On Incompleteness of Some Integrable Rational Maps

TL;DR

This paper explores the limitations of classical descriptions of nonlinear maps, linking the transition from integrable to nonintegrable maps with logical principles like Lawvere's fixed point theorem, revealing inherent incompleteness.

Contribution

It introduces a novel connection between integrability transitions in discrete maps and foundational logical theorems, highlighting their impact on predictability.

Findings

Classical descriptions of nonlinear maps are incomplete.

Transition from integrable to nonintegrable maps relates to Lawvere's fixed point theorem.

Highlights inherent unpredictability in nonlinear map dynamics.

Abstract

An argument is given to associate integrable nonintegrable transition of discrete maps with the transition of Lawvere's fixed point theorem to its own contrapositive. We show that the classical description of nonlinear maps is neither complete nor totally predictable.