# Differential Transform Method as an Effective Tool for Investigating Fractional Dynamical Systems
## 1 Introduction  
Real technical and physical systems are usually best described by differential equations with nonlinear terms. In many cases, obtaining a satisfactory consistency with the real system requires the use of terms with fractional-order derivatives in the mathematical model [1, 2]. These terms implement the influence of the previous states of the system on its current solution in the model. As a result, the integration of the fractional-order differential equation (FODE) using the method based on the definition of fractional derivative requires considering, in each step, the values obtained in the previous steps; theoretically, these are all values obtained from the beginning of motion of the system. This makes such calculations very time-consuming and requires a great deal of computing power. Therefore, various methods have been developed to increase quality and speed up such calculations, including the Adomian Decomposition method [3], [PERSON]'s Homotopy perturbation method [4] and the generalized Adams-Bashforth-Moulton method [5]. One of these methods is the differential transform method, which is derived from the Taylor series expansion, but it does not require that the derivatives be evaluated symbolically. The DTM was introduced in 1986 by [PERSON] [6] to study electrical circuits described by linear and nonlinear equations of the integer order. Many subsequent studies demonstrated that DTM could be used to obtain exact solutions to differential equations through simple iteration diagrams [7, 8, 9, 10]. [PERSON] and [PERSON] [11] extended the DTM by introducing a fractional derivative transform, which made it possible to obtain good quality numerical solutions of fractional equations by using simple iterative DTM algorithms. It should be noted, however, that in a broader sense of topological dynamics, the FODE solution written in the Caputo sense causes some problems with interpretation. The attractor of a dynamical system defined by the integer-order ordinary differential equation \(\hat{x}=f(t,x0)\), where \(f\): \(JxR^{d}\to R^{d}\) is a continuous vector-valued function for a finite time interval \(J:=[0,T]\) and, for integer \(d\), is located in the system variable space \(R^{d}\). However, when we turn to solving fractional differential equations in the Caputo sense,  
\[D^{\alpha}_{0+}x(t)=f(x,t), \tag{1}\]where the fractional order of the equation \(\alpha\in(0,1)\), \(x\) is a vector in \(R^{d}\) and \(f\): \(JxR^{d}\to R^{d}\), the obtained solution is not a dynamic system in \(R^{d}\)[12]. The second condition defining a dynamical system (semi-group) [13; 14]  
\[x(x_{0},t+s)=x(x(x_{0},s),t) \tag{2}\]  
is not satisfied. Thus, the attractor of the dynamic system, defined, e.g., in [14; 15; 16], cannot be discussed. Formally, this makes it impossible to study solutions of the FODE written in the Caputo sense using nonlinear dynamics methods. As was shown in [17], FODEs written in the Caputo sense can generate dynamic systems in an infinitely dimensional Banach space. Given these topological conditions, solutions (including attractors) obtained by the DTM method should be treated as projections of global objects into the \(R^{d}\) space, where \(d=3\) in the case under consideration. The aim of this study is to assess the applicability of the DTM to the study of changes in the dynamics of the fractional Rossler system, following a change in the bifurcation parameter \(a\). In preliminary tests of the integer-order systems (Section 3.1), the optimal integration step of the DTM is determined. In this section, the DTM is compared with other integration schemes (RK4 and ode45), and the evaluation is based on comparing the convergence times of solutions obtained with these different methods. In Section 3.2, we examine the convergence between different cases of the DTM. Based on the obtained results, optimal orders of the DTM are determined for solving systems with different characteristics: integer-order periodic, integer-order chaotic, and fractional-order chaotic.