Data-Driven Approach for Determining Stability of Fractional-Order Delay-Differential Equations Using Orthonormal History Functions

Abstract It is well known that delay-differential equations (DDEs) are infinite dimensional, thus making their stability determination difficult. Incorporating a fractional-order term, which is also infinite dimensional, in a DDE compounds this difficulty. The resulting equation is known as a fractional-order delay-differential equation (FODDE). In this work, we present a method to determine the linear stability of a time-periodic FODDE of any order. Our approach is motivated by the Floquet theory, a popular method to determine the linear stability of time-periodic ordinary-differential equation (ODE) systems. Analogous to the Floquet theory, we consider one basis function at a time as the history function and compute the corresponding solutions by integrating the FODDE. These solutions are then projected onto the basis functions to construct the approximate Floquet transition matrix, the magnitude of whose largest eigenvalue determines the stability of the FODDE. Notably, our method does not rely on an explicit knowledge of the FODDE and merely requires the response of the system governed by the FODDE, to prescribed inputs, making our approach suitable for data-driven stability of delayed systems with fractional damping. To establish the validity, accuracy, and efficiency of our approach, we determine the stability of six linear time-periodic second-order fractional delay differential equations of varying nature. We show, in particular, that our framework can handle various types of delays, such as constant, time-periodic, discrete, and distributed, with equal ease. Keywords: fractional derivative, delays, Floquet theory, Eigenvalues, stability charts