Dynamics of weakly coupled parametrically forced oscillators

Area of Science:

• Nonlinear Dynamics
• Oscillations and Waves
• Fluid Dynamics

Background:

• Parametric oscillators exhibit complex dynamics near instability thresholds.
• Coupling between oscillators introduces rich phenomena dependent on symmetries and forcing.
• Understanding these instabilities is crucial for predicting system behavior.

Purpose of the Study:

• To investigate the dynamics of two weakly coupled parametric oscillators near primary subharmonic instability.
• To analyze the influence of permutation symmetries and forcing phases on instability nature.
• To map detailed bifurcation sets and compare model predictions with experimental data.

Main Methods:

• Analysis of coupled parametric oscillators in the vicinity of subharmonic instability.
• Calculation of detailed bifurcation sets, including Bogdanov-Takens points.
• Comparison of theoretical predictions with direct numerical simulations and experimental results on modulated cross waves.

Main Results:

• Instability nature critically depends on remaining permutation symmetries and relative phases of forcing terms.
• Complex transition series organized by Bogdanov-Takens points were identified.
• Hopf bifurcation and subsequent saddle-node heteroclinic bifurcation were confirmed for out-of-phase forcing.

Conclusions:

• The study elucidates the critical role of symmetry and phase in coupled parametric oscillator dynamics.
• The findings provide a detailed understanding of bifurcations in such systems.
• Experimental validation confirms the theoretical model's accuracy for modulated cross waves.