Lyapunov exponents of fractional-order systems: a definition from the diffusive representation, the ageing of memory, and what memory truncation measures

Technical Summary: Lyapunov Exponents of Fractional-Order Systems

Problem Statement
Lyapunov exponents are routinely computed for fractional-order systems (specifically those using the Caputo derivative) to detect chaos, yet their mathematical definition has remained ill-posed. Standard definitions rely on Oseledets' Multiplicative Ergodic Theorem, which requires a dynamical system to be a flow on a finite-dimensional space with an invariant measure and a linearized evolution that forms a cocycle. Fractional-order systems violate these conditions because:

1. Non-locality: The state at time $t$ depends on the entire history, not just a point in $\mathbb{R}^n$.
2. Lack of Time Translation Invariance: The system "ages"; shifting a solution in time changes the lower limit of the integral, resulting in a different equation.
3. Failure of the Cocycle Property: The linearized evolution of perturbations depends on the history of the perturbation, preventing the formation of a standard cocycle.

Consequently, existing numerical codes (e.g., those using predictor-corrector schemes with QR renormalization) compute quantities whose physical meaning is unclear. Empirical observations, such as the absence of a zero exponent in autonomous flows and the dependence of estimates on integration horizons, have remained unexplained.

Methodology
The paper resolves these issues by utilizing the diffusive representation of the Caputo operator.

1. Augmented System Formulation: The power-law kernel of the Caputo derivative is expressed as the Laplace transform of a positive measure. This allows the fractional system to be rewritten as an autonomous semilinear evolution equation on a Hilbert space $X = \mathbb{R}^n \times H$, where $H$ is a space of continuous memory modes $z(\omega, t)$ relaxing at rates $\omega$.
2. Cocycle Construction: On this infinite-dimensional Hilbert space, the linearized evolution of the augmented system satisfies the cocycle property, allowing the application of the Multiplicative Ergodic Theorem (specifically the versions by Ruelle, Mañé, and Thieullen for asymptotically compact cocycles).
3. Numerical Implementation: A Benettin algorithm is implemented on the discretized augmented system. The memory continuum is approximated using a sum-of-exponentials quadrature (Gauss-Jacobi for the singularity at $\omega=0$ and Gauss-Legendre for higher rates). The algorithm tracks tangent vectors in the subspace $\{\delta a = 0\}$ (perturbations tangent to the invariant set, excluding shifts in initial age).
4. Validation: The method is validated against the exact solution of linear systems involving the Mittag-Leffler function and applied to chaotic fractional Chen and Rössler systems.

Key Contributions and Theoretical Results

1. Definition of Exponents: The author defines Lyapunov exponents for fractional systems as the exponents of the autonomous augmented system restricted to the invariant set of stationary solutions. The Caputo trajectory is treated as an algebraically decaying perturbation ($O(t^{-\nu})$) of this stationary dynamics.
2. System Aging: It is proven that even if the observable trajectory $x(t)$ is bounded, the memory state $z(\cdot, t)$ grows unbounded as $t^{(1-\nu)/2}$. The system never forgets its initial condition; the "age" of the system manifests as a power-law drift in the memory norm.
3. Spectrum Structure:
   • Positive Exponents: Strictly positive exponents are well-defined, discrete, and independent of memory truncation. They characterize the true chaotic nature of the system.
   • Non-Positive Spectrum: The remainder of the spectrum is an essential band at zero. There is no exponential contraction in the fractional system for any $\nu < 1$; perturbations decay algebraically ($t^{-\nu}$).
   • The Zero Exponent: A zero exponent exists corresponding to the time-translation direction of the augmented system, but it is embedded in the essential band and cannot be separated by value in numerical computations.
4. Analysis of Truncation: The paper distinguishes between two types of memory truncation:
   • Truncating the Derivative (Grünwald-Letnikov): Applying the short-memory principle to the derivative cuts a summable kernel. This leaves the maximal exponent and the attractor geometry essentially unchanged.
   • Truncating the Integral (Volterra/Fixed-Memory): Truncating the integral form or removing slow modes in finite-mode realizations cuts a non-summable kernel. This fundamentally alters the system, deforming the attractor, reducing the maximal exponent by 25–50%, and potentially suppressing chaos entirely. In finite-mode realizations, this introduces spurious negative exponents equal to $-1/\tau_m$ (the inverse of the memory window).

Numerical Results

• Fractional Chen and Rössler Systems: The maximal Lyapunov exponent is reproducible and insensitive to the resolution of the memory band ($\omega_{min}$). For the Chen system at $\nu=0.9$, the maximal exponent stabilizes around 2.25.
• Vanishing Negative Exponents: All non-positive exponents converge to zero as the memory resolution increases. The residual negative values observed in finite-time computations are artifacts of the discretization and decay algebraically, not exponentially.
• Comparison with Existing Codes: Codes that reset memory (e.g., via QR renormalization) measure "memory-reset exponents," which are contraction rates of the observable over the renormalization interval, not true system exponents. These values drift toward zero as the interval increases.
• Impact of Truncation: Numerical experiments confirm that fixed-memory Volterra codes and finite-mode realizations with short windows ($\tau_m \approx 100$) destroy the chaotic attractor, whereas the short-memory principle applied to the derivative preserves it.

Significance and Claims
The paper claims to provide the first rigorous definition of Lyapunov exponents for fractional-order systems that is independent of numerical truncation. It clarifies that:

• The "chaos" of a fractional system is defined by its positive exponents, which are robust.
• The negative exponents reported by standard codes are not properties of the system but artifacts of the memory treatment (either the renormalization interval or the memory window).
• The "aging" of the system is a fundamental feature: the memory of the initial condition grows indefinitely, meaning the system is never truly stationary.
• Practical implications for the field are significant: results obtained by truncating the integral form (common in fixed-memory codes) describe a different, less chaotic system than the intended model. To obtain reliable results, one must either use the short-memory principle on the derivative or employ the augmented-system approach with a sufficiently resolved memory band.

The author notes that the existence of the compact invariant set for chaotic fractional systems remains a hypothesis (though numerically supported), and the rate of convergence of finite-time estimates is conjectured to include a memory term proportional to $t^{-\nu}$, which is difficult to isolate numerically for $\nu > 0.5$.