**Appendix A. Expressions of the Coefficients Used in the Algorithm for Solving the SDEOFBVM for the DMFFBVM of the Tulip and the Bowl in the PPR Region**

$$\mathbf{C}\_{1} = \frac{\mathbf{a}\_{1}}{1 + \mathbf{a}\_{1}} \boldsymbol{\chi}\_{\text{nT}} \tag{A1}$$

$$\mathbf{C}\_{2} = \frac{\mathbf{a}\_{1}\mathbf{b}\_{1}}{\mathbf{J}\_{\mathbf{Z}\_{2}\mathbf{T}\mathbf{A}} + \mathbf{b}\_{1}(1+\mathbf{a}\_{1})} \boldsymbol{\chi}\_{\text{nT}\text{\textquotedblleft}}\tag{A2}$$

$$\mathbf{C}\_3 = \frac{\mathbf{a}\_2}{1 + \mathbf{a}\_2} \chi\_{\mathbf{n}\mathbf{B}\prime} \tag{A3}$$

$$\mathbf{C\_4} = \frac{\mathbf{a\_2b\_2}}{\mathbf{J\_{Z\_2BA}} + \mathbf{b\_2(1+a\_2)}} \chi\_{\text{nB}\_{\text{V}}} \tag{A4}$$

$$
\lambda\_1 = 1 - \frac{\eta\_1}{2\Omega\_1}, \lambda\_2 = \frac{3}{8} \frac{\Gamma\_1}{\Omega\_1^{2^\prime}} \tag{A5}
$$

$$
\lambda\_3 = \frac{5}{16} \frac{\Gamma\_2}{\Omega\_1^2}, \lambda\_4 = \xi^2 - \mu^2 \left(\frac{\Omega\_1}{\eta\_1}\right)^2,\tag{A6}
$$

$$
\lambda\_5 = \lambda\_{3\prime}^2 \,\lambda\_6 = 2\lambda\_2 \lambda\_{3\prime} \tag{A7}
$$

$$
\lambda\_7 = \lambda\_2^2 + 2\lambda\_1\lambda\_{3\prime}\ \lambda\_8 = 2\lambda\_1\lambda\_{2\prime}\ \lambda\_9 = \lambda\_1^2 + \lambda\_{4\prime} \tag{A8}
$$

$$
\lambda\_{10} = 1 - \frac{\eta\_3}{2\Omega\_3}, \lambda\_{20} = \frac{3}{8} \frac{\Gamma\_3}{\Omega\_3^2} \tag{A9}
$$

$$
\lambda\_{30} = \frac{5}{16} \frac{\Gamma\_4}{\Omega\_3^2}, \lambda\_{40} = \xi^2 - \mu^2 \left(\frac{\Omega \mathfrak{Z}}{\mathfrak{I}\_3}\right)^2,\tag{A10}
$$

$$
\lambda\_{50} = \lambda\_{30\prime}^2 \ \lambda\_{60} = 2\lambda\_{20}\lambda\_{30\prime} \tag{A11}
$$

$$
\lambda\_{70} = \lambda\_{20}^2 + 2\lambda\_{10}\lambda\_{30\prime}\ \lambda\_{80} = 2\lambda\_{10}\lambda\_{20\prime}\lambda\_{90} = \lambda\_{10}^2 + \lambda\_{40\prime}\tag{A12}
$$
