On the Convergence Rate of the Caputo Fractional Difference Logistic Map of Nilpotent Matrices

The convergence rate of the Caputo fractional difference logistic map of nilpotent matrices is investigated in this paper. The divergence rate of the auxiliary parameters governing the dynamics of nilpotents is exponential and is multiple to the Lyapunov exponent of the scalar non-fractional map. However, the convergence of the Caputo fractional difference logistic map of nilpotent matrices to a stable fixed point is governed by the interplay between the convergence rate of the scalar fractional map (the power law rate) and the exponential convergence induced by the nilpotent matrices. It is demonstrated that convergence is determined by the competition between the power law and exponential mechanisms, a feature not captured by scalar fractional maps, with higher-order auxiliary parameters diverging exponentially at increasingly higher rates. This paper provides insight into the complex dynamics of fractional maps of nilpotent matrices.