On the Stability of Fractal Differential Equations

TL;DR

This paper reviews fractal calculus and investigates the stability of fractal differential equations, providing conditions for stability, boundedness, and convergence, supported by examples and graphical illustrations.

Contribution

It introduces a Lyapunov-based stability analysis framework for fractal differential equations, extending classical stability concepts to fractal calculus.

Findings

Sufficient conditions for stability of fractal differential equations

Criteria for uniform boundedness and convergence of solutions

Examples demonstrating stability and behavior of solutions

Abstract

In this paper, we give a review of fractal calculus which is an expansion of standard calculus. Fractal calculus is applied for functions which are not differentiable or integrable on totally disconnected fractal sets such as middle-$\mu$ Cantor sets. Analogues of the Lyapunov functions and features are given for asymptotic behaviors of fractal differential equations. Stability of fractal differentials in the sense of Lyapunov is defined. For the suggested fractal differential equations, sufficient conditions for the stability and uniform boundedness and convergence of the solutions are presented and proved. We present examples and graphs for more details of the results.