Matrix logistic map: Fractal spectral distributions and transfer of chaos

The standard logistic map, x'=ax(1-x), serves as a paradigmatic model to demonstrate how apparently simple nonlinear equations lead to complex and chaotic dynamics. In this work, we introduce and investigate its matrix analogue defined for an arbitrary matrix X of a given order N. We show that for an arbitrary initial ensemble of Hermitian random matrices with a continuous level density supported on the interval [0,1], the asymptotic level density converges to the invariant measure of the logistic map. Depending on the parameter a, the constructed measure may be either singular, fractal or described by a continuous density. In a wider class of the map, the multiplication by a scalar logistic parameter a is replaced by transforming aX(I-X) into BX(I-X)B†, where A=BB† is a fixed positive matrix of order N. This approach generalizes the known model of coupled logistic maps and allows us to study the transition to chaos in complex networks and multidimensional systems. In particular, by associating the matrix B with a given graph, we demonstrate the gradual transfer of chaos between subsystems corresponding to vertices of a graph and coupled according to its edges.