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.

Figure 1

(a) Snapshot of $(w,v)$ for phase zero in the entire domain. Variables $w$ and $v$ are represented by the solid line and broken lines, respectively. (b) Eight snapshots obtained from solutions during one cycle in the domain $[5.0,L]$.

Figure 2

(a) Phase sensitivity vector for $w$, shown in the domain $[4,L]$. (b) Same as (a), but for $v$.

Figure 3

Components of the phase sensitivity vectors, $Q_w$ and $Q_v$. (a) Phase zero. (b) Phase 2. (c) Phase 4. (d) Phase 6.

Figure 4

(a) Computational grid and target regions. Target regions are displayed by rectangles with dotted lines A, dashed lines B, broken lines C, and solid lines D, respectively. Rectangles are slightly shifted such that each region can be clearly distinguished. (b) Velocity and vorticity fields. Target regions are shown as well.

Figure 5

Phase sensitivity vectors obtained using projection method. Colors indicate the magnitude of $\left|\mathit{Q}\right|$. Regions A–C correspond to those shown in Fig. 4.

Figure 6

Distributions of phase sensitivity functions obtained using the projection method along cross lines. (a) ${\tilde{Z}}_w$ along line $x=x_c(=2.5)$. (b) ${\tilde{Z}}_v$ along line $x=x_c$. (c) ${\tilde{Z}}_w$ along line $y=y_c(=2.0)$. (d) ${\tilde{Z}}_v$ along line $y=y_c$.