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Automation of Mode Locking in a Nonlinear Polarization Rotation Fiber Laser through Output Polarization Measurements

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Frequency locking and complex dynamics near a periodically forced robust heteroclinic cycle.

J H P Dawes¹, T L Tsai

¹DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, United Kingdom. J.H.P.Dawes@damtp.cam.ac.uk

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|February 7, 2007

Summary

Robust heteroclinic cycles in nonlinear systems exhibit frequency locking when near instability under periodic forcing. However, strong stability prevents this phenomenon, leading to chaotic dynamics instead.

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Area of Science:

Nonlinear Dynamics
Mathematical Ecology
Fluid Mechanics

Background:

Robust heteroclinic cycles are common in nonlinear differential equations with invariant hyperplanes.
These cycles frequently appear in ecological dynamics and fluid mechanics models.

Purpose of the Study:

To investigate the impact of small-amplitude, time-periodic forcing on robust heteroclinic cycles.
To analyze the resulting dynamics and identify conditions for frequency locking.

Main Methods:

Reduction of complex dynamics to a two-dimensional map.
Analysis of system behavior in different stability regimes of the heteroclinic cycle.

Main Results:

Frequency locking intervals emerge as the heteroclinic cycle approaches the loss of asymptotic stability.
When the heteroclinic cycle is strongly stable, the system exhibits chaotic dynamics without frequency locking.

Conclusions:

Time-periodic forcing can induce frequency locking in systems with heteroclinic cycles near instability.
The stability of the heteroclinic cycle is critical in determining the presence or absence of frequency locking and chaos.