Hierarchies of Arnold Tongues Generated by High-Dimensional Nilpotent Matrices

Arnold tongues are wedge-shaped regions in parameter space associated with mode locking and synchronization phenomena in nonlinear dynamical systems. The Caputo fractional standard map extends the classical standard map by incorporating long-memory effects through fractional derivatives and is known to generate Arnold tongue structures as the fractionality parameter approaches unity. In this paper, we investigate the fractional standard map applied to matrix-valued state variables, with particular emphasis on systems governed by high-dimensional nilpotent matrices. We show that the interplay between fractional memory and nilpotent algebra produces hierarchical families of Arnold tongues associated with divergent dynamics. This phenomenon is not observed in either the classical standard map or the non-fractional standard map of nilpotent matrices alone. For idempotent matrices, the fractional standard map retains the same level of dynamical complexity as its scalar counterpart. For nilpotent matrices, higher-order terms induce coupling between the map coefficients, giving rise to substantially richer dynamical behavior. This combination of fractional memory and nilpotent algebra provides a systematic framework for studying higher-dimensional nonlinear dynamics beyond the scalar setting. To support numerical investigations, an efficient computational scheme for the auxiliary parameters is derived and calibrated using the H-rank algorithm, which provides a concise measure of algebraic complexity in sequences generated by dynamical systems. Numerical simulations reveal hierarchical structures of Arnold tongues of divergence together with characteristic divergence rates of the auxiliary parameters. The hierarchical level of a given auxiliary parameter is identified as a key quantity determining the algebraic complexity of the transient dynamics, with potential implications for information encoding in applications exploiting transient dynamical processes.