On the formulation and solution of the isochronal synchronization stability problem in delay-coupled complex networks

Area of Science:

• Complex Systems
• Nonlinear Dynamics
• Network Science

Background:

• Oscillator networks are fundamental in various scientific fields.
• Achieving synchronized behavior in such networks is a key challenge.
• Existing methods for synchronization can be complex to formulate.

Purpose of the Study:

• To introduce a novel framework for formulating the isochronal synchronization problem.
• To simplify the definition of the error system for delay-coupled oscillators.
• To provide a method for solving synchronization problems based on established theorems.

Main Methods:

• Utilizing a linear transformation to alter the network state vector coordinates.
• Defining an error system based on the transformed coordinates.
• Applying the Lyapunov-Krasovskii theorem to solve the synchronization problem.

Main Results:

• The proposed framework allows for a straightforward definition of the error system.
• The dimension of the error system can be equal to or less than the network state vector.
• Successful formulation and solution of the isochronal synchronization problem.

Conclusions:

• The new framework offers a more accessible approach to oscillator network synchronization.
• This method enhances the understanding and control of complex dynamical networks.
• The approach is applicable to various networks of delay-coupled oscillators.