Phase reduction technique on a target region.

Area of Science:

• Dynamical Systems Theory
• Computational Fluid Dynamics
• Numerical Analysis

Background:

• Phase reduction theory is crucial for analyzing oscillatory systems.
• Calculating the phase sensitivity function (PSF) is computationally intensive for systems with many degrees of freedom and global coupling, such as incompressible fluid systems.
• Existing numerical methods like the direct and adjoint methods are insufficient for these complex systems.

Purpose of the Study:

• To propose a novel phase reduction technique for accurately computing the phase sensitivity function (PSF) in target regions.
• To address the computational challenges in calculating the PSF for large, globally coupled systems.
• To provide a method that is computationally efficient and yields good approximations of the PSF.

Main Methods:

• A hybrid approach combining a Jacobian-free algorithm with the Rayleigh-Ritz procedure is introduced.
• This method significantly reduces computational cost compared to traditional algorithms.
• The accuracy of the approximation is validated using Ritz values.

Main Results:

• The proposed method successfully approximates the phase sensitivity function (PSF) in specific regions of interest.
• Demonstrated efficacy on a reaction-diffusion system's breathing solution and flow past a flat plate.
• The computational cost is substantially reduced, making PSF calculation feasible for complex systems.

Conclusions:

• The proposed Jacobian-free and Rayleigh-Ritz based phase reduction technique offers an efficient and accurate solution for computing the phase sensitivity function (PSF).
• This method overcomes the limitations of existing algorithms for large-scale, globally coupled systems.
• The technique is applicable to diverse fields, including fluid dynamics and reaction-diffusion systems, enabling deeper analysis of system dynamics.