Clocking divergence of iterative maps of matrices

Abstract

A theoretical framework describing the dynamics of nonlinear iterative maps of matrices is derived in this paper. It is shown that an iterative map of matrices does exhibit the effect of exponential divergence if the following two conditions are satisfied simultaneously. The first condition is that at least one of the multiplicity indexes of the eigenvalues of the matrix of initial conditions must be greater than one. The second condition is that the Lyapunov exponent of the corresponding scalar iterative map is greater than zero. The concept of packing and divergence codes is introduced to characterize the rate of the divergence of nonlinear iterative maps of matrices. Theoretical derivations and computational simulations yield counterintuitive results that the divergence rate of the logistic map of matrices is the same as of the circle map of matrices – even though the higher-order derivatives of the circle mapping function do not vanish.