Matrix fractional systems

•Generalization of fractional dynamics to matrix models.•Generalization of Cole–Cole, Davidson–Cole, and Havriliak–Negami transfer functions.•Formulation of the concept of fraction of a system.

This paper addresses the matrix representation of dynamical systems in the perspective of fractional calculus. Fractional elements and fractional systems are interpreted under the light of the classical Cole–Cole, Davidson–Cole, and Havriliak–Negami heuristic models. Numerical simulations for an electrical circuit enlighten the results for matrix based models and high fractional orders. The conclusions clarify the distinction between fractional elements and fractional systems.