## 1 Introduction  
A pattern is a generic term for any type of repeated, often regular, arrangement [1]. It is also a design in which lines, shapes, forms, or colours are repeated. The repeated part is called a motif. A motif can be repeated and arranged in many ways to create different types of patterns. In regular patterns, the motif (or motifs) is repeated predictably [2]. It could be the same each time, or it could change in regularly repeating ways. Motifs can be arranged in many ways to create a regular pattern. An irregular pattern is one in which the motif changes or the way it is repeated is unpredictable [3]. How complicated a pattern is depends on what is repeated and the way in which it is repeated.  
Before the invention of computers, patterns were generated by hand through drawings and many other skills. It takes a very long time to generate a single pattern, and sometimes, after all the labour, the pattern may not meet the requirements of the designer or the consumers. With the use of computers, it is easy to generate millions of different patterns in just a minute [4]. In a bid to satisfy human wants, researchers have found out that patterns can be generated using mathematical equations or artificial intelligence methods.  
In the literature, we can find various methods for generating patterns that use mathematical equations. Among the methods, a popular approach is the use of fractals, e.g., inversion fractals [5] and Mandelbox [6]. Furthermore, other non-Euclidean geometries are used in the generation of patterns. For instance, in [7], the authors used spherical geometry, whereas in [8], hyperbolic geometry was used. In the generation of patterns, not only various types of non-Euclidean geometries are used. Mathematical objects such as whirls [9] or spirals [10, 11] are commonly used in obtaining artistic patterns. Another group of methods are the methods that are based on the dynamics of discrete dynamical systems [12, 13, 14]. Not only the dynamics but also the orbits of discrete dynamical systems can generate aesthetic patterns [15]. Based on the results presented in [15] and the orbit trap method [16], [PERSON] et al. in [17] introduced a method that is the main subject of this paper, namely, the orbit trap method that generates patterns with wallpaper symmetry.  
In this paper, we introduce modifications of the orbit trap method presented by [PERSON] et al. in [17]. The first modification relies on using the S-iteration [18] instead of the Picard one that is used in the original algorithm. We also extend the parameters in the S-iteration from scalar to vector ones. Moreover, we propose the use of various metrics for defining orbit traps and introduce some new orbit traps. Additionally, we present examples showing that we can obtain a local fractal structure in the generated pattern when using an appropriate mapping and orbit traps.  
The paper is organised as follows. In Section 2, we briefly present the orbit trap method. Then, in Section 3, we introduce modifications to the orbit trap method from Section 2. Some examples showing the potential of the proposed modifications are presented in Section 4. Moreover, we present examples showing the local fractal structure of some of the patterns. Finally, in Section 5, we provide some concluding remarks and the directions for future work.