Review Reports - Differential Transform Method as an Effective Tool for Investigating Fractional Dynamical Systems

Round 1

Reviewer 1 Report

In this article, on the example of the fractional-order Rossler system (with Caputo fractional-order differential operator), the authors study the application of the differential transform method (DTM) for numerical integration of the system and show that the DTM results are more resistant to changes in the "fractionality" of the system.

Overall material of the article is presented pretty tidy, all the descriptions and results of experiments are mostly clear for understanding.
Nevertheless, I have a general yet important remark for the authors, which I hope will be clarified.

According to definition, in order to say that a solution x(t, x_0) of some ODE \dot{x} = f(x) generates a dynamical system (DS), one should check several properties (see e.g. [1,2]), i.e.:
1) x(0, x_0) = x_0 ;
2) x(t+s, x_0) = x(t, x(s, x_0)), \forall t,s \in [0,+\infty) [semi-group property];
3) x(t, x_0) - continuous function w.r.t time t, and initial coordinate x_0;

It is known (see e.g. [3]) that fractional-order differential equations in the Caputo sense do not generate a DS in \mathbb{R}^n, because the semi-group property 2) is not satisfied.
This also means that without properly defined DS in \mathbb{R}^n, it is not correct to call the objects in Fig. 4 as "attractors" (according to definition (see e.g. [1,2]), attractor is a compact invariant locally (globally) attractive set of a DS).

In fairness, it is possible to prove (see e.g. [4]) that a fractional-order differential equation in the Caputo sense generates a DS, but in a (infinite-dimensional) Banach space and (possible) attractors of such DS will be in Banach space as well.

In my opinion, the authors should be careful when using the terms "dynamic system", "attractors" and, at least, they should discuss this issue in their article. Or maybe even point out the connections between the considered mathematical objects such as system (4), chaotic sets in Fig. 4 and "rigorous" dynamical system and attractors in Banach space, generated by system (4).

Summarizing all written above, I find this article interesting and worth of being published in the MDPI Applied Sciences journal, but after a minor revision.

References:
[1] G.A. Leonov, N.V. Kuznetsov, T.N. Mokaev. Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion. The European Physical Journal Special Topics, 224(8):1421–1458, 2015.

[2] N.V. Kuznetsov, T.N. Mokaev, O.A. Kuznetsova, E.V. Kudryashova. The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension. Nonlinear Dynamics, 102:713–732, 2020.

[3] N.D. Cong, H.T. Tuan. Generation of nonlocal fractional dynamical systems by fractional differential equations. J. Integral Equ. Appl. 29, 585–608, 2017.

[4] T.S. Doan, P.E. Kloeden. Semi-dynamical systems generated by autonomous Caputo fractional differential equations. Vietnam Journal of Mathematics, 1-11, 2021.