Matricial representation of period doubling cascade

TL;DR

This paper develops a matrix-based representation of the period doubling cascade, providing a systematic way to analyze the order and structure of periodic orbits through permutation matrices.

Contribution

It introduces a novel matricial framework for representing the period doubling cascade using permutation matrices derived from cycle permutations.

Findings

Explicit matrix representation of period doubling cascade obtained

Recurrence relations for the matrices derived

Provides a new tool for analyzing bifurcation structures

Abstract

Starting from the cycle permutation sigma_(2^k) associated with the (2^k)-periodic orbit of the period doubling cascade we obtain the inverse permutation (sigma_(2^k))^-1. Then we build a matrix permutation related to (sigma_(2^k))^-1, which includes the visiting order of the (2^k)-periodic orbit points. After some manipulations a recurrence relation of matricial representation of period doubling cascade is obtained. Finally the explicit matricial representation is reached.