dots, r_N$, the tuple $(\lambda C_\alpha)^{r_1}, \dots, (\lambda C_\alpha)^{r_N}$ is disjoint mixing if and only if $|\lambda|\rho > 1$. This is achieved through quantitative error estimates involving coefficient tails and geometric decay.
• Relation to Fréchet Spaces: The paper demonstrates that the Banach realizations $X^p_{\alpha, \rho}$ embed compactly and densely into the Mittag–Leffler–Caputo Fréchet space. This connection allows the recovery of the dynamical properties (mixing, chaos) for the Fréchet space realization, provided the radius parameter $\rho$ is chosen such that $|\lambda|\rho > 1$.

Significance and Claims
The paper claims to provide a "quantitative evaluation" of the local approximation condition, which serves as a norm formulation of classical transitivity. By converting coefficient estimates into explicit approximation costs, the work bridges the gap between abstract dynamical properties and computable metrics.
The authors state that their results "complement" previous continuous realizations by determining optimal costs in a specific Banach norm, whereas prior work described approximation via neighborhoods of graph norms. The explicit formulas for minimizing perturbations and the sharp thresholds for chaos are presented as the central results.
The study emphasizes that the "shift conjugacy" fixes the norm in which approximation costs are calculated, allowing for the derivation of exact formulas for balanced and exact-arrival problems. The paper concludes that these bounded realizations, connected via compact embeddings to the Fréchet space, offer a complete and computable framework for understanding the dynamics of Caputo differentiation, including the existence of dense stable invariant subspaces within chaotic realizations.