Double Periodicity and Frequency-Locking in the Langford Equation

Details Double Periodicity and Frequency-Locking in the Langford Equation Double Periodicity and Frequency-Locking in the Langford Equation

2007-07-05

arxiv.org/abs/0707.0769v1

Physics

Fluid Dynamics

9 pages, 10 figures, 2 tables, submitted to JJIAM

Scientific paper

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.

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