| Abstract [eng] |
The analysis of Arnold tongues in nonlinear discrete systems is an important research direction in modern dynamical systems theory. Fractional–order models allow memory effects to be incorporated, while matrix structures enable the study of multidimensional and hierarchical dynamics. The interplay between these two aspects constitutes a complex research object that has so far been addressed only fragmentarily in literature. This work aims to investigate how fractional–order memory and matrix structure affect the formation of Arnold tongues and their internal structural organization in discrete systems. The object of study is the Caputo fractional standard map extended to matrix form, with particular emphasis on the structure of nilpotent matrices. The theoretical part of the work covers the foundations of nonlinear discrete systems, the circle map, and the structure of Arnold tongues in parameter space. To incorporate memory effects, fractional–order models and discrete analogues of the Caputo derivative are examined. The theory of matrix iterative models is presented through the algebraic properties of idempotent and nilpotent matrices, which form the basis for the extension of the model. The Caputo fractional matrix standard map is formulated, and the H–rank algorithm is applied to assess the algebraic complexity of sequences generated by the system. The results show that in the case of idempotent matrices the system retains the properties of the scalar model, whereas nilpotent matrices give rise to a hierarchical structure. It is established that the interaction between fractional memory and nilpotent structure generates multilayered Arnold tongues and their divergence hierarchies. The obtained results extend classical synchronization theory and are illustrated by a practical application, demonstrating that different hierarchical layers can be used as independent keys in signal encoding schemes. |
