Phase reduction technique on a target region

We propose a phase reduction technique that provides the phase sensitivity function, which is one of the essential functions in phase reduction theory, on a target region. A system with a large degree of freedom and global coupling, such as an incompressible fluid system, is emphasized. Such a system poses challenges for the numerical calculation of the phase sensitivity function, which cannot be resolved using known algorithms such as the direct method or the adjoint method. A combination of the Jacobian-free algorithm and the Rayleigh-Ritz procedure is proposed to significantly reduce the computational cost and obtain a good approximation of the phase sensitivity function in a particular region of interest. In addition, the approximation can be assessed using the Ritz value. The breathing solution of a reaction-diffusion system and the flow past a flat plate are used to analyze the proposed methods, and the characteristics of the proposed method are discussed.