Double Periodicity and Frequency-Locking in the Langford Equation

TL;DR

This paper investigates the complex bifurcation structures of the Langford equation, revealing various periodic and chaotic behaviors, and explores frequency-locking phenomena and bifurcation similarities with the sine-circle map.

Contribution

It provides a detailed numerical analysis of the Langford equation's bifurcation structure, including the construction of Poincaré sections for triple periodicity and comparison with known maps.

Findings

Identification of periodic, doubly-periodic, and chaotic solutions.

Observation of frequency-locking corresponding to Farey sequences.

Similarity of bifurcation structures with the sine-circle map.

Abstract

The bifurcation structure of the Langford equation is studied numerically in detail. Periodic, doubly-periodic, and chaotic solutions and the routes to chaos via coexistence of double periodicity and period-doubling bifurcations are found by the Poincar\'e plot of successive maxima of the first mode $x_1$. Frequency-locked periodic solutions corresponding to the Farey sequence $F_n$ are examined up to $n=14$. Period-doubling bifurcations appears on some of the periodic solutions and the similarity of bifurcation structures between the sine-circle map and the Langford equation is shown. A method to construct the Poincar\'e section for triple periodicity is proposed.