# Differential Transform Method as an Effective Tool for Investigating Fractional Dynamical Systems
## 3 Results
### Finding the Optimal Order of DTM  
The results discussed in Section 3.1 were obtained for the integer-order chaotic (\(a=0.4079\)) Rossler system. Since the chaoticity and fractionality of the system influence its dynamic characteristics, the optimal DTM order was determined separately for three cases: (a) an integer-order periodic (a = 0.4) system, (b) integer-order chaotic (\(a=0.4079\)) system, and (c) fractional-order (\(a=0.8\)) chaotic (\(a=0.4079\)) system. The optimal order of the DTM was determined each time by obtaining convergence for solutions whose orders differed by 1. Taking into account the nonchaotic integer-order system, we noticed that, for the third-order solution (DTM-3), the results were almost identical to the DTM-4 results. The DTM-4 and DTM-3 results are exactly the same in a 2D phase space (Figure 3b), but they differ in a 3D phase space (Figure 3c). This fact results from a phase difference between the two solutions (Figure 3a). Basically, the two results on the 3D plot match. In Figure 3b, we can only observe a local color change resulting from a different order of drawing the curves by the Matlab graphical function, which results from a phase difference between the two solutions. The phase difference occurs because DTM3 and DTM4 have different accuracies. As a result, their trajectories diverge and are attracted to the attractor at different locations of the phase space. For this case, a full agreement is obtained between the DTM-5 and DTM-6 results, which means that the optimal order of the DTM for integer-order periodic systems is 5. A comparison of different solutions of the integer-order chaotic Rossler system showed an exact convergence between the DTM-11 and DTM-12 results. Therefore, it can be claimed that the DTM order equal to 11 is optimal for this case. For the fractional and chaotic Rossler system, a total convergence was obtained between the DTM-14 and DTM-15 results. This proves that the greatest accuracy of solutions is generally obtained for the DTM order of 14. Despite the fact that the best solution of the chaotic fractional system is obtained for the DTM of order 14, the solutions of the lower orders are qualitatively similar to it. Figure 2: (**a**) Duration of convergence between the DTM and optimal ODE (\(h_{max}=0.002\) s). (**b**) Duration of convergence between the ODE and optimal DTM (order 10, \(h_{DTM}=0.01\) s). (**c**) Duration of convergence between the RK4 and optimal ODE result. In all cases, the best results are marked with red points. Figure 4a-c,f show the 2D phase diagrams of the results obtained by the DTM of the orders 2, 3, 5 and 10, respectively. Figure 4d,f compare the 3D phase diagrams of the DTM-10 result with DTM-2 and DTM-5, respectively. Despite the fact that the DTM-2 result in the 2D phase diagram (Figure 4a) differs from the DTM-10 result (Figure 4f), the shape of both attractors in the 3D phase diagram is similar (Figure 4d). In turn, a comparison of the DTM-5 and DTM-10 results reveals that they are similar both in the 2D (Figure 4c,f) and 3D (Figure 4e) phase diagrams. All calculations presented in this paper were made in MATLAB, with an accuracy of 16 significant digits. While performing preliminary and actual test calculations, it was observed that changing one numerical formula to another (mat  
Figure 4: Results obtained for the fractional chaotic Rössler system, \(h_{DTM}\) = 0.01 s. Results in a two-dimensional phase space: (**a**) DTM of the second order, (**b**) DTM of the third order, (**c**) DTM of the fifth order, and (**f**) 10 th order DTM. Results in a three-dimensional phase space: (**d**) comparison of DTM-2 and DTM-10 results, (**e**) comparison of DTM-5 and DTM-10 results. Figure 3: (**a**) DTM3 vs. DTM4 for the periodic Rössler system of the integer order (\(h_{DTM}\) = 0.01 s). Two identical results shifted in phase. (**b**) In a 2D phase space, the DTM3 result agrees with the DTM4 result. (**c**) In a 3D space, the phase shift makes a difference. would change the simulation results.