[Paper Review] Fractional Stability

Vasily E. Tarasov|arXiv (Cornell University)|Nov 14, 2007

Fractional Differential Equations Solutions6 references7 citations

TL;DR

This paper introduces a novel concept of fractional stability for dynamical systems using fractional-order variations derived from Caputo derivatives. By redefining stability through fractional perturbations, it shows that systems unstable under classical Lyapunov criteria can exhibit asymptotic stability under fractional variation, thereby extending the scope of stability analysis to non-integer order dynamics.

ABSTRACT

A fractional generalization of variations is used to define a stability of non-integer order. Fractional variational derivatives are suggested to describe the properties of dynamical systems at fractional perturbations. We formulate stability with respect to motion changes at fractional changes of variables. Note that dynamical systems, which are unstable "in sense of Lyapunov", can be stable with respect to fractional variations.

Motivation & Objective

• To extend classical stability theory by introducing a fractional-order generalization of variation and stability.
• To address the limitation of Lyapunov stability in characterizing systems with long-range or fractal-like dynamics.
• To define stability with respect to fractional changes in system variables, enabling analysis of systems with non-integer perturbation responses.
• To demonstrate that fractional stability can capture stable behavior in systems deemed unstable under standard Lyapunov analysis.

Proposed method

• Uses Caputo fractional derivatives to define fractional variations of order α (m−1 < α ≤ m) via the expression δ^αf(x) = [δx]^α · C D^α_x f(x).
• Derives equations of motion for fractional variations by applying fractional variation to the system's differential equations, leading to D^1_t δ^αx_i = C D^α_x_j F_i(x) [δx_j]^α.
• Expresses the fractional power of variation [δx_j]^α in terms of the fractional variation δ^αx_j using the inverse of the Caputo derivative of the coordinate: [δx_j]^α = (C D^α_x_j x_j)^{-1} δ^αx_j.
• Transforms the system into a linear matrix differential equation: D^1_t Z(t) = A_α Z(t), where Z(t) is the vector of fractional variations and A_α is a matrix of fractional variational derivatives.
• Defines stability via the eigenvalues of A_α: asymptotic stability holds if all eigenvalues have negative real parts.
• Establishes a new stability criterion based on the sign of the real parts of eigenvalues of the fractional variational matrix A_α, generalizing Lyapunov stability to non-integer order.

Experimental results

Research questions

• RQ1 Can a dynamical system unstable under classical Lyapunov stability be stable under fractional variation?
• RQ2 How can fractional-order variations be mathematically defined using Caputo derivatives for dynamical systems?
• RQ3 What is the form of the differential equation governing fractional variations in a system of ODEs?
• RQ4 How does the stability of a system under fractional variation differ from its stability under integer-order variation?
• RQ5 Can fractional stability theory encompass classical Lyapunov stability as a special case when α = 1?

Key findings

• Fractional stability is defined via the eigenvalues of the matrix A_α derived from fractional variational derivatives, with asymptotic stability occurring when all eigenvalues have negative real parts.
• The system is asymptotically stable with respect to fractional variations if lim_{t→∞} ||δ^αx(t,α)|| = 0, which is determined by the spectral properties of A_α.
• A system that is unstable under Lyapunov stability (e.g., with positive real parts in the linearization) can still be asymptotically stable under fractional variation if the eigenvalues of A_α have negative real parts.
• The fractional variation δ^αx_i is expressed as δ^αx_i = [δx_i]^α · C D^α_x_i x_i, linking the fractional variation to the fractional derivative of the coordinate.
• The matrix A_α(α) = (C D^α_x_j x_j)^{-1} C D^α_x_j F_i(x) fully determines the evolution of fractional variations and thus the stability behavior.
• Fractional stability generalizes classical stability: when α = 1, the fractional stability condition reduces to the standard Lyapunov stability condition.