## 2 Orbit Trap Method
In the example, we used the following parameters: \(\beta~{}=~{}[0.5,0]^{T}\) in the S-iteration, orbit traps from Figure 1c with Euclidean metric, area \([-2\pi,2\pi]^{2}\), \(M~{}=~{}100\), \(E~{}=~{}16\), colour map from Figure 4d, and function \(f\) given by (1) with the p4 symmetry defined by the coefficients (again we give only the non-zero elements of the matrix) \(P_{03}~{}=~{}[0.215989,-0.150289]^{T}\), \(P_{04}~{}=~{}[0.0748772,0.670663]^{T}\), \(P_{13}~{}=~{}[0.741719,543639]^{T}\), \(P_{14}~{}=~{}[0.709028,-0.822211]^{T}\), \(P_{23}~{}=~{}[0.708072,-0.814562]^{T}\), \(P_{24}~{}=~{}[0.100835,0.40511]^{T}\), \(P_{30}~{}=~{}[-0.150289,-0.215989]^{T}\), \(P_{31}~{}=~{}[0.543639,-0.741719]^{T}\), \(P_{32}~{}=~{}[-0.814562,-0.708072]^{T}\), \(P_{40}~{}=~{}[0.670663,-0.0748772]^{T}\), \(P_{41}~{}=~{}[-0.822211,-0.709028]^{T}\), and \(P_{42}~{}=~{}[0.40511,-0.100835]^{T}\). In Figure 9, we see patterns generated with the following values of the \(\alpha\) parameter: (a) \([1.1,0.5]^{T}\), (b) \([1.1,0.8]^{T}\), (c) \([1.1,-0.8]^{T}\), and (d) \([1.1,4]^{T}\). Similar to the scalar case, when varying the \(y\) coordinate of the \(\alpha\) parameter, we are able to obtain a great variety of patterns. The generated patterns vary not only in shape but also in colouring. In Figure 9, we see a greater variety of colours than in Figure 8.

In the end, we present an interesting observation. When using orbit traps in which we have an empty space in the neighbourhood of the origin that is surrounded by the orbit traps (see orbit traps in Figures 1b,c and 3b) and function \(f\) with compatible symmetry type, then in the area near the origin of the generated pattern, we find self-similar structures. Thus, the patterns have a local fractal structure. To observe this, in Figure 10, we present magnification of the central part of the patterns from Figures 5c, 8c and 9a. If we magnify these areas further, then we get the same images repeatedly, so we clearly see the local self-similarity property. ## 5 Conclusions

In this paper, we proposed extensions of the orbit trap method introduced in [17]. The extensions rely on the use of (1) new orbit traps, (2) various metrics instead of the Euclidean one, and (3) the S-iteration. Moreover, we extended the S-iteration from scalar to vector parameters. The presented examples showed that we are able to generate patterns, which we had not been able to obtain previously. In our extensions, we used only the S-iteration. In the literature, there are many other iteration methods. For example, in [26], we can find a review of 17 different iterations and their dependencies. Moreover, there are iteration methods that use several mappings [14]. Thus, an interesting direction for further study would be the use of other iteration methods. Not all values of the parameters defining the \(f\) function that satisfy the appropriate symmetry condition give rise to interesting and artistic patterns [15]. To find the values that will give artistic patterns, [PERSON] et al. in [15] used a simple Monte Carlo method, so the evaluation of the pattern was made by the authors. Therefore, another direction for future studies would be developing an automatic method for finding patterns with artistic features. For instance, we could try to use genetic algorithms for this task because this type of method has proven itself in various methods that generate artistic images [27; 28; 29]. Conceptualization, K.G. and H.A. ; methodology, K.G. ; software, K.G. ; investigation, K.G. ; writing--original draft preparation, K.G. ; writing--review and editing, K.G. and H.A. ; visualization, K.G. All authors have read and agreed to the published version of the manuscript. This research was funded by Natural Science Foundation of China grant number 62062042. Figure 10: Magnifications of the areas near the origin of patterns from Figures 5c, 8c and 9a. These magnifications show the local self-similarity property of the patterns. (**a**) Figure 5c; (**b**) Figure 8c; and (**c**) Figure 9a.