Convergence of Periodically-Forced Rank-Type Equations

TL;DR

This paper proves that certain periodically-forced rank-type difference equations converge to a periodic limit, which is independent of initial conditions and matches the forcing period, extending understanding of non-autonomous difference equations.

Contribution

It establishes convergence properties of non-autonomous rank-type difference equations with periodic forcing, showing the limit's independence from initial conditions and its period.

Findings

Convergence to a periodic limit is proven for a broad class of such equations.

The limit's period equals the forcing period, regardless of look-back depth.

The limit is independent of initial conditions.

Abstract

Consider a difference equation which takes the k-th largest output of m functions of the previous m terms of the sequence. If the functions are also allowed to change periodically as the difference equation evolves this is analogous to a differential equation with periodic forcing. A large class of such non-autonomous difference equations are shown to converge to a periodic limit which is independent of the initial condition. The period of the limit does not depend on how far back each term is allowed to look back in the sequence, and is in fact equal to the period of the forcing.