Optimizing stability of mutual synchronization between a pair of limit-cycle oscillators with weak cross coupling

Area of Science:

• Dynamical Systems and Control Theory
• Nonlinear Dynamics
• Complex Systems

Background:

• Coupled oscillators are fundamental to many natural and engineered systems.
• Synchronized states in weakly coupled limit-cycle oscillators are susceptible to instability.
• Cross-coupling, where different state variables interact, offers a potential mechanism for enhanced stability.

Purpose of the Study:

• To develop a method for optimizing the linear stability of synchronized states in a pair of weakly coupled limit-cycle oscillators.
• To investigate the role of cross-coupling between different state variable components in enhancing oscillator synchronization.
• To derive a coupling matrix that maximizes linear stability under specific constraints.

Main Methods:

• Application of phase reduction theory to analyze coupled oscillator dynamics.
• Derivation of a coupling matrix that optimizes linear stability.
• Analysis of stability under constraints on coupling intensity and phase difference.
• Numerical illustration using diverse limit-cycle oscillator models.

Main Results:

• A method to derive a cross-coupling matrix that maximizes linear stability of synchronized states was successfully developed.
• The derived coupling matrix allows for significant improvement in the stability of synchronized states.
• The effectiveness of the method was demonstrated across various types of limit-cycle oscillators.

Conclusions:

• Optimized cross-coupling provides a powerful strategy to enhance the stability of synchronized states in weakly coupled oscillators.
• The derived coupling matrix offers a practical approach for designing more robust oscillatory systems.
• This work contributes to the understanding and control of synchronization phenomena in complex systems.