Appendix 1: The averaged drift and diffusion coefficients
The coefficients in Eq. (52) are as follows:
$$\begin{aligned} \overline{{m}}_i&= \overline{{H}}_i (\mathbf{A})+F_i (\mathbf{A}) \quad (i=1,2),\end{aligned}$$
(62)
$$\begin{aligned} \overline{{m}}_3&= \overline{{H}}_3 (\mathbf{A})+\Omega _{11} -b_{01} ({A}_1 )\nonumber \\&\quad +\frac{E_{11} \cos \Gamma [2b_{01} (A_1 )-b_{21} (A_1 )]}{[4(\alpha _1 A_1 ^{2}+\omega _1^2 )]} \end{aligned}$$
(63)
$$\begin{aligned} F_1 (A_1 ,\Gamma )&= \frac{E_1 \sin \Gamma \left[ 2b_{01} (A_1 )\!-\!b_{21} (A_1 )\right] }{[4(\alpha _1 A_1 ^{2}\!+\!\omega _{10}^2 )]} \nonumber \\&-\,\frac{A_1 \left[ \beta _{10} (16\omega _1^2 \!+\!10\alpha _1 A_1^2 )\!+\!A_1^2 \beta _{11} (4\omega _1^2 \!+\!3\alpha A_1^2 )\!+\!A_2^2 \beta _{12} (16\omega _1^2 \!+\!10\alpha A_1^2 )\right] }{[32(\omega _1^2 \!+\!\alpha _1 A_1^2 )]} \\ F_2 (A_2 )&= -\frac{A_2 \left[ \beta _{20} (16\omega _2^2 +10\alpha _2 A_2^2 )+A_1^2 \beta _{21} (16\omega _2^2 +10\alpha _2 A_2^2 )+A_2^2 \beta _{22} (4\omega _2^2 +3\alpha _2 A_2^2 )\right] }{[32(\omega _2^2 +\alpha _2 A_2^2 )]} \end{aligned}$$
$$\begin{aligned} \overline{{H}}_i (A_i )&= \overline{{m}}_{i1} +\overline{{m}}_{i2} +\overline{{m}}_{i3} +\overline{{m}}_{i4} \quad (i=1,2),\\ \overline{{H}}_3 (A_1 )&= \overline{{m}}_{31} +\overline{{m}}_{32} +\overline{{m}}_{33} +\overline{{m}}_{34} ,\\ \overline{{m}}_{i1}&= \overline{{m}}_{i11} S_{i1} (\omega _i (A_i))+\overline{{m}}_{i13} S_{i1} (3\omega _i (A_i))\\&+\, \overline{{m}}_{i15} S_{i1} (5\omega _i (A_i))+\overline{{m}}_{i17} S_{i1} (7\omega _i (A_i )),\\ \overline{{m}}_{i3}&= \overline{{m}}_{i31} S_{i1} (\omega _i (A_i ))+\overline{{m}}_{i33} S_{i1} (3\omega _i (A_i ))\\&+\,\overline{{m}}_{i35} S_{i1} (5\omega _i (A_i ))+\overline{{m}}_{i37} S_{i1} (7\omega _i (A_i )),\\ \overline{{m}}_{i2}&= \overline{{m}}_{i22} S_{i2} (2\omega _i (A_i ))+\overline{{m}}_{i24} S_{i2} (4\omega _i (A_i ))\\&+\,\overline{{m}}_{i26} S_{i2} (6\omega _i (A_i ))+\overline{{m}}_{i28} S_{i2} (8\omega _i (A_i )),\\ \overline{{m}}_{i4}&= \overline{{m}}_{i42} S_{i2} (2\omega _i (A_i ))+\overline{{m}}_{i44} S_{i2} (4\omega _i (A_i ))\\&+\,\overline{{m}}_{i46} S_{i2} (6\omega _i (A_i ))+\overline{{m}}_{i48} S_{i2} (8\omega _i (A_i )),\\ \overline{{m}}_{31}&= \overline{{m}}_{311} I_1 (\omega _i (A_i ))+\overline{{m}}_{313} I_1 (3\omega _i (A_i ))\\&+\,\overline{{m}}_{315} I_1 (5\omega _i (A_i ))+\overline{{m}}_{317} I_1 (7\omega _i (A_i )),\\ \overline{{m}}_{33}&= \overline{{m}}_{331} I_1 (\omega _i (A_i ))+\overline{{m}}_{333} I_1 (3\omega _i (A_i ))\\&+\,\overline{{m}}_{335} I_1 (5\omega _i (A_i ))+\overline{{m}}_{337} I_1 (7\omega _i (A_i )),\\ \end{aligned}$$
$$\begin{aligned} \overline{{m}}_{32}&= \overline{{m}}_{322} I_2 (2\omega _i (A_i ))+\overline{{m}}_{324} I_2 (4\omega _i (A_i ))\\&+\,\overline{{m}}_{326} I_2 (6\omega _i (A_i ))+\overline{{m}}_{328} I_2 (8\omega _i (A_i )),\\ \overline{{m}}_{34}&= \overline{{m}}_{342} I_2 (2\omega _i (A_i ))+\overline{{m}}_{344} I_2 (4\omega _i (A_i ))\\&+\,\overline{{m}}_{346} I_2 (6\omega _i (A_i ))+\overline{{m}}_{348} I_2 (8\omega _i (A_i )),\\ I_i (\omega )&= \frac{D_{1i} }{\omega _{1i} }\frac{\omega }{\omega ^{2}+\omega _{1i}^2 }\quad { (i=1,2)} \end{aligned}$$
$$\begin{aligned} \overline{{m}}_{i11}&= \pi \left[ b_{2i} (A_i )-2b_{0i} (A_i )\right] \frac{\left\{ 2\alpha A_i \left[ 2b_{0i} (A_i )-b_{2i} (A_i )\right] +(A_i^2 \alpha _i +\omega _i^2 )\left[ db_{2i} (A_i )/dA_i -2(db_{0i} (A_i )/dA_i )\right] \right\} }{\left[ 8(A_i^2 \alpha _i +\omega _i^2 )^{3}\right] }\\ \overline{{m}}_{i13}&= \pi [b_{2i} (A_i )-b_{4i} (A_i )] \frac{\left\{ 2\alpha A_i [b_{4i} (A_i )-b_{2i} (A_i )]+(A_i^2 \alpha _i +\omega _i^2 )\left[ db_{2i} (A_i )/dA_i -2(db_{4i} (A_i )/dA_i )\right] \right\} }{[8(A_i^2 \alpha _i +\omega _i^2 )^{3}]} \\ \overline{{m}}_{i15}&= \pi \left[ b_{4i} (A_i )-2b_{6i} (A_i )\right] \frac{\left\{ 2\alpha A_i \left[ b_{6i} (A_i )-b_{4i} (A_i )\right] +(A_i^2 \alpha _i +\omega _i^2 )\left[ db_{4i} (A_i )/dA_i -2(db_{6i} (A_i )/dA_i )\right] \right\} }{\left[ 8(A_i^2 \alpha _i +\omega _i^2 )^{3}\right] } \\ \overline{{m}}_{i17}&= \frac{\pi b_{6i} (A_i )\left\{ -2\alpha A_i b_{6i} (A_i )+(A_i^2 \alpha _i +\omega _i^2 )(db_{6i} (A_i )/dA_i )\right\} }{\left[ 8(A_i^2 \alpha _i +\omega _i^2 )^{3}\right] }\\ \overline{{m}}_{i22}&= \pi A_i \left[ 2b_{0i} (A_i )-b_{4i} (A_i )\right] \frac{\left\{ \left[ 2b_{0i} (A_i )-b_{4i} (A_i )\right] (A_i^2 \alpha _i -\omega _i^2 )-A_i (A_i^2 \alpha _i -\omega _i^2 )\left[ 2db_{0i} (A_i )/dA_i -(db_{4i} (A_i )/dA_i )\right] \right\} }{\left[ 32(A_i^2 \alpha _i +\omega _i^2 )^{3}\right] } \\ \overline{{m}}_{i24}&= \pi A_i \left[ b_{2i} (A_i )-b_{6i} (A_i )\right] \frac{\left\{ \left[ b_{6i} (A_i )-b_{2i} (A_i )\right] (A_i^2 \alpha _i -\omega _i^2 )-A_i (A_i^2 \alpha _i -\omega _i^2 )\left[ db_{2i} (A_i )/dA_i -(db_{6i} (A_i )/dA_i )\right] \right\} }{\left[ 32(A_i^2 \alpha _i +\omega _i^2 )^{3}\right] } \\ \overline{{m}}_{i26}&= \frac{\pi A_i b_{4i} (A_i )\left\{ b_{4i} (A_i )(A_i^2 \alpha _i -\omega _i^2 )-A_i (A_i^2 \alpha _i -\omega _i^2 )(db_{4i} (A_i )/dA_i )\right\} }{\left[ 32(A_i^2 \alpha _i +\omega _i^2 )^{3}\right] }\\ \overline{{m}}_{i28}&= \frac{\pi A_i b_{6i} (A_i )\left\{ b_{6i} (A_i )(A_i^2 \alpha _i -\omega _i^2 )-A_i (A_i^2 \alpha _i -\omega _i^2 )(db_{6i} (A_i )/dA_i )\right\} }{\left[ 32(A_i^2 \alpha _i +\omega _i^2 )^{3}\right] } \end{aligned}$$
$$\begin{aligned} \overline{{m}}_{i31}&= \frac{-\pi [b_{2i}^2 (A_i )-4b_{0i}^2 (A_i )]}{[8A_i (A_i^2 \alpha _i +\omega _i^2 )^{2}]},\\ \overline{{m}}_{i33}&= \frac{-3\pi [b_{4i}^2 (A_i )-b_{2i}^2 (A_i )]}{[8A_i (A_i^2 \alpha _i +\omega _i^2 )^{2}]},\\ \overline{{m}}_{i35}&= \frac{-5\pi [b_{6i}^2 (A_i )-b_{4i}^2 (A_i )]}{[8A_i (A_i^2 \alpha _i +\omega _i^2 )^{2}]},\\ \overline{{m}}_{i37}&= \frac{7\pi b_{6i}^2 (A_i )}{[8A_i (A_i^2 \alpha _i +\omega _i^2 )^{2}]}, \end{aligned}$$
$$\begin{aligned} \overline{{m}}_{i42}&= \frac{\pi A_i [2b_{0i} (A_i )-b_{4i} (A_i )][2b_{0i} (A_i )+2b_{2i} (A_i )+b_{4i} (A_i )]}{[16(A_i^2 \alpha _i +\omega _i^2 )^{2}]},\\ \overline{{m}}_{i44}&= \frac{\pi A_i [b_{2i} (A_i )-b_{6i} (A_i )][b_{2i} (A_i )+2b_{4i} (A_i )+b_{6i} (A_i )]}{[8(A_i^2 \alpha _i +\omega _i^2 )^{2}]}, \end{aligned}$$
$$\begin{aligned} \overline{{m}}_{i46}&= \frac{3\pi A_i b_{4i} (A_i )[b_{4i} (A_i )+2b_{6i} (A_i )]}{[16(A_i^2 \alpha _i +\omega _i^2 )^{2}]},\\ \overline{{m}}_{i48}&= \frac{\pi A_i b_{6i}^2 (A_i )}{[4(A_i^2 \alpha _i +\omega _i^2 )^{2}]},\\ \overline{{m}}_{311}&= \frac{[2b_{01} (A_1 )+b_{21} (A_1 )]^{2}}{[8A_1^2 (A_1^2 \alpha _1 +\omega _1^2 )^{2}]},\\ \end{aligned}$$
$$\begin{aligned} \overline{{m}}_{313}&= \frac{3[b_{21} (A_1 )+b_{41} (A_1 )]^{2}}{[8A_1^2 (A_1^2 \alpha _1 +\omega _1^2 )^{2}]},\\ \overline{{m}}_{315}&= \frac{5[b_{41} (A_1 )+b_{61} (A_1 )]^{2}}{[8A_1^2 (A_1^2 \alpha _1 +\omega _1^2 )^{2}]},\\ \overline{{m}}_{317}&= \frac{7[b_{61} (A_1 )]^{2}}{[8A_1^2 (A_1^2 \alpha _1 +\omega _1^2 )^{2}]},\\ \overline{{m}}_{322}&= \frac{[2b_{01} (A_1 )+2b_{21} (A_1 )+b_{41} (A_1 )]^{2}}{[16(A_1^2 \alpha _1 +\omega _1^2 )^{2}]},\\ \overline{{m}}_{324}&= \frac{[b_{21} (A_1 )+2b_{41} (A_1 )+b_{61} (A_1 )]^{2}}{[8(A_1^2 \alpha _1 +\omega _1^2 )^{2}]},\\ \end{aligned}$$
$$\begin{aligned} \overline{{m}}_{326}&= \frac{3[b_{41} (A_1 )+2b_{61} (A_1 )]^{2}}{[16(A_1^2 \alpha _1 +\omega _1^2 )^{2}]},\\ \overline{{m}}_{328}&= \frac{[b_{61} (A_1 )]^{2}}{[4(A_1^2 \alpha _1 +\omega _1^2 )^{2}]}, \end{aligned}$$
$$\begin{aligned} \overline{{m}}_{331}&= \left[ 2b_{01} (A_1 )-b_{21} (A_1 )\right] \frac{\left\{ -\left[ b_{21} (A_1 )+2b_{01} (A_1 )\right] (3A_1^2 \alpha _1 +\omega _1^2 )+A_1 (\alpha _1 A_1^2 +\omega _1^2 )\left[ 2(db_{01} (A_1 )/dA_1 )+(db_{21} (A_1 )/dA_1 )\right] \right\} }{\left[ 8(A_1^2 \alpha _1 +\omega _1^2 )^{3}\right] }, \\ \overline{{m}}_{333}&= \left[ b_{21} (A_1 )-b_{41} (A_1 )\right] \frac{\left\{ -\left[ b_{21} (A_1 )+b_{41} (A_1 )\right] (3A_1^2 \alpha _1 +\omega _1^2 )+A_1 (\alpha _1 A_1^2 +\omega _1^2 )\left[ (db_{21} (A_1 )/dA_1 )+(db_{41} (A_1 )/dA_1 )\right] \right\} }{\left[ 8(A_1^2 \alpha _1 +\omega _1^2 )^{3}\right] }, \\ \overline{{m}}_{335}&= \left[ b_{41} (A_1 )-b_{61} (A_1 )\right] \frac{\left\{ -\left[ b_{41} (A_1 )+b_{61} (A_1 )\right] (3A_1^2 \alpha _1 +\omega _1^2 )+A_1 (\alpha _1 A_1^2 +\omega _1^2 )\left[ (db_{41} (A_1 )/dA_1 )+(db_{61} (A_1 )/dA_1 )\right] \right\} }{\left[ 8(A_1^2 \alpha _1 +\omega _1^2 )^{3}\right] }, \\ \overline{{m}}_{337}&= \frac{b_{61} (A_1 )\left\{ -b_{61} (A_1 )(3A_1^2 \alpha _1 +\omega _1^2 )+A_1 (\alpha _1 A_1^2 +\omega _1^2 )\left[ (db_{61} (A_1 )/dA_1 )\right] \right\} }{\left[ 8(A_1^2 \alpha _1 +\omega _1^2 )^{3}\right] } \end{aligned}$$
$$\begin{aligned} \overline{{m}}_{342}&= \big \{-2A_1 \alpha _1 \left[ 2b_{01} (A_1 )+2b_{21} (A_1 )+b_{41} (A_1 )\right] \\&+\,(A_1^2 \alpha _1 +\omega _1^2 )\Bigg [\frac{2db_{01} (A_1 )}{dA_1 }+\frac{2db_{21} (A_1 )}{dA_1 }\\&+\frac{2db_{41} (A_1 )}{dA_1 }\Bigg ] \frac{A_1 [(2b_{01} (A_1 )-b_{41} (A_1 )\big \}}{[32(A_1^2 \alpha _1 +\omega _1^2 )^{3}]} \\ \overline{{m}}_{344}&= \big \{-2A_1 \alpha _1 [b_{21} (A_1 )+2b_{41} (A_1 )+b_{61} (A_1 )]\\&+\,(A_1^2 \alpha _1 +\omega _1^2 )\Bigg [\frac{2db_{21} (A_1 )}{dA_1 }+\frac{2db_{41} (A_1 )}{dA_1 }\\&+\,\frac{2db_{61} (A_1 )}{dA_1 }\Bigg ] \frac{A_1 [(2b_{21} (A_1 )-b_{61} (A_1 )\big \}}{[32(A_1^2 \alpha _1 +\omega _1^2 )^{3}]} \end{aligned}$$
$$\begin{aligned} \overline{{m}}_{346}&= Ab_{41} (A_1 ) \times \frac{\left\{ -2A_1 \alpha _1 [b_{41} (A_1 )+2b_{61} (A_1 )]+(A_1^2 \alpha _1 +\omega _1^2 )[(db_{41} (A_1 )/dA_1 +(2db_{61} (A_1 )/dA_1 )]\right\} }{[32(A_1^2 \alpha _1 +\omega _1^2 )^{3}]}, \\ \overline{{m}}_{348}&= \frac{A_1 b_{61} (A_1 )\left\{ -2A_1 \alpha _1 b_{61} (A_1 )+(A_1^2 \alpha _1 +\omega _1^2 )db_{61} (A_1 )/dA_1 )\right\} }{[32(A_1^2 \alpha _1 +\omega _1^2 )^{3}]}, \end{aligned}$$
$$\begin{aligned} \overline{{b}}_{ii}&= \overline{{b}}_{ii1} +\overline{{b}}_{ii2} , \quad (i=1,2)\end{aligned}$$
(64)
$$\begin{aligned} \overline{{b}}_{33}&= \overline{{b}}_{331} +\overline{{b}}_{332} , \end{aligned}$$
(65)
$$\begin{aligned}&\overline{{b}}_{ij} =0, \quad i\ne j,\\ \overline{{b}}_{ii1}&= \overline{{b}}_{ii11} S_1 (\omega _i (A_i ))+\overline{{b}}_{ii13} S_1 (3\omega _i (A_i))\\&\quad +\,\overline{{b}}_{ii15} S_1 (5\omega _i (A_i ))+\overline{{b}}_{ii17} S_1 (7\omega _i (A_i )),\\ \overline{{b}}_{ii2}&= \overline{{b}}_{ii22} S_2 (\omega _i (A_i ))+\overline{{b}}_{ii24} S_2 (4\omega _i (A_i))\\&\quad +\,\overline{{b}}_{ii26} S_1 (6\omega _i (A_i ))+\overline{{b}}_{ii28} S_1 (8\omega _i (A_i )),\\ \overline{{b}}_{331}&= \overline{{b}}_{3311} S_1 (\omega _1 (A_1 ))+\overline{{b}}_{3313} S_1 (3\omega _1 (A_1))\\&\quad +\,\overline{{b}}_{3315} S_1 (5\omega _1 (A_1 ))+\overline{{b}}_{3317} S_1 (7\omega _1 (A_1 )),\\ \overline{{b}}_{333}&= \overline{{b}}_{3330} S_2 (0)\!+\!\overline{{b}}_{3332} S_2 (2\omega _1 (A_1 ))\!+\!\overline{{b}}_{3334} S_2 (4\omega _1 (A_1))\\&\quad +\,\overline{{b}}_{3336} S_2 (6\omega _1 (A_1 ))+\overline{{b}}_{3338} S_2 (8\omega _1 (A_1 )),\\ \end{aligned}$$
$$\begin{aligned} b_{ii11}&= \frac{\pi [b_{2i} (A_i )-2b_{0i} (A_i )]^{2}}{[4(\alpha _i A_i^2 +\omega _{0i}^2 )^{2}]},\\ b_{ii13}&= \frac{\pi [b_{2i} (A_i )-2b_{4i} (A_i )]^{2}}{[4(\alpha _i A_i^2 +\omega _{0i}^2 )^{2}]},\\ b_{ii15}&= \frac{\pi [b_{4i} (A_i )-b_{6i} (A_i )]^{2}}{[4(\alpha _i A_i^2 +\omega _{0i}^2 )^{2}]},\\ b_{ii17}&= \frac{\pi b_{6i}^2 (A_i )}{[4(\alpha _i A_i^2 +\omega _{0i}^2 )^{2}]},\\ b_{ii22}&= \frac{\pi A_i^2 [b_{4i} (A_i )-2b_{0i} (A_i )]^{2}}{[16(\alpha _i A_i^2 +\omega _{0i}^2 )^{2}]},\\ b_{ii24}&= \frac{\pi A_i^2 [b_{2i} (A_i )-b_{6i} (A_i )]^{2}}{[16(\alpha _i A_i^2 +\omega _{0i}^2 )^{2}]},\\ b_{ii26}&= \frac{\pi A_i^2 b_{4i}^2 (A_i )}{[16(\alpha _i A_i^2 +\omega _{0i}^2 )^{2}]},\\ b_{ii28}&= \frac{\pi A_i^2 b_{6i}^2 (A_i )}{[16(\alpha _i A_i^2 +\omega _{0i}^2 )^{2}]},\\ b_{3311}&= \frac{\pi [2b_{02} (A_1 )+b_{22} (A_1 )]^{2}}{[4A_1^2 (\alpha _1 A_1^2 +\omega _{01}^2 )^{2}]},\\ b_{3313}&= \frac{\pi [b_{21} (A_1 )+b_{41} (A_1 )]^{2}}{[4A_1^2 (\alpha _1 A_1^2 +\omega _{01}^2 )^{2}]},,\\ b_{3315}&= \frac{\pi [b_{41} (A_1 )+b_{61} (A_1 )]^{2}}{[4A_1^2 (\alpha _1 A_1^2 +\omega _{01}^2 )^{2}]},\\ b_{3317}&= \frac{\pi b_{61} ^{2}(A_1 )}{[4A_1^2 (\alpha _1 A_1^2 +\omega _{01}^2 )^{2}]},\\ b_{3320}&= \frac{\pi [2b_{01} (A_1 )+b_{21} (A_1 )]^{2}}{[8(\alpha _1 A_1^2 +\omega _{01}^2 )^{2}]},\\ b_{3322}&= \frac{\pi [2b_{01} (A_1 )+2b_{21} (A_1 )+b_{41} (A_1 )]^{2}}{[16(\alpha _1 A_1^2 +\omega _{01}^2 )^{2}]},\\ b_{3324}&= \frac{\pi [b_{21} (A_1 )+2b_{41} (A_1 )+b_{61} (A_1 )]^{2}}{[16(\alpha _1 A_1^2 +\omega _{01}^2 )^{2}]},\\ b_{3326}&= \frac{\pi [b_{41} (A_1 )+2b_{61} (A_1 )]^{2}}{[16(\alpha _1 A_1^2 +\omega _{01}^2 )^{2}]},\\ b_{3328}&= \frac{\pi b_{61}^2 (A_1 )}{[16(\alpha _1 A_1^2 +\omega _{01}^2 )^{2}]}. \end{aligned}$$
Appendix 2: An approximate solution of final dynamical programming equation
For ergodic control of 2-DOF Duffing system, the final dynamical programming equation is of the form:
$$\begin{aligned} \eta _c&= L_1 (\mathbf{H},\Gamma )+\tilde{m}_1 (\mathbf{H},\Gamma )\frac{\partial V}{\partial H_1 }+\tilde{m}_2 (\mathbf{H})\frac{\partial V}{\partial H_2 }\nonumber \\&\quad +\,\tilde{m}_3 (\mathbf{H},\Gamma )\frac{\partial V}{\partial \Gamma }-\frac{m_{11} (\mathbf{H})}{4R_1 }\left( {\frac{\partial V}{\partial H_1 }} \right) ^{2} \nonumber \\&\quad -\,\frac{m_{22} (\mathbf{H})}{4R_2 }\left( {\frac{\partial V}{\partial H_2 }} \right) ^{2}-\frac{m_{33} (\mathbf{H})}{4R_1 }\left( {\frac{\partial V}{\partial \Gamma }} \right) ^{2}\nonumber \\&\quad +\,\frac{b_{11} }{2}\frac{\partial ^{2}V}{\partial H_1^2 }+\frac{b_{22} }{2}\frac{\partial ^{2}V}{\partial H_2^2 }+\frac{b_{33} }{2}\frac{\partial ^{2}V}{\partial \Gamma ^{2}} \end{aligned}$$
(66)
Suppose that \(L_1 (\mathbf{H},\Gamma )\) can be expanded into Fourier series with respect to\(\Gamma \),
$$\begin{aligned} L_1 (\mathbf{H},\Gamma )&= L_{10} (\mathbf{H})+\sum _{i=1}^\infty (L_{1i}^c (\mathbf{H})\cos i\Gamma \nonumber \\&\quad +\, L_{1i}^s (\mathbf{H})\sin i\Gamma ) \end{aligned}$$
(67)
Then the solution of Eq. (66) can be assumed of the form of the following Fourier series:
$$\begin{aligned} V(\mathbf{H},\Gamma )=V_0 (\mathbf{H})+\sum _{i=1}^\infty {(V_i^c (\mathbf{H})\cos i\Gamma +V_i^s (\mathbf{H})\sin i\Gamma )}\nonumber \\ \end{aligned}$$
(68)
Note that coefficients \(\tilde{m}_1 , \tilde{m}_2 \) and \(\tilde{m}_3 \) in Eq. (54) are of the form
$$\begin{aligned} \tilde{m}_1 (\mathbf{H},\Gamma )&= m_{10} (\mathbf{H})+m_{11}^s (\mathbf{H})\sin \Gamma \nonumber \\ \tilde{m}_2 (\mathbf{H})&= m_{20} (\mathbf{H})\nonumber \\ \tilde{m}_3 (\mathbf{H},\Gamma )&= m_{30} (\mathbf{H})+m_{31}^c (\mathbf{H})\cos \Gamma \end{aligned}$$
(69)
and \(m_{i0} (i=1,2,3),m_{11}^s ,m_{31}^c \) are all independent of \(\Gamma \). Substituting Eqs. (67)–(69), the following series of equation can be obtained:
$$\begin{aligned}&\eta _c =L_{10} (\mathbf{H})+m_{10} (\mathbf{H})\frac{\partial V_0 }{\partial H_1 }+m_{20} (\mathbf{H})\frac{\partial V_0 }{\partial H_2 }\nonumber \\&\quad -\,\frac{1}{2}m_{11}^s (\mathbf{H})\frac{\partial V_1^c }{\partial H_1 }+\frac{1}{2}m_{31}^c (\mathbf{H})V_1^s\nonumber \\&\quad -\,\frac{m_{11} (\mathbf{H})}{4R_1 }\!\left\{ {\left( \!{\frac{\partial V_0 }{\partial H_1 }}\!\right) ^{2}\!+\!\frac{1}{2}\sum _{i=1}^\infty {\left[ \!\!{\left( \!{\frac{\partial V_i^c }{\partial H_1 }} \right) ^{2}\!+\! \left( \!{\frac{\partial V_i^s }{\partial H_1 }} \right) ^{2}}\!\right] }}\!\right\} \nonumber \\&\quad -\,\frac{m_{22} (\mathbf{H})}{4R_2 }\!\left\{ {\left( \! {\frac{\partial V_0 }{\partial H_2 }} \!\right) ^{2}\!+\!\frac{1}{2}\sum _{i=1}^\infty {\!\left[ {\!\left( \! {\frac{\partial V_i^c }{\partial H_2 }} \!\right) ^{2}\!+\!\left( \! {\frac{\partial V_i^s }{\partial H_2 }} \!\right) ^{2}} \!\right] } }\!\right\} \nonumber \\&\quad -\,\frac{m_{33} (\mathbf{H})}{8R_1 }\!\left[ {\sum _{i=1}^\infty {i^{2}\!\left( \!{V_i^{c^{2}}\!+\!V_i^{s^{2}} } \right) } }\!\right] \!+\!\frac{b_{11} }{2}\frac{\partial ^{2}V_0 }{\partial H_1^2 }\!+\!\frac{b_{22} }{2}\frac{\partial ^{2}V_0 }{\partial H_2^2 }\nonumber \\\end{aligned}$$
(70)
$$\begin{aligned}&0=L_{11}^c (\mathbf{H})+m_{10} (\mathbf{H})\frac{\partial V_1^c }{\partial H_1 }+m_{20} (\mathbf{H})\frac{\partial V_1^c }{\partial H_2 }\nonumber \\&\quad +\,m_{30} (\mathbf{H})V_1^s (\mathbf{H})+\frac{1}{2}m_{11}^s (\mathbf{H})\frac{\partial V_2^c }{\partial H_1 }+m_{31}^c (\mathbf{H})V_2^s \nonumber \\&\quad -\,\frac{m_{11} (\mathbf{H})}{4R_1 }\!\left[ {2\frac{\partial V_0 }{\partial H_1 }\frac{\partial V_1^c }{\partial H_1 }\!+\!\!\sum _{i=1}^\infty {\left( {\frac{\partial V_i^c }{\partial H_1 }\frac{\partial V_{i+1}^c }{\partial H_1 }+\!\frac{\partial V_i^s }{\partial H_1 }\frac{\partial V_{i+1}^s }{\partial H_1 }} \right) }}\! \right] \nonumber \\&\quad -\,\frac{m_{22} (\mathbf{H})}{4R_2 }\!\left[ {2\frac{\partial V_0 }{\partial H_2 }\frac{\partial V_1^c }{\partial H_2 }\!+\!\!\sum _{i=1}^\infty {\!\left( {\frac{\partial V_i^c }{\partial H_2 }\frac{\partial V_{i+1}^c }{\partial H_2 }\!+\!\frac{\partial V_i^s }{\partial H_2 }\frac{\partial V_{i+1}^s }{\partial H_2 }} \right) }} \!\right] \nonumber \\&\quad -\,\frac{m_{33} (\mathbf{H})}{4R_1 }\left[ {\sum _{i=1}^\infty {i(i+1)\left( {V_i^c V_{i+1}^c +V_i^s V_{i+1}^s } \right) } } \right] \nonumber \\&\quad +\,\frac{b_{11} }{2}\frac{\partial ^{2}V_1^c }{\partial H_1^2 }+\frac{b_{22} }{2}\frac{\partial ^{2}V_1^c }{\partial H_2^2 }-\frac{b_{33} }{2}V_1^c \end{aligned}$$
(71)
$$\begin{aligned}&0=L_{11}^c (\mathbf{H})+m_{10} (\mathbf{H})\frac{\partial V_1^s }{\partial H_1 }+m_{20} (\mathbf{H})\frac{\partial V_1^s }{\partial H_2 }\nonumber \\&\quad -\, m_{30} (\mathbf{H})V_1^c (\mathbf{H})+m_{11}^s (\mathbf{H})\frac{\partial V_0 }{\partial H_1 }\nonumber \\&+\frac{1}{2}m_{11}^s (\mathbf{H})\frac{\partial V_2^c }{\partial H_1 }-m_{31}^c (\mathbf{H})V_2^c \nonumber \\&\quad -\,\frac{m_{11} (\mathbf{H})}{4R_1 }\!\left[ \!{2\frac{\partial V_0 }{\partial H_1 }\frac{\partial V_1^s }{\partial H_1 }\!+\!\!\sum _{i=1}^\infty {\left( {\frac{\partial V_i^c }{\partial H_1 }\frac{\partial V_{i+1}^s }{\partial H_1 }+\frac{\partial V_i^s }{\partial H_1 }\frac{\partial V_{i+1}^c }{\partial H_1 }} \!\right) } } \!\!\right] \nonumber \\&\quad -\,\frac{m_{22} (\mathbf{H})}{4R_2 }\!\left[ \!{2\frac{\partial V_0 }{\partial H_2 }\frac{\partial V_1^s }{\partial H_2 }\!+\!\!\sum _{i=1}^\infty {\left( {\frac{\partial V_i^c }{\partial H_2 }\frac{\partial V_{i+1}^s }{\partial H_2 }+\!\frac{\partial V_i^s }{\partial H_2 }\frac{\partial V_{i+1}^c }{\partial H_2 }} \right) } }\! \!\right] \nonumber \\&\quad -\,\frac{m_{33} (\mathbf{H})}{4R_1 }\left[ {\sum _{i=1}^\infty {i(i+1)\left( {V_i^s V_{i+1}^c +V_i^c V_{i+1}^s } \right) } } \right] \nonumber \\&\quad +\,\frac{b_{11} }{2}\frac{\partial ^{2}V_1^s }{\partial H_1^2 }+\frac{b_{22} }{2}\frac{\partial ^{2}V_1^s }{\partial H_2^2 }-\frac{b_{33} }{2}V_1^s \end{aligned}$$
(72)
The higher harmonic terms in the solution can be neglected. An approximated solution of Eq. (66)
$$\begin{aligned} V(\mathbf{H},\Gamma )\!=\!V_0 (\mathbf{H})\!+\!V_1^c (\mathbf{H})\cos \Gamma \!+\! V_1^s (\mathbf{H})\sin \Gamma \end{aligned}$$
(73)
To evaluate \(V_0 (\mathbf{H}), V_1^c (\mathbf{H})\) and \(V_1^s (\mathbf{H})\) from the coupled differential equations (70)–(72), the Fourier coefficients are further expanded into Taylor series:
$$\begin{aligned} m_{10}&= m_{101} H_1 +m_{102} H_1^3 +m_{103} H_2^2 H_1 +\cdots \nonumber \\ m_{20}&= m_{201} H_2 +m_{202} H_1^2 H_2 +m_{203} H_2^3 +\cdots \nonumber \\ m_{30}&= m_{300} +m_{302} H_1^2 +\cdots \nonumber \\ m_{11}^s&= m_{111}^s +m_{113}^s H_1^2 +\cdots \nonumber \\ m_{31}^c&= m_{311}^c H_1^{-1} +m_{313}^c H_1 +\cdots \nonumber \\ m_{11}&= m_{110} +m_{112} H_1^2 +\cdots \nonumber \\ m_{22}&= m_{220} +m_{222} H_2^2 +\cdots \nonumber \\ m_{33}&= m_{330} H_1^{-2} +m_{332} +\cdots \nonumber \\ b_{11}&= b_{110} +b_{112} H_1^2 +\cdots \nonumber \\ b_{22}&= b_{220} +b_{222} H_2^2 +\cdots \nonumber \\ b_{33} :&= b_{330} H_1^{-2} +b_{332} +\cdots \end{aligned}$$
(74)
$$\begin{aligned} L_{10}&= L_{100} H_1^4 +L_{101} H_1^3 +(L_{102} H_2^2 +L_{103} )H_1^2\nonumber \\&+\, L_{104} H_1 +L_{105} H_2^4 +L_{106} H_2^2 +L_{107} +\cdots \nonumber \\ L_{11}^c&= L_{200} H_1^4 +(L_{201} H_2^2 +L_{202} )H_1^2 +L_{203} +\cdots \nonumber \\ L_{11}^s&= L_{300} H_1^4 +L_{301} H_1^3 +(L_{302} H_2^2\nonumber \\&+\, L_{303} )H_1^2 +L_{304} H_1 +L_{305} +\cdots \end{aligned}$$
(75)
$$\begin{aligned} V_0&= V_{011} H_1 +V_{012} H_2 +V_{021} H_1^2\nonumber \\&+V_{022} H_2^2 +V_{023} H_1 H_2 +\cdots \nonumber \\ V_1^c&= V_{11}^c H_1 +V_{12}^c H_2 +V_{13}^c H_1^2\nonumber \\&+V_{14}^c H_2^2 +V_{15}^c H_1 H_2 +\cdots \nonumber \\ V_1^s&= V_{11}^s H_1 +V_{12}^s H_2 +V_{13}^s H_1^2\nonumber \\&+V_{14}^s H_2^2 +V_{15}^s H_1 H_2 +\cdots \end{aligned}$$
(76)
Substituting Eqs. (74)–(76) into Eqs. (70)–(72)and letting the coefficients of the same power of \(H_1 ,H_2 \) vanish yield the coefficients of polynomials \(V_0,V_1^c \) and \(V_1^s \).
