The differential pattern

Bibliogr. s. 58

The aim of this work is to present the prosperities of a certain iterative pattern, which we have called the "differential pattern". It operates through a subtraction of the two adjoining elements of the sequence and returning of their absolute difference, used subsequently in the next steps. The whole procedure can be prolonged, enabling the investigation of the generated sequences. The pattern generates two-dimensional matrixes and the numerical structures of a higher level. The evolution of the pattern leads to various possible behaviours, among which characteristic attractors may be mentioned. They can be the limit cycles, i.e. oscillations appearing after the certain number of iterations, which may be constant or continuously silenced. Another type of a possible attractor is a constant number, which is usually zero. Interestingly enough, the type of attractor toward which the pattern leads may depend on the number of elements in a single sequence, or the assumed edge conditions. It is a peculiar pseudobifurcation dependent on the parameters of generated structure, appearing regardless of the value of the elements filling the created numerical structure. During the pattern evolution, complex oscillations have also been observed, i.e. those exerting a different frequency. For instance, in a 5-element sequence, a distinct frequency of oscillations tends to appear on the third position. The visualisation of the pattern has been attempted with the use of R-packet, so it was possible to observe that the pattern generates more complex structures, exerting some level of order. Some behaviours are emerging only after reaching the specific level of complexity. These characteristic objects are dychotomic forks resembling the lightings, or other behaviours whoch we have called 'ping-pong objects'. The last aspect of this work is the presentation of the differential pattern as a potential candidate for a one-way function, i.e. the procedure possible to be applied in data coding. A distinct section of this article has also been devoted to discuss the pattern as a particular type of cellular automata, about which the authors did not know until the article's review.