Integrable Difference Analogue of the Logistic Equation and B\"acklund Transformation of the KP Hierarchy

TL;DR

This paper introduces an integrable difference analogue of the logistic equation derived from Hirota's bilinear difference equation, linking it to the B"acklund transformation of the KP hierarchy and exploring its dynamics.

Contribution

It presents a novel integrable difference equation related to the logistic map and a scheme to interpolate between integrable and nonintegrable systems.

Findings

The integrable difference map preserves key symmetries.

The scheme allows comparison of integrable and nonintegrable dynamics.

Analysis of Julia sets at the transition point.

Abstract

A difference analogue of the logistic equation, which preserves integrability, is derived from Hirota's bilinear difference equation. The integrability of the map is shown to result from the large symmetry associated with the B\"acklund transformation of the KP hierarchy. We introduce a scheme which interpolates between this map and the standard logistic map and enables us to study integrable and nonintegrable systems on an equal basis. In particular we study the behavior of Julia set at the point where the nonintegrable map passes to the integrable map.