Application of Floquet theory to dynamical systems with memory.

We extend the recently developed generalized Floquet theory [Phys. Rev. Lett.

Memcomputing NP-complete problems in polynomial time using polynomial resources and collective states.

Memcomputing is a novel non-Turing paradigm of computation that uses interacting memory cells (memprocessors for short) to store and process information on the same physical platform. It was recently proven mathematically that memcomputing machines have the same computational power of nondeterministic Turing machines. Therefore, they can solve NP-complete problems in polynomial time and, using the appropriate architecture, with resources that only grow polynomially with the input size.

Generalized Floquet theory: application to dynamical systems with memory and Bloch's theorem for nonlocal potentials.

Floquet theory is a powerful tool in the analysis of many physical phenomena, and extended to spatial coordinates provides the basis for Bloch's theorem. However, in its original formulation it is limited to linear systems with periodic coefficients. Here, we extend the theory by proving a theorem for the general class of systems including linear operators commuting with the period-shift operator.