Analysis of supercritical pitchfork bifurcation in active magnetic bearing-rotor system with current saturation

Abstract

The bifurcation characteristics of the active magnetic bearing-rotor system subjected to the external excitation were investigated analytically when it was operating at a speed far away from its natural frequencies. During operation of the system, some nonlinear factors may be prominent, for example, the nonlinearity of bearing force and current saturation. Nonlinear factors can lead to some complicated behaviors, which have negative effects on the operating performance and stability. To analyze the bifurcations happening at the speed far away from harmonic resonances, an approximate analytical method that can be applicable to the bifurcation analysis of the forced vibration system was proposed. By applying it to the active magnetic bearing-rotor system, multiple static equilibriums and periodic solutions were obtained, and then, the stability analysis was conducted based on Floquet theory. The validity and accuracy of the approximate analytical method were verified by the numerical integration method and generalized cell mapping digraph method. It was found that there was supercritical pitchfork bifurcation of static equilibrium in the active magnetic bearing-rotor system. The influences of external excitation and controller parameters on dynamical characteristics were discussed. Based on analysis results, controller parameters were also improved to prevent nonlinear behaviors and improve system performance.

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Acknowledgements

This work was supported by the National S&T Major Project (Grant No. ZX069).

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Correspondence to Zhe Sun.

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Appendices

Appendix A

The coefficients \(k_{m,n}\) in Eq. (7):

\(k_{1,0}=0.02287{{\tilde{K}_{\mathrm{p}}}}-0.01033 , k_{3,0}=-0.02308{{\tilde{K}_{\mathrm{p}}}}^3-0.02852{{\tilde{K}_{\mathrm{p}}}}^2+0.06079{{\tilde{K}_{\mathrm{p}}}}-0.05111, k_{5,0}=0.01302{{\tilde{K}_{\mathrm{p}}}}^5+0.02609{{\tilde{K}_{\mathrm{p}}}}^4-0.06087{{\tilde{K}_{\mathrm{p}}}}^3 -0.0412{{\tilde{K}_{\mathrm{p}}}}^2+0.1919{{\tilde{K}_{\mathrm{p}}}}+0.3458 , k_{7,0}=0.003566{{\tilde{K}_{\mathrm{p}}}}^7-0.009379{{\tilde{K}_{\mathrm{p}}}}^6+0.02636{{\tilde{K}_{\mathrm{p}}}}^5 +0.0287{{\tilde{K}_{\mathrm{p}}}}^4-0.0713{{\tilde{K}_{\mathrm{p}}}}^3-0.05661{{\tilde{K}_{\mathrm{p}}}}^2 -0.1839{{\tilde{K}_{\mathrm{p}}}}-1.968 , k_{9,0}=0.000455{{\tilde{K}_{\mathrm{p}}}}^9+0.001425{{\tilde{K}_{\mathrm{p}}}}^8-0.004855{{\tilde{K}_{\mathrm{p}}}}^7 -0.00675{{\tilde{K}_{\mathrm{p}}}}^6+0.02344{{\tilde{K}_{\mathrm{p}}}}^5+0.02136{{\tilde{K}_{\mathrm{p}}}}^4-0.07812{{\tilde{K}_{\mathrm{p}}}}^3 -0.008971{{\tilde{K}_{\mathrm{p}}}}^2+0.4281{{\tilde{K}_{\mathrm{p}}}}+4.045 , k_{11,0}=-0.000022{{\tilde{K}_{\mathrm{p}}}}^{11}-0.000077{{\tilde{K}_{\mathrm{p}}}}^{10}+0.00031{{\tilde{K}_{\mathrm{p}}}}^9 +0.000514{{\tilde{K}_{\mathrm{p}}}}^8-0.002422{{\tilde{K}_{\mathrm{p}}}}^7-0.00299{{\tilde{K}_{\mathrm{p}}}}^6+0.01789{{\tilde{K}_{\mathrm{p}}}}^5 +0.016{{\tilde{K}_{\mathrm{p}}}}^4-0.117{{\tilde{K}_{\mathrm{p}}}}^3-0.08791{{\tilde{K}_{\mathrm{p}}}}^2+0.5638{{\tilde{K}_{\mathrm{p}}}}-3.368 , k_{0,1}=0.02287{{\tilde{K}_\mathrm{d}}} , k_{2,1}=-0.06924{{\tilde{K}_{\mathrm{p}}}}^2{{\tilde{K}_\mathrm{d}}}-0.05704{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}+0.06079{{\tilde{K}_\mathrm{d}}} , k_{4,1}=0.0651{{\tilde{K}_{\mathrm{p}}}}^4{{\tilde{K}_\mathrm{d}}}+0.10436{{\tilde{K}_{\mathrm{p}}}}^3{{\tilde{K}_\mathrm{d}}} - 0.18261{{\tilde{K}_{\mathrm{p}}}}^2{{\tilde{K}_\mathrm{d}}} - 0.0824{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}+0.1919{{\tilde{K}_\mathrm{d}}} ,\) \(k_{6,1}=- 0.024962{{\tilde{K}_{\mathrm{p}}}}^6{{\tilde{K}_\mathrm{d}}}- 0.056274{{\tilde{K}_{\mathrm{p}}}}^5{{\tilde{K}_\mathrm{d}}} + 0.1318{{\tilde{K}_{\mathrm{p}}}}^4{{\tilde{K}_\mathrm{d}}}+0.1148{{\tilde{K}_{\mathrm{p}}}}^3{{\tilde{K}_\mathrm{d}}} - 0.2139{{\tilde{K}_{\mathrm{p}}}}^2{{\tilde{K}_\mathrm{d}}}- 0.11322{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}- 0.1839{{\tilde{K}_\mathrm{d}}},\) \(k_{8,1}=0.004094{{\tilde{K}_{\mathrm{p}}}}^8{{\tilde{K}_\mathrm{d}}} + 0.0114{{\tilde{K}_{\mathrm{p}}}}^7{{\tilde{K}_\mathrm{d}}} - 0.033985{{\tilde{K}_{\mathrm{p}}}}^6{{\tilde{K}_\mathrm{d}}}- 0.0405{{\tilde{K}_{\mathrm{p}}}}^5{{\tilde{K}_\mathrm{d}}} + 0.1172{{\tilde{K}_{\mathrm{p}}}}^4{{\tilde{K}_\mathrm{d}}}+ 0.08544{{\tilde{K}_{\mathrm{p}}}}^3{{\tilde{K}_\mathrm{d}}} - 0.23436{{\tilde{K}_{\mathrm{p}}}}^2{{\tilde{K}_\mathrm{d}}}- 0.017942{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}} + 0.428{{\tilde{K}_\mathrm{d}}},\) \(k_{10,1}=- 0.000238{{\tilde{K}_{\mathrm{p}}}}^{10}{{\tilde{K}_\mathrm{d}}} - 0.000767{{\tilde{K}_{\mathrm{p}}}}^9{{\tilde{K}_\mathrm{d}}} + 0.0027891{{\tilde{K}_{\mathrm{p}}}}^8{{\tilde{K}_\mathrm{d}}}+ 0.004114{{\tilde{K}_{\mathrm{p}}}}^7{{\tilde{K}_\mathrm{d}}} - 0.016954{{\tilde{K}_{\mathrm{p}}}}^6{{\tilde{K}_\mathrm{d}}} - 0.01794{{\tilde{K}_{\mathrm{p}}}}^5{{\tilde{K}_\mathrm{d}}} + 0.08945{{\tilde{K}_{\mathrm{p}}}}^4{{\tilde{K}_\mathrm{d}}} + 0.064{{\tilde{K}_{\mathrm{p}}}}^3{{\tilde{K}_\mathrm{d}}} - 0.351{{\tilde{K}_{\mathrm{p}}}}^2{{\tilde{K}_\mathrm{d}}} - 0.17582{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}+ 0.5638{{\tilde{K}_\mathrm{d}}}, k_{1,2}=- 0.06924{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}^2-0.02852{{\tilde{K}_\mathrm{d}}}^2, k_{3,2}=0.1302{{\tilde{K}_{\mathrm{p}}}}^3{{\tilde{K}_\mathrm{d}}}^2 + 0.15654{{\tilde{K}_{\mathrm{p}}}}^2{{\tilde{K}_\mathrm{d}}}^2 - 0.18261{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}^2 - 0.0412{{\tilde{K}_\mathrm{d}}}^2, k_{5,2}=- 0.074886{{\tilde{K}_{\mathrm{p}}}}^5{{\tilde{K}_\mathrm{d}}}^2 - 0.140685{{\tilde{K}_{\mathrm{p}}}}^4{{\tilde{K}_\mathrm{d}}}^2 + 0.2636{{\tilde{K}_{\mathrm{p}}}}^3{{\tilde{K}_\mathrm{d}}}^2 + 0.1722{{\tilde{K}_{\mathrm{p}}}}^2{{\tilde{K}_\mathrm{d}}}^2 - 0.2139{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}^2 - 0.05661{{\tilde{K}_\mathrm{d}}}^2 ,\) \(k_{7,2}=0.016376{{\tilde{K}_{\mathrm{p}}}}^7{{\tilde{K}_\mathrm{d}}}^2 +0.0399{{\tilde{K}_{\mathrm{p}}}}^6{{\tilde{K}_\mathrm{d}}}^2 - 0.101955{{\tilde{K}_{\mathrm{p}}}}^5{{\tilde{K}_\mathrm{d}}}^2- 0.10125{{\tilde{K}_{\mathrm{p}}}}^4 {{\tilde{K}_\mathrm{d}}}^2 + 0.2344{{\tilde{K}_{\mathrm{p}}}}^3{{\tilde{K}_\mathrm{d}}}^2 + 0.12816{{\tilde{K}_{\mathrm{p}}}}^2{{\tilde{K}_\mathrm{d}}}^2 - 0.23436{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}^2 - 0.008971{{\tilde{K}_{\mathrm{p}}}}^2, k_{9,2}=-0.00119{{\tilde{K}_{\mathrm{p}}}}^9{{\tilde{K}_\mathrm{d}}}^2 - 0.00345{{\tilde{K}_{\mathrm{p}}}}^8{{\tilde{K}_\mathrm{d}}}^2 + 0.011156{{\tilde{K}_{\mathrm{p}}}}^7{{\tilde{K}_\mathrm{d}}}^2+ 0.0144{{\tilde{K}_{\mathrm{p}}}}^6{{\tilde{K}_\mathrm{d}}}^2 - 0.050862{{\tilde{K}_{\mathrm{p}}}}^5{{\tilde{K}_\mathrm{d}}}^2 - 0.04485{{\tilde{K}_{\mathrm{p}}}}^4{{\tilde{K}_\mathrm{d}}}^2 + 0.1789{{\tilde{K}_{\mathrm{p}}}}^3{{\tilde{K}_\mathrm{d}}}^2 + 0.096{{\tilde{K}_{\mathrm{p}}}}^2{{\tilde{K}_\mathrm{d}}}^2 - 0.351{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}^2 - 0.08791{{\tilde{K}_\mathrm{d}}}^2 ,\) \(k_{0,3}=-0.02308{{\tilde{K}_\mathrm{d}}}^3 , k_{2,3}=0.1302{{\tilde{K}_{\mathrm{p}}}}^2{{\tilde{K}_\mathrm{d}}}^3 + 0.10436{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}^3 - 0.06087{{\tilde{K}_\mathrm{d}}}^3 , k_{4,3}=- 0.12481{{\tilde{K}_{\mathrm{p}}}}^4{{\tilde{K}_\mathrm{d}}}^3 - 0.18758{{\tilde{K}_{\mathrm{p}}}}^3{{\tilde{K}_\mathrm{d}}}^3 + 0.2636{{\tilde{K}_{\mathrm{p}}}}^2{{\tilde{K}_\mathrm{d}}}^3 + 0.1148{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}^3 - 0.0713{{\tilde{K}_\mathrm{d}}}^3 ,\) \(k_{6,3}=0.038212{{\tilde{K}_{\mathrm{p}}}}^6{{\tilde{K}_\mathrm{d}}}^3+ 0.0798{{\tilde{K}_{\mathrm{p}}}}^5{{\tilde{K}_\mathrm{d}}}^3 - 0.169925{{\tilde{K}_{\mathrm{p}}}}^4{{\tilde{K}_\mathrm{d}}}^3 - 0.135{{\tilde{K}_{\mathrm{p}}}}^3{{\tilde{K}_\mathrm{d}}}^3 + 0.2344{{\tilde{K}_{\mathrm{p}}}}^2{{\tilde{K}_\mathrm{d}}}^3 + 0.08544{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}^3 - 0.07812{{\tilde{K}_\mathrm{d}}}^3 , k_{8,3}=- 0.003571{{\tilde{K}_{\mathrm{p}}}}^8{{\tilde{K}_\mathrm{d}}}^3 - 0.0092{{\tilde{K}_{\mathrm{p}}}}^7{{\tilde{K}_\mathrm{d}}}^3 + 0.026032{{\tilde{K}_{\mathrm{p}}}}^6{{\tilde{K}_\mathrm{d}}}^3 + 0.028801{{\tilde{K}_{\mathrm{p}}}}^5{{\tilde{K}_\mathrm{d}}}^3 - 0.08477{{\tilde{K}_{\mathrm{p}}}}^4{{\tilde{K}_\mathrm{d}}}^3 - 0.0598{{\tilde{K}_{\mathrm{p}}}}^3{{\tilde{K}_\mathrm{d}}}^3 + 0.1789{{\tilde{K}_{\mathrm{p}}}}^2{{\tilde{K}_\mathrm{d}}}^3 + 0.064{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}^3 - 0.117{{\tilde{K}_\mathrm{d}}}^3 , k_{1,4}=0.0651{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}^4+ 0.02609{{\tilde{K}_\mathrm{d}}}^4 , k_{3,4}=- 0.12481{{\tilde{K}_{\mathrm{p}}}}^3{{\tilde{K}_\mathrm{d}}}^4 - 0.140685{{\tilde{K}_{\mathrm{p}}}}^2{{\tilde{K}_\mathrm{d}}}^4 + 0.1318{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}^4 + 0.0287{{\tilde{K}_\mathrm{d}}}^4 , k_{5,4}=0.057317{{\tilde{K}_{\mathrm{p}}}}^5{{\tilde{K}_\mathrm{d}}}^4 + 0.09975{{\tilde{K}_{\mathrm{p}}}}^4{{\tilde{K}_\mathrm{d}}}^4 - 0.169925{{\tilde{K}_{\mathrm{p}}}}^3{{\tilde{K}_\mathrm{d}}}^4 - 0.10125{{\tilde{K}_{\mathrm{p}}}}^2{{\tilde{K}_\mathrm{d}}}^4 + 0.1172{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}^4 + 0.02136{{\tilde{K}_\mathrm{d}}}^4 ,\) \(k_{7,4}=- 0.007141{{\tilde{K}_{\mathrm{p}}}}^7{{\tilde{K}_\mathrm{d}}}^4 - 0.016101{{\tilde{K}_{\mathrm{p}}}}^6{{\tilde{K}_\mathrm{d}}}^4 + 0.039047{{\tilde{K}_{\mathrm{p}}}}^5{{\tilde{K}_\mathrm{d}}}^4 + 0.036001{{\tilde{K}_{\mathrm{p}}}}^4{{\tilde{K}_\mathrm{d}}}^4 - 0.08477{{\tilde{K}_{\mathrm{p}}}}^3{{\tilde{K}_\mathrm{d}}}^4 - 0.04485{{\tilde{K}_{\mathrm{p}}}}^2{{\tilde{K}_\mathrm{d}}}^4 + 0.08945{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}^4 + 0.016{{\tilde{K}_\mathrm{d}}}^4 ,\) \(k_{0,5}=0.01302{{\tilde{K}_\mathrm{d}}}^5 , k_{2,5}=- 0.074886{{\tilde{K}_{\mathrm{p}}}}^2{{\tilde{K}_\mathrm{d}}}^5 - 0.056274{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}^5 + 0.02636{{\tilde{K}_\mathrm{d}}}^5 ,\) \(k_{4,5}=0.057317{{\tilde{K}_{\mathrm{p}}}}^4 {{\tilde{K}_\mathrm{d}}}^5 + 0.0798{{\tilde{K}_{\mathrm{p}}}}^3{{\tilde{K}_\mathrm{d}}}^5 - 0.101955{{\tilde{K}_{\mathrm{p}}}}^2{{\tilde{K}_\mathrm{d}}}^5 - 0.0405{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}^5 + 0.02344 {{\tilde{K}_\mathrm{d}}}^5 , k_{6,5}=- 0.009998{{\tilde{K}_{\mathrm{p}}}}^6{{\tilde{K}_\mathrm{d}}}^5 - 0.019321{{\tilde{K}_{\mathrm{p}}}}^5{{\tilde{K}_\mathrm{d}}}^5 + 0.039047{{\tilde{K}_{\mathrm{p}}}}^4{{\tilde{K}_\mathrm{d}}}^5 + 0.028801{{\tilde{K}_{\mathrm{p}}}}^3{{\tilde{K}_\mathrm{d}}}^5 - 0.050862{{\tilde{K}_{\mathrm{p}}}}^2{{\tilde{K}_\mathrm{d}}}^5 - 0.01794{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}^5 + 0.01789{{\tilde{K}_\mathrm{d}}}^5 , k_{1,6}=- 0.024962{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}^6 - 0.009379{{\tilde{K}_\mathrm{d}}}^6 , k_{3,6}=0.038212{{\tilde{K}_{\mathrm{p}}}}^3{{\tilde{K}_\mathrm{d}}}^6 + 0.0399{{\tilde{K}_{\mathrm{p}}}}^2{{\tilde{K}_\mathrm{d}}}^6 - 0.033985{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}^6 - 0.00675{{\tilde{K}_\mathrm{d}}}^6 , k_{5,6}=- 0.009998{{\tilde{K}_{\mathrm{p}}}}^5 {{\tilde{K}_\mathrm{d}}}^6- 0.016101{{\tilde{K}_{\mathrm{p}}}}^4{{\tilde{K}_\mathrm{d}}}^6 + 0.026032{{\tilde{K}_{\mathrm{p}}}}^3{{\tilde{K}_\mathrm{d}}}^6 + 0.0144{{\tilde{K}_{\mathrm{p}}}}^2{{\tilde{K}_\mathrm{d}}}^6 - 0.016954{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}^6 - 0.00299{{\tilde{K}_\mathrm{d}}}^6 , k_{0,7}=-0.003566{{\tilde{K}_\mathrm{d}}}^7 , k_{2,7}=0.016376{{\tilde{K}_{\mathrm{p}}}}^2{{\tilde{K}_\mathrm{d}}}^7 + 0.0114{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}^7 - 0.004855{{\tilde{K}_\mathrm{d}}}^7 , k_{4,7}=- 0.007141{{\tilde{K}_{\mathrm{p}}}}^4{{\tilde{K}_\mathrm{d}}}^7 - 0.0092{{\tilde{K}_{\mathrm{p}}}}^3{{\tilde{K}_\mathrm{d}}}^7 + 0.011156{{\tilde{K}_{\mathrm{p}}}}^2{{\tilde{K}_\mathrm{d}}}^7 + 0.004114{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}^7 - 0.002422{{\tilde{K}_\mathrm{d}}}^7 , k_{1,8}=0.004094{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}^8 + 0.001425{{\tilde{K}_\mathrm{d}}}^8 ,\) \(k_{3,8}=- 0.003571{{\tilde{K}_{\mathrm{p}}}}^3{{\tilde{K}_\mathrm{d}}}^8 - 0.00345{{\tilde{K}_{\mathrm{p}}}}^2{{\tilde{K}_\mathrm{d}}}^8 + 0.002789{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}^8 + 0.000514{{\tilde{K}_\mathrm{d}}}^8 ,\) \(k_{0,9}=0.000455{{\tilde{K}_\mathrm{d}}}^9 , k_{2,9}=- 0.00119{{\tilde{K}_{\mathrm{p}}}}^2 {{\tilde{K}_\mathrm{d}}}^9- 0.000767{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}^9 + 0.00031{{\tilde{K}_\mathrm{d}}}^9 , k_{1,10}=- 0.000238{{\tilde{K}_{\mathrm{p}}}}{{\tilde{K}_\mathrm{d}}}^{10} - 0.000077{{\tilde{K}_\mathrm{d}}}^{10} , k_{0,11}=-0.000022{{\tilde{K}_\mathrm{d}}}^{11} .\)

Appendix B

The coefficients in Eq. (16): \(b_1=-k_{1,0} -6 k_{3,0} a\bar{a}-30 k_{5,0}a^2\bar{a}^2-140 k_{7,0} a^3\bar{a}^3 -630 k_{9,0} a^4\bar{a}^4-2772 k_{11,0}a^5\bar{a}^5-2 k_{1,2}{\tilde{\Omega }} ^2 a\bar{a}-6 k_{3,2} {\tilde{\Omega }} ^2 a^2\bar{a}-20 k_{5,2} {\tilde{\Omega }} ^2 a^3\bar{a}^3-70 k_{7,2} {\tilde{\Omega }} ^2 a^4\bar{a}^4 -252 k_{9,2} {\tilde{\Omega }} ^2a^5\bar{a}^5-6 k_{1,4} {\tilde{\Omega }} ^4 a^2\bar{a}^2 -12 k_{3,4} {\tilde{\Omega }} ^4 a^3\bar{a}^3-30 k_{5,4} {\tilde{\Omega }} ^4 a^4\bar{a}^4-84 k_{7,4} {\tilde{\Omega }} ^4 a^5\bar{a}^5-20 k_{1,6} {\tilde{\Omega }} ^6 a^3\bar{a}^3-30 k_{3,6} {\tilde{\Omega }} ^6 a^4\bar{a}^4-60 k_{5,6} {\tilde{\Omega }} ^6 a^5\bar{a}^5-70 k_{1,8} {\tilde{\Omega }} ^8 a^4 \bar{a}^4-84 k_{3,8} {\tilde{\Omega }} ^8 a^5\bar{a}^5-252 k_{1,10} {\tilde{\Omega }} ^{10} a^5\bar{a}^5 ,\)

\(b_2=-k_{3,0}-20k_{5,0} a\bar{a}-210k_{7,0}a^2\bar{a}^2-1680 k_{9,0} a^3 \bar{a}^3-11550 k_{11,0} a^4\bar{a}^4-2k_{3,2}{\tilde{\Omega }} ^2a\bar{a} -20 k_{5,2} {\tilde{\Omega }} ^2 a^2\bar{a}^2-140 k_{7,2}{\tilde{\Omega }} ^2a^3 \bar{a}^3-840 k_{9,2}{\tilde{\Omega }}^2 a^4\bar{a}^4-6 k_{3,4}{\tilde{\Omega }} ^4 a^2\bar{a}^2-40k_{5,4}{\tilde{\Omega }} ^4a^3\bar{a}^3-210 k_{7,4}{\tilde{\Omega }} ^4 a^4\bar{a}^4-20 k_{3,6}{\tilde{\Omega }} ^6 a^3\bar{a}^3 -100 k_{5,6} {\tilde{\Omega }} ^6 a^4\bar{a}^4-70k_{3,8}{\tilde{\Omega }} ^8a^4\bar{a}^4 ,\)

\(b_3=-k_{5,0}- 42 k_{7,0} a\bar{a} - 756 k_{9,0} a^2\bar{a}^2 - 9240 k_{11,0}a^3 \bar{a}^3 - 2 k_{5,2}{\tilde{\Omega }} ^2 a\bar{a}- 42 k_{7,2} {\tilde{\Omega }} ^2 a^2\bar{a}^2 -504 k_{9,2} {\tilde{\Omega }} ^2 a^3\bar{a}^3- 6 k_{5,4} {\tilde{\Omega }} ^4 a^2\bar{a}^2 - 84 k_{7,4} {\tilde{\Omega }} ^4 a^3\bar{a}^3 - 20 k_{5,6}{\tilde{\Omega }} ^6 a^3\bar{a}^3 ,\)

\(b_4=-k_{7,0}-72 k_{9,0} a\bar{a}-1980 k_{11,0} a^2\bar{a}^2 -2 k_{7,2} {\tilde{\Omega }} ^2 a\bar{a}-72 k_{9,2} {\tilde{\Omega }} ^2 a^2\bar{a}^2-6 k_{7,4} {\tilde{\Omega }} ^4 a^2\bar{a}^2 ,\)

\(b_5=-k_{9,0}-110 k_{11,0}a\bar{a} -2 k_{9,2} {\tilde{\Omega }} ^2 a\bar{a} ,\)

\(b_6=-k_{11,0} ,\)

\(b_7=-k_{1,0}a-k_{3,0}a^2\bar{a}-3k_{3,0}a{C_0}^2-10k_{5,0}a^3\bar{a}^2-30k_{5,0} a^2\bar{a}{C_0}^2-5k_{5,0}a{C_0}^4-35k_{7,0}a^4\bar{a}^3-210k_{7,0}a^3\bar{a}^2 {C_0}^2-105k_{7,0}a^2\bar{a}{C_0}^4-7k_{7,0}a{C_0}^6-126k_{9,0}a^5\bar{a}^4 -1260k_{9,0}a^4\bar{a}^3{C_0}^2-1260k_{9,0}a^3\bar{a}^2{C_0}^4-252k_{9,0}a^2 \bar{a}{C_0}^6-9k_{9,0}a{C_0}^8-462k_{11,0}a^6\bar{a}^5-6930k_{11,0}a^5 \bar{a}^4{C_0}^2-11550k_{11,0} a^4\bar{a}^3{C_0}^4-4620k_{11,0} a^3\bar{a}^2 {C_0}^6-495k_{11,0}a^2\bar{a}{C_0}^8-11k_{11,0}a{C_0}^{10}+5\mathrm{j}k_{6,1} {\tilde{\Omega }} a^4\bar{a}^3+30\mathrm{j}k_{6,1}{\tilde{\Omega }} a^3\bar{a}^2{C_0}^2 +15\mathrm{j}k_{6,1}{\tilde{\Omega }} a^2\bar{a}{C_0}^4+\mathrm{j}k_{6,1}{\tilde{\Omega }} a{C_0}^6+14\mathrm{j}k_{6,1}{\tilde{\Omega }} a^5\bar{a}^4+140\mathrm{j}k_{8,1}{\tilde{\Omega }} a^4\bar{a}^3{C_0}^2+140\mathrm{j}k_{8,1}{\tilde{\Omega }} a^3\bar{a}^2{C_0}^4+28\mathrm{j} k_{8,1}{\tilde{\Omega }} a^2\bar{a}{C_0}^6+\mathrm{j}k_{8,1}{\tilde{\Omega }} a{C_0}^8 +42\mathrm{j}k_{10,1}{\tilde{\Omega }} a^6\bar{a}^5+630\mathrm{j}k_{10,1}{\tilde{\Omega }} a^5\bar{a}^4{C_0}^2+1050\mathrm{j}k_{10,1}{\tilde{\Omega }} a^4\bar{a}^3{C_0}^4 +420\mathrm{j}k_{10,1}{\tilde{\Omega }} a^3\bar{a}^2{C_0}^6+45\mathrm{j}k_{10,1}{\tilde{\Omega }} a^2\bar{a}{C_0}^8+\mathrm{j}k_{10,1}{\tilde{\Omega }} a{C_0}^{10}+\mathrm{j}k_{0,1} {\tilde{\Omega }} a+\mathrm{j}k_{2,1}{\tilde{\Omega }} a^2\bar{a}+\mathrm{j}k_{2,1} {\tilde{\Omega }} a{C_0}^2+2\mathrm{j}k_{4,1}{\tilde{\Omega }} a^3\bar{a}^2+6\mathrm{j} k_{4,1}{\tilde{\Omega }} a^2\bar{a}{C_0}^2+\mathrm{j}k_{4,1}{\tilde{\Omega }} a {C_0}^4-k_{1,2}{{\tilde{\Omega }}}^2 a^2 \bar{a}-2k_{3,2}{{\tilde{\Omega }}}^2 a^3 \bar{a}^2-3k_{3,2}{{\tilde{\Omega }}}^2 a^2 \bar{a} {C_0}^2-5k_{5,2}{{\tilde{\Omega }}}^2 a^4 \bar{a}^3-20k_{5,2}{{\tilde{\Omega }}}^2 a^3 \bar{a}^2 {C_0}^2 -5k_{5,2} {{\tilde{\Omega }}}^2 a^2 \bar{a} {C_0}^4-14k_{7,2}{{\tilde{\Omega }}}^2 a^5 \bar{a}^4 -105k_{7,2}{{\tilde{\Omega }}}^2 a^4 \bar{a}^3 {C_0}^2 -70k_{7,2} {{\tilde{\Omega }}}^2 a^3 \bar{a}^2 {C_0}^4 -7k_{7,2}{{\tilde{\Omega }}}^2 a^2 \bar{a} {C_0}^6-42k_{9,2}{{\tilde{\Omega }}}^2 a^6 \bar{a}^5-504k_{9,2} {{\tilde{\Omega }}}^2 a^5 \bar{a}^4 {C_0}^2 -630k_{9,2}{{\tilde{\Omega }}}^2 a^4 \bar{a}^3 {C_0}^2 -168k_{9,2}{{\tilde{\Omega }}}^2 a^3 \bar{a}^2 {C_0}^6 -9k_{9,2}{{\tilde{\Omega }}}^2 a^2 \bar{a}{C_0}^8+ 3\mathrm{j}k_{0,3} {{\tilde{\Omega }}}^3 a^2\bar{a}+ 2\mathrm{j}k_{2,3} {{\tilde{\mathrm{\Omega }}}}^3 a^3 \bar{a}^2+3\mathrm{j}k_{2,3} {{\tilde{\Omega }}}^3 a^2\bar{a} {C_0}^2+3\mathrm{j}k_{4,3} {{\tilde{\Omega }}}^3 a^4\bar{a}^3+12\mathrm{j}k_{4,3} {{\tilde{\Omega }}}^3 a^3 \bar{a}^2{C_0}^2+3\mathrm{j}k_{4,3} {{\tilde{\Omega }}}^3 a^2\bar{a}{C_0}^4+6\mathrm{j} k_{6,3} {{\tilde{\Omega }}}^3 a^5\bar{a}^4+45\mathrm{j}k_{6,3} {{\tilde{\Omega }}}^3 a^4\bar{a}^3{C_0}^2+30\mathrm{j}k_{6,3} {{\tilde{\Omega }}}^3 a^2\bar{a}{C_0}^4 +3\mathrm{j}k_{6,3} {{\tilde{\Omega }}}^3 a^2\bar{a}{C_0}^6 +14\mathrm{j}k_{8,3} {{\tilde{\Omega }}}^3 a^6\bar{a}^5+168\mathrm{j}k_{8,3} {{\tilde{\Omega }}}^3 a^5 \bar{a}^4{C_0}^2+210\mathrm{j}k_{8,3} {{\tilde{\Omega }}}^3 a^4\bar{a}^3{C_0}^4 +56\mathrm{j}k_{8,3} {{\tilde{\Omega }}}^3 a^3\bar{a}^2{C_0}^6+3\mathrm{j}k_{8,3} {{\tilde{\Omega }}}^3 a^2\bar{a}{C_0}^8-2k_{1,4} {{\tilde{\Omega }}}^4 a^3\bar{a}^2 -3k_{3,4} {{\tilde{\Omega }}}^4 a^4\bar{a}^3-6k_{3,4} {{\tilde{\Omega }}}^4 a^3 \bar{a}^2{C_0}^2-6k_{5,4} {{\tilde{\Omega }}}^4 a^5\bar{a}^4-30k_{5,4} {\Omega }^4 a^4\bar{a}^3{C_0}^2 -10k_{5,4} {{\tilde{\Omega }}}^4 a^3\bar{a}^2{C_0}^4-14k_{7,4} {{\tilde{\Omega }}}^4 a^6\bar{a}^5-126k_{7,4} {{\tilde{\Omega }}}^4 a^5\bar{a}^4{C_0}^2 -10k_{7,4} {{\tilde{\Omega }}}^4 a^4\bar{a}^3{C_0}^4-14k_{7,4} {{\tilde{\Omega }}}^4 a^3\bar{a}^2{C_0}^6+10\mathrm{j}k_{0,5}{{\tilde{\Omega }}}^5 a^3 \bar{a}^2+5\mathrm{j}k_{2,5} {{\tilde{\Omega }}}^5 a^4 \bar{a}^3+10\mathrm{j}k_{2,5}{{\tilde{\Omega }}}^5 a^3 \bar{a}^2{C_0}^2+6\mathrm{j}k_{4,5}{{\tilde{\Omega }}}^5 a^5 \bar{a}^4+30\mathrm{j} k_{4,5}{{\tilde{\Omega }}}^5 a^4 \bar{a}^3{C_0}^2+10\mathrm{j}k_{4,5} {{\tilde{\Omega }}}^5 a^3 \bar{a}^2{C_0}^4+10\mathrm{j}k_{6,5}{{\tilde{\Omega }}}^5 a^6 \bar{a}^5+90\mathrm{j}k_{6,5}{{\tilde{\Omega }}}^5 a^4 \bar{a}^4{C_0}^2+75\mathrm{j} k_{6,5}{{\tilde{\Omega }}}^5 a^4 \bar{a}^3{C_0}^4+10\mathrm{j}k_{6,5}{{\tilde{\Omega }}}^5 a^3 \bar{a}^2{C_0}^6-5k_{1,6}{{\tilde{\Omega }}}^6 a^4 \bar{a}^3-6k_{3,6} {{\tilde{\Omega }}}^6 a^5 \bar{a}^4-15k_{3,6}{{\tilde{\Omega }}}^6 a^4 \bar{a}^3 {C_0}^2-10k_{5,6}{{\tilde{\Omega }}}^6 a^6 \bar{a}^5-60k_{5,6}{{\tilde{\Omega }}}^6 a^5 \bar{a}^4{C_0}^2-25k_{5,6}{{\tilde{\Omega }}}^6 a^4 \bar{a}^3{C_0}^4+35\mathrm{j} k_{0,7}{{\tilde{\Omega }}}^7 a^4 \bar{a}^3+14\mathrm{j}k_{2,7}{{\tilde{\Omega }}}^7 a^5 \bar{a}^4+35\mathrm{j}k_{2,7}{{\tilde{\Omega }}}^7 a^4 \bar{a}^3{C_0}^2+14\mathrm{j}k_{4,7} {{\tilde{\Omega }}}^7 a^6 \bar{a}^5+84\mathrm{j}k_{4,7}{{\tilde{\Omega }}}^7 a^5 \bar{a}^4{C_0}^2+35\mathrm{j}k_{4,7}{{\tilde{\Omega }}}^7 a^4 \bar{a}^3{C_0}^4-14k_{1,8} {{\tilde{\Omega }}}^8 a^5 \bar{a}^4-14k_{3,8}{{\tilde{\Omega }}}^8 a^6 \bar{a}^5-42k_{3,8} {{\tilde{\Omega }}}^8 a^5 \bar{a}^4{C_0}^2+126\mathrm{j}k_{0,9}{{\tilde{\Omega }}}^9 a^5 \bar{a}^4+42\mathrm{j}k_{2,9}{{\tilde{\Omega }}}^9 a^6 \bar{a}^5+126\mathrm{j}k_{2,9} {{\tilde{\Omega }}}^9 a^5 \bar{a}^4{C_0}^2-42k_{1,10}{{\tilde{\Omega }}}^{10} a^6 \bar{a}^5+462\mathrm{j}k_{0,11}{{\tilde{\mathrm{\Omega }}}}^{11} a^6 \bar{a}^5 ,\)

\(b_8=-3k_{3,0}a^2{C_0}-20k_{5,0}a^3\bar{a}{C_0}^3-10k_{5,0}a^2{C_0}^3-105k_{7,0}a^4 \bar{a}^2{C_0}-140k_{7,0}a^3\bar{a}{C_0}^3-210k_{7,0}a^2{C_0}^5-504k_{9,0}a^5 \bar{a}^3{C_0}-1260k_{9,0}a^4\bar{a}^2{C_0}^3-504k_{9,0}a^3\bar{a}{C_0}^5-36k_{9,0} a^2{C_0}^7-2310k_{11,0}a^6\bar{a}{C_0}-9240k_{11,0}a^5\bar{a}^3{C_0}^3-6930k_{11,0} a^4\bar{a}^2{C_0}^5-1320k_{11,0}a^3\bar{a}{C_0}^7-55k_{11,0}a^2{C_0}^9+30\mathrm{j}k_{6,1} {\tilde{\Omega }} a^4\bar{a}^2{C_0}+40\mathrm{j}k_{6,1}{\tilde{\Omega }} a^3\bar{a}{C_0}^3 +6\mathrm{j}k_{6,1}{\tilde{\Omega }} a^2{C_0}^5+112\mathrm{j}k_{8,1}{\tilde{\Omega }} a^5 \bar{a}^3{C_0}+280\mathrm{j}k_{8,1}{\tilde{\Omega }} a^4\bar{a}^2{C_0}^3+112\mathrm{j}k_{8,1} {\tilde{\Omega }} a^3\bar{a}{C_0}^5+8\mathrm{j}k_{8,1}{\tilde{\Omega }} a^2{C_0}^7 +420\mathrm{j}k_{10,1}{\tilde{\Omega }} a^6\bar{a}^4{C_0}+1680\mathrm{j}k_{10,1} {\tilde{\Omega }} a^5\bar{a}^3{C_0}^3+1260\mathrm{j}k_{10,1}{\tilde{\Omega }} a^4 \bar{a}^2{C_0}^5+240\mathrm{j}k_{10,1}{\tilde{\Omega }} a^3\bar{a}{C_0}^7+10\mathrm{j}k_{10,1} {\tilde{\Omega }} a^2{C_0}^9+2\mathrm{j}k_{2,1}{\tilde{\Omega }} a^2{C_0}+8\mathrm{j}k_{4,1} {\tilde{\Omega }} a^3\bar{a}{C_0}+4\mathrm{j}k_{4,1}{\tilde{\Omega }} a^2{C_0}^3+k_{1,2} {\tilde{\Omega }}^2 a^2{C_0}+k_{3,2}{\tilde{\Omega }}^2 a^2{C_0}^3-5k_{5,2} {\tilde{\Omega }}^2 a^4\bar{a}^2{C_0}+k_{5,2}{\tilde{\Omega }}^2 a^2{C_0}^5-128k_{7,2} {\tilde{\Omega }}^2 a^5\bar{a}^3{C_0}-35k_{7,2}{\tilde{\Omega }}^2 a^4\bar{a}^2{C_0}^3 +k_{7,2}{\tilde{\Omega }}^2 a^2{C_0}^7-126k_{9,2}{\tilde{\Omega }}^2 a^6\bar{a}^4{C_0} -336k_{9,2}{\tilde{\Omega }}^2 a^5\bar{a}^3{C_0}^3-126k_{9,2}{\tilde{\Omega }}^2 a^4 \bar{a}^2{C_0}^5+k_{9,2}{\tilde{\Omega }}^2 a^2{C_0}^9+4\mathrm{j}k_{2,3} {\tilde{\Omega }}^3 a^3\bar{a}{C_0}+12\mathrm{j}k_{4,3}{\tilde{\Omega }}^3 a^4\bar{a}^2{C_0} +8\mathrm{j}k_{4,3}{\tilde{\Omega }}^3 a^3\bar{a}{C_0}^3+36\mathrm{j}k_{6,3}{\tilde{\Omega }}^3 a^5\bar{a}^3{C_0}+60\mathrm{j}k_{6,3}{\tilde{\Omega }}^3 a^4\bar{a}^2{C_0}^3+12\mathrm{j}k_{6,3} {\tilde{\Omega }}^3 a^3\bar{a}{C_0}^5+112\mathrm{j}k_{8,3}{\tilde{\Omega }}^3 a^6\bar{a}^4{C_0} +336\mathrm{j}k_{8,3}{\tilde{\Omega }}^3 a^5\bar{a}^3{C_0}^3+168\mathrm{j}k_{8,3} {\tilde{\Omega }}^3 a^4\bar{a}^2{C_0}^5+16\mathrm{j}k_{8,3}{\tilde{\Omega }}^3 a^3 \bar{a}{C_0}^7+4k_{1,4}{\tilde{\Omega }}^4 a^3\bar{a}{C_0}+3k_{3,4} {\tilde{\Omega }}^4 a^4\bar{a}^2{C_0}+4k_{3,4}{\tilde{\Omega }}^4 a^3\bar{a}{C_0}^3 +10k_{5,4}{\tilde{\Omega }}^4 a^4\bar{a}^2{C_0}^3+4k_{5,4}{\tilde{\Omega }}^4 a^3 \bar{a}{C_0}^5-14k_{7,4}{\tilde{\Omega }}^4 a^6\bar{a}^4{C_0}+21k_{7,4}{\tilde{\Omega }}^4 a^4\bar{a}^2{C_0}^5+4k_{7,4}{\tilde{\Omega }}^4 a^3\bar{a}{C_0}^7+10\mathrm{j}k_{2,5} {\tilde{\Omega }}^5 a^4\bar{a}^2{C_0}+24\mathrm{j}k_{4,5}{\tilde{\Omega }}^5 a^5 \bar{a}^3{C_0}+20\mathrm{j}k_{4,5}{\tilde{\Omega }}^5 a^4\bar{a}^2{C_0}^3+60\mathrm{j}k_{6,5} {\tilde{\Omega }}^5 a^6\bar{a}^4{C_0}+120\mathrm{j}k_{6,5}{\tilde{\Omega }}^5 a^5 \bar{a}^3{C_0}^3+30\mathrm{j}k_{6,5}{\tilde{\Omega }}^5 a^4\bar{a}^2{C_0}^5+15k_{1,6} {\tilde{\Omega }}^6 a^4\bar{a}^2{C_0}+12k_{3,6}{\tilde{\Omega }}^6 a^5\bar{a}^3{C_0} +12k_{3,6}{\tilde{\Omega }}^6 a^5\bar{a}^3{C_0}+15k_{3,6}{\tilde{\Omega }}^6 a^4 \bar{a}^2{C_0}^3+10k_{5,6}{\tilde{\Omega }}^6 a^6\bar{a}^4{C_0}+40k_{5,6} {\tilde{\Omega }}^6 a^5\bar{a}^3{C_0}^3+15k_{5,6}{\tilde{\Omega }}^6 a^4\bar{a}^2{C_0}^5 +28\mathrm{j}k_{2,7}{\tilde{\Omega }}^7 a^5\bar{a}^3{C_0}+56\mathrm{j}k_{4,7}{\tilde{\Omega }}^7 a^6\bar{a}^4{C_0}+56\mathrm{j}k_{4,7}{\tilde{\Omega }}^7 a^5\bar{a}^3{C_0}^3+56k_{1,8} {\tilde{\Omega }}^8 a^5\bar{a}^3{C_0}+42k_{3,8}{\tilde{\Omega }}^8 a^6\bar{a}^4{C_0} +56k_{3,8}{\tilde{\Omega }}^8 a^5\bar{a}^3{C_0}^3+84\mathrm{j}k_{2,9}{\tilde{\Omega }}^9 a^6\bar{a}^4{C_0}+210k_{1,10}{\tilde{\Omega }}^{10} a^6\bar{a}^4{C_0} ,\)

\(b_9=-k_{3,0}a^3-5k_{5,0}a^4\bar{a}-10k_{5,0}a^3{C_0}^2-21k_{7,0}a^5\bar{a}^2 -105k_{7,0}a^4\bar{a}{C_0}^2-35k_{7,0}a^3{C_0}^4-84k_{9,0}a^6\bar{a}^3-756k_{9,0} a^5\bar{a}^2{C_0}^2-630k_{9,0}a^4\bar{a}{C_0}^4-84k_{9,0}a^3{C_0}^6-330k_{11,0}a^7 \bar{a}^4-4620k_{11,0}a^6\bar{a}^3{C_0}^2-6930k_{11,0}a^5\bar{a}^2{C_0}^4-2310k_{11,0} a^4\bar{a}{C_0}^6-165k_{11,0}a^3{C_0}^8+9\mathrm{j}k_{6,1}{\tilde{\Omega }} a^5\bar{a}^2 +45\mathrm{j}k_{6,1}{\tilde{\Omega }} a^4\bar{a}{C_0}^2+15\mathrm{j}k_{6,1}{\tilde{\Omega }} a^3{C_0}^4+28\mathrm{j}k_{8,1}{\tilde{\Omega }} a^6\bar{a}^3+252\mathrm{j}k_{8,1}{\tilde{\Omega }} a^5\bar{a}^2{C_0}^2+210\mathrm{j}k_{8,1}{\tilde{\Omega }} a^4\bar{a}{C_0}^4+28\mathrm{j}k_{8,1} {\tilde{\Omega }} a^3{C_0}^6+90\mathrm{j}k_{10,1}{\tilde{\Omega }} a^7\bar{a}^4+1260\mathrm{j} k_{10,1}{\tilde{\Omega }} a^6\bar{a}^3{C_0}^2+1890\mathrm{j}k_{10,1}{\tilde{\Omega }} a^5 \bar{a}^2{C_0}^4+630\mathrm{j}k_{10,1}{\tilde{\Omega }} a^4\bar{a}{C_0}^6+45\mathrm{j}k_{10,1} {\tilde{\Omega }} a^3{C_0}^8+\mathrm{j}k_{2,1}{\tilde{\Omega }} a^3+3\mathrm{j}k_{4,1} {\tilde{\Omega }} a^4\bar{a}+6\mathrm{j}k_{4,1}{\tilde{\Omega }} a^3{C_0}^2+k_{1,2} {\tilde{\Omega }}^2 a^3+k_{3,2}{\tilde{\Omega }}^2 a^4 \bar{a}+3k_{3,2}{\tilde{\Omega }}^2 a^3{C_0}^2+k_{5,2}{\tilde{\Omega }}^2 a^5 \bar{a}^2+10k_{5,2}{\tilde{\Omega }}^2 a^4 \bar{a}{C_0}^2+5k_{5,2}{\tilde{\Omega }}^2 a^3 {C_0}^4+21k_{7,2}{\tilde{\Omega }}^2 a^5 \bar{a}^2{C_0}^2+35k_{7,2}{\tilde{\Omega }}^2 a^4 \bar{a}{C_0}^4+7k_{7,2} {\tilde{\Omega }}^2 a^3{C_0}^6-6k_{9,2}{\tilde{\Omega }}^2 a^7 \bar{a}^4+126k_{9,2} {\tilde{\Omega }}^2 a^5 \bar{a}^2{C_0}^4+84k_{9,2}{\tilde{\Omega }}^2 a^4 \bar{a}{C_0}^6 +9k_{9,2}{\tilde{\Omega }}^2 a^3 {C_0}^8-\mathrm{j}k_{0,3}{\tilde{\Omega }}^3 a^3+\mathrm{j} k_{2,3}{\tilde{\Omega }}^3 a^4\bar{a}-\mathrm{j}k_{2,3}{\tilde{\Omega }}^3 a^3{C_0}^2 +3\mathrm{j}k_{4,3}{\tilde{\Omega }}^3 a^5\bar{a}^2+6\mathrm{j}k_{4,3}{\tilde{\Omega }}^3 a^4\bar{a}{C_0}^2-\mathrm{j}k_{4,3}{\tilde{\Omega }}^3 a^3{C_0}^4+8\mathrm{j}k_{6,3} {\tilde{\Omega }}^3 a^6\bar{a}^3+45\mathrm{j}k_{6,3}{\tilde{\Omega }}^3 a^5\bar{a}^2{C_0}^2 +15\mathrm{j}k_{6,3}{\tilde{\Omega }}^3 a^4\bar{a}{C_0}^4-\mathrm{j}k_{6,3}{\tilde{\Omega }}^3 a^3{C_0}^6+22\mathrm{j}k_{8,3}{\tilde{\Omega }}^3 a^7\bar{a}^4+224\mathrm{j}k_{8,3} {\tilde{\Omega }}^3 a^6\bar{a}^3{C_0}^2+210\mathrm{j}k_{8,3}{\tilde{\Omega }}^3 a^5 \bar{a}^2{C_0}^4+28\mathrm{j}k_{8,3}{\tilde{\Omega }}^3 a^4\bar{a}{C_0}^6-\mathrm{j}k_{8,3} {\tilde{\Omega }}^3 a^3{C_0}^8+3k_{1,4}{\tilde{\Omega }}^4 a^4\bar{a}+3k_{3,4} {\tilde{\Omega }}^4 a^5\bar{a}^2+9k_{3,4}{\tilde{\Omega }}^4 a^4\bar{a}{C_0}^2+4k_{5,4} {\tilde{\Omega }}^4 a^6\bar{a}^3+30k_{5,4}{\tilde{\Omega }}^4 a^5\bar{a}^2{C_0}^2+15k_{5,4} {\tilde{\Omega }}^4 a^4\bar{a}{C_0}^4+6k_{7,4}{\tilde{\Omega }}^4 a^7\bar{a}^4+84k_{7,4} {\tilde{\Omega }}^4 a^6\bar{a}^3{C_0}^2+105k_{7,4}{\tilde{\Omega }}^4 a^5\bar{a}^2{C_0}^4 +21k_{7,4}{\tilde{\Omega }}^4 a^4\bar{a}{C_0}^6-5\mathrm{j}k_{0,5}{\tilde{\Omega }}^5 a^4 \bar{a}+\mathrm{j}k_{2,5}{\tilde{\Omega }}^5 a^5\bar{a}^2-5\mathrm{j}k_{2,5}{\tilde{\Omega }}^5 a^4\bar{a}{C_0}^2+4\mathrm{j}k_{4,5}{\tilde{\Omega }}^5 a^6\bar{a}^3+6\mathrm{j}k_{4,5} {\tilde{\Omega }}^5 a^5\bar{a}^2{C_0}^2-5\mathrm{j}k_{4,5}{\tilde{\Omega }}^5 a^4\bar{a} {C_0}^4+10\mathrm{j}k_{6,5}{\tilde{\Omega }}^5 a^7\bar{a}^4+60\mathrm{j}k_{6,5} {\tilde{\Omega }}^5 a^6\bar{a}^3{C_0}^2+15\mathrm{j}k_{6,5}{\tilde{\Omega }}^5 a^5 \bar{a}^2{C_0}^4-5\mathrm{j}k_{6,5}{\tilde{\Omega }}^5 a^4\bar{a}{C_0}^6+9k_{1,6} {\tilde{\Omega }}^6 a^5\bar{a}^2+8k_{3,6}{\tilde{\Omega }}^6 a^6\bar{a}^3+27k_{3,6} {\tilde{\Omega }}^6 a^5\bar{a}^2{C_0}^2+10k_{5,6}{\tilde{\Omega }}^6 a^7\bar{a}^4 +80k_{5,6}{\tilde{\Omega }}^6 a^6\bar{a}^3{C_0}^2+45k_{5,6}{\tilde{\Omega }}^6 a^5 \bar{a}^2{C_0}^4-21\mathrm{j}k_{0,7}{\tilde{\Omega }}^7 a^5\bar{a}^2-21\mathrm{j}k_{2,7} {\tilde{\Omega }}^7 a^5\bar{a}^2{C_0}^2+6\mathrm{j}k_{4,7}{\tilde{\Omega }}^7 a^4-21\mathrm{j} k_{4,7}{\tilde{\Omega }}^7 a^5\bar{a}^2{C_0}^4+28k_{0,7}{\tilde{\Omega }}^8 a^6 \bar{a}^3+22k_{3,8}{\tilde{\Omega }}^8 a^7\bar{a}^4+84k_{3,8}{\tilde{\Omega }}^8 a^6\bar{a}^3{C_0}^2-84\mathrm{j}k_{0,9}{\tilde{\Omega }}^9 a^6\bar{a}^3-6\mathrm{j} k_{2,9}{\tilde{\Omega }}^9 a^7\bar{a}^4-84\mathrm{j}k_{2,9}{\tilde{\Omega }}^9 a^6 \bar{a}^3{C_0}^2+90k_{1,10}{\tilde{\Omega }}^{10} a^7\bar{a}^4-330\mathrm{j}k_{0,11} {\tilde{\Omega }}^{11} a^7\bar{a}^4 ,\)

\(b_{10}=-5k_{5,0}a^4C_0-42k_{7,0}a^5\bar{a}C_0-35k_{7,0}a^4{C_0}^3-252k_{9,0}a^6 \bar{a}^2C_0-504k_{9,0}a^5\bar{a}{C_0}^3-126k_{9,0}a^4{C_0}^5-1320k_{11,0}a^7 \bar{a}^3{C_0}-4620k_{11,0}a^6\bar{a}^2{C_0}^3-2772k_{11,0}a^5\bar{a}{C_0}^5-330 k_{11,0}a^4{C_0}^7+24\mathrm{j}k_{6,1}{\tilde{\Omega }} a^5\bar{a}{C_0}+20\mathrm{j}k_{6,1} {\tilde{\Omega }} a^4{C_0}^3+112\mathrm{j}k_{8,1}{\tilde{\Omega }} a^6\bar{a}^2{C_0} +224\mathrm{j}k_{8,1}{\tilde{\Omega }} a^5\bar{a} {C_0}^3+56\mathrm{j}k_{8,1}{\tilde{\Omega }} a^4{C_0}^5+480\mathrm{j}k_{10,1}{\tilde{\Omega }} a^7\bar{a}^3{C_0}+1680\mathrm{j}k_{10,1} {\tilde{\Omega }} a^6\bar{a}^2{C_0}^3+1008\mathrm{j}k_{10,1}{\tilde{\Omega }} a^5\bar{a} {C_0}^5+120\mathrm{j}k_{10,1}{\tilde{\Omega }} a^4{C_0}^7+4\mathrm{j}k_{4,1}{\tilde{\Omega }} a^4{C_0}+3k_{3,2}{\tilde{\Omega }}^2 a^4{C_0}+10k_{5,2}{\tilde{\Omega }}^2 a^5\bar{a} {C_0}+10k_{5,2}{\tilde{\Omega }}^2 a^4 {C_0}^3+28k_{7,2}{\tilde{\Omega }}^2 a^6\bar{a}^2 {C_0}+70k_{7,2}{\tilde{\Omega }}^2 a^5\bar{a} {C_0}^3+21k_{7,2}{\tilde{\Omega }}^2 a^4 {C_0}^5+72k_{9,2}{\tilde{\Omega }}^2 a^7\bar{a}^3 {C_0}+336k_{9,2}{\tilde{\Omega }}^2 a^6\bar{a}^2 {C_0}^3+252k_{9,2}{\tilde{\Omega }}^2 a^5\bar{a} {C_0}^5+36k_{9,2} {\tilde{\Omega }}^2 a^4 {C_0}^7-2\mathrm{j}k_{2,3}{\tilde{\Omega }}^3 a^4{C_0}-4\mathrm{j} k_{4,3}{\tilde{\Omega }}^3 a^4{C_0}^3+12\mathrm{j}k_{6,3}{\tilde{\Omega }}^3 a^6 \bar{a}^2 {C_0}-6\mathrm{j}k_{6,3}{\tilde{\Omega }}^3 a^4 {C_0}^5+64\mathrm{j}k_{8,3} {\tilde{\Omega }}^3 a^7 \bar{a}^3 {C_0}+112\mathrm{j}k_{8,3}{\tilde{\Omega }}^3 a^6 \bar{a}^2 {C_0}^3-8\mathrm{j}k_{8,3}{\tilde{\Omega }}^3 a^4 {C_0}^7-k_{1,4} {\tilde{\Omega }}^4 a^4 {C_0}+6k_{3,4}{\tilde{\Omega }}^4 a^5 \bar{a} {C_0}-k_{3,4} {\tilde{\Omega }}^4 a^4 {C_0}^3+20k_{5,4}{\tilde{\Omega }}^4 a^6 \bar{a}^2 {C_0} +20k_{5,4}{\tilde{\Omega }}^4 a^5 \bar{a} {C_0}^3-k_{5,4}{\tilde{\Omega }}^4 a^4 {C_0}^5+56k_{7,4}{\tilde{\Omega }}^4 a^7 \bar{a}^3 {C_0}+140k_{7,4}{\tilde{\Omega }}^4 a^6 \bar{a}^2 {C_0}^3+42k_{7,4}{\tilde{\Omega }}^4 a^5 \bar{a} {C_0}^5-k_{7,4} {\tilde{\Omega }}^4 a^4 {C_0}^7-8\mathrm{j}k_{2,5}{\tilde{\Omega }}^5 a^5 \bar{a}{C_0} -8\mathrm{j}k_{4,5}{\tilde{\Omega }}^5 a^6 \bar{a}^2 {C_0}-16\mathrm{j}k_{4,5}{\tilde{\Omega }}^5 a^5 \bar{a} {C_0}^3-40\mathrm{j}k_{6,5}{\tilde{\Omega }}^5 a^6 \bar{a}^2 {C_0}^3-24\mathrm{j} k_{6,5}{\tilde{\Omega }}^5 a^5 \bar{a} {C_0}^5-6k_{1,6}{\tilde{\Omega }}^6 a^6 \bar{a} {C_0}+12k_{3,6}{\tilde{\Omega }}^6 a^6 \bar{a}^2 {C_0}-6k_{3,6}{\tilde{\Omega }}^6 a^5 \bar{a} {C_0}^3+40k_{5,6}{\tilde{\Omega }}^6 a^5 \bar{a} {C_0}^3+40k_{5,6} {\tilde{\Omega }}^6 a^7 \bar{a}^3 {C_0}+40k_{5,6}{\tilde{\Omega }}^6 a^6 \bar{a}^2 {C_0}^3-6k_{5,6}{\tilde{\Omega }}^6 a^5 \bar{a} {C_0}^5-28\mathrm{j}k_{2,7} {\tilde{\Omega }}^7 a^6 \bar{a}^2 {C_0}-32\mathrm{j}k_{4,7}{\tilde{\Omega }}^7 a^7 \bar{a}^3 {C_0}-56\mathrm{j}k_{4,7}{\tilde{\Omega }}^7 a^6 \bar{a}^2 {C_0}^3-28k_{1,8} {\tilde{\Omega }}^8 a^6 \bar{a}^2 {C_0}+24k_{3,8}{\tilde{\Omega }}^8 a^7 \bar{a}^3 {C_0}-28k_{3,8}{\tilde{\Omega }}^8 a^6 \bar{a}^2 {C_0}^3-96\mathrm{j}k_{2,9}{\tilde{\Omega }}^9 a^7 \bar{a}^3 {C_0}-120k_{1,10}{\tilde{\Omega }}^{10} a^7 \bar{a}^3 {C_0} ,\)

\(b_{11}=-k_{5,0}a^5-7k_{7,0}a^6\bar{a}-21k_{7,0}a^5{C_0}^2-36k_{9,0}a^7\bar{a}^2-252 k_{9,0}a^6\bar{a}{C_0}^2-126k_{9,0}a^5{C_0}^4-165k_{11,0}a^8\bar{a}^3-1980k_{11,0}a^7 \bar{a}^2{C_0}^2-2310k_{11,0}a^6\bar{a}{C_0}^4-462k_{11,0}a^5{C_0}^6+5\mathrm{j}k_{6,1} a^6 \bar{a}+15\mathrm{j}k_{6,1}{\tilde{\Omega }} a^5 {C_0}^2+20\mathrm{j}k_{8,1}{\tilde{\Omega }} a^7 \bar{a}^2+140\mathrm{j}k_{8,1} a^6 \bar{a} {C_0}^2+70\mathrm{j}k_{8,1}{\tilde{\Omega }} a^5 {C_0}^4+75\mathrm{j}k_{10,1}{\tilde{\Omega }} a^8 \bar{a}^3 +900\mathrm{j}k_{10,1} {\tilde{\mathrm{\Omega }}} a^7 \bar{a}^2 {C_0}^2+1050\mathrm{j}k_{10,1}{\tilde{\Omega }} a^6 \bar{a}^2 {C_0}^4+210\mathrm{j}k_{10,1}{\tilde{\Omega }} a^5 {C_0}^6+\mathrm{j}k_{4,1} {\tilde{\Omega }} a^5+k_{3,2}{\tilde{\Omega }}^2 a^5+3k_{5,2}{\tilde{\Omega }}^2 a^6 \bar{a} +10k_{5,2}{\tilde{\Omega }}^2 a^5 {C_0}^2+8k_{7,2}{\tilde{\Omega }}^2 a^7 \bar{a}^2+63k_{7,2}{\tilde{\Omega }}^2 a^6 \bar{a} {C_0}^2+35k_{7,2}{\tilde{\Omega }}^2 a^5 {C_0}^4+21k_{9,2}{\tilde{\Omega }}^2 a^8 \bar{a}^3+288k_{9,2}{\tilde{\Omega }}^2 a^7 \bar{a}^2 {C_0}^2+378k_{9,2}{\tilde{\Omega }}^2 a^6 \bar{a} {C_0}^4+84k_{9,2} {\tilde{\Omega }}^2 a^5 {C_0}^6-\mathrm{j}k_{2,3}{\tilde{\Omega }}^3 a^5-\mathrm{j}k_{4,3} {\tilde{\Omega }}^3 a^6 \bar{a} -6\mathrm{j}k_{4,3} {\tilde{\Omega }}^3 a^5 {C_0}^2-15\mathrm{j} k_{6,3} {\tilde{\Omega }}^3 a^6 \bar{a} {C_0}^2-15\mathrm{j}k_{6,3} {\tilde{\Omega }}^3 a^5 {C_0}^4+5\mathrm{j}k_{8,3} {\tilde{\Omega }}^3 a^8 \bar{a}^3-70\mathrm{j}k_{8,3} {\tilde{\Omega }}^3 a^6 \bar{a} {C_0}^4-28\mathrm{j}k_{8,3} {\tilde{\Omega }}^3 a^5 {C_0}^6-k_{1,4}{\tilde{\Omega }}^4 a^5+k_{3,4}{\tilde{\Omega }}^4 a^6 \bar{a}-3k_{3,4} {\tilde{\Omega }}^4 a^5 {C_0}^2+4k_{5,4}{\tilde{\Omega }}^4 a^7 \bar{a}^2+10k_{5,4} {\tilde{\Omega }}^4 a^6 \bar{a}{C_0}^2-5k_{5,4}{\tilde{\Omega }}^4 a^5 {C_0}^4+11k_{7,4} {\tilde{\Omega }}^4 a^8 \bar{a}^3+84k_{7,4}{\tilde{\Omega }}^4 a^7 \bar{a}^2{C_0}^2 +35k_{7,4}{\tilde{\Omega }}^4 a^6 \bar{a}{C_0}^4-7k_{7,4}{\tilde{\Omega }}^4 a^5 {C_0}^6 +\mathrm{j}k_{0,5} {\tilde{\Omega }}^5 a^5-3\mathrm{j}k_{2,5} {\tilde{\Omega }}^5 a^6\bar{a} +\mathrm{j}k_{2,5} {\tilde{\Omega }}^5 a^5{C_0}^2-4\mathrm{j}k_{4,5} {\tilde{\Omega }}^5 a^7 \bar{a}^2-18\mathrm{j}k_{4,5} {\tilde{\Omega }}^5 a^6\bar{a}{C_0}^2+\mathrm{j}k_{4,5} {\tilde{\Omega }}^5 a^5{C_0}^4-5\mathrm{j}k_{6,5} {\tilde{\Omega }}^5 a^8\bar{a}^3-60 \mathrm{j}k_{6,5} {\tilde{\Omega }}^5 a^7\bar{a}^2{C_0}^2-45\mathrm{j}k_{6,5} {\tilde{\Omega }}^5 a^6\bar{a}{C_0}^4+\mathrm{j}k_{6,5} {\tilde{\Omega }}^5 a^5{C_0}^6-5k_{1,6} {\tilde{\Omega }}^6 a^6\bar{a}-15k_{3,6} {\tilde{\Omega }}^6 a^6\bar{a}{C_0}^2+5k_{5,6} {\tilde{\Omega }}^6 a^8\bar{a}^3-25k_{5,6} {\tilde{\Omega }}^6 a^6\bar{a}{C_0}^4 +7\mathrm{j}k_{0,7} {\tilde{\Omega }}^7 a^6\bar{a}-8\mathrm{j}k_{2,7} {\tilde{\Omega }}^7 a^7\bar{a}^2+7\mathrm{j}k_{2,7} {\tilde{\Omega }}^7 a^6\bar{a}{C_0}^2-11\mathrm{j}k_{4,7} {\tilde{\Omega }}^7 a^8\bar{a}^3-48\mathrm{j}k_{4,7} {\tilde{\Omega }}^7 a^7\bar{a}^2{C_0}^2 +7\mathrm{j}k_{4,7} {\tilde{\Omega }}^7 a^6\bar{a}{C_0}^4-20k_{1,8} {\tilde{\Omega }}^8 a^7 \bar{a}^2-5k_{3,8} {\tilde{\Omega }}^8 a^8\bar{a}^3-60k_{3,8} {\tilde{\Omega }}^8 a^7 \bar{a}^2{C_0}^2+36\mathrm{j}k_{0,9} {\tilde{\Omega }}^9 a^7\bar{a}^2-21\mathrm{j}k_{2,9} {\tilde{\Omega }}^9 a^8\bar{a}^3+36\mathrm{j}k_{2,9} {\tilde{\Omega }}^9 a^7\bar{a}^2{C_0}^2 -75k_{1,10} {\tilde{\Omega }}^{10} a^8\bar{a}^3+165\mathrm{j}k_{0,11} {\tilde{\Omega }}^{11} a^8\bar{a}^3 ,\)

\(b_{12}=-7k_{7,0}a^6{C_0}-72k_{9,0}a^7\bar{a}{C_0}-84k_{9,0}a^6{C_0}^3-495k_{11,0} a^8\bar{a}^2{C_0}-1320k_{11,0}a^7\bar{a}{C_0}^3-462k_{11,0}a^6{C_0}^5+6\mathrm{j}k_{6,1} {\tilde{\Omega }} a^6{C_0}+48\mathrm{j}k_{8,1}{\tilde{\Omega }} a^7\bar{a}{C_0}+56\mathrm{j}k_{8,1} {\tilde{\Omega }} a^6{C_0}^3+270\mathrm{j}k_{10,1}{\tilde{\Omega }} a^8\bar{a}^2{C_0}+720\mathrm{j} k_{10,1}{\tilde{\Omega }} a^7\bar{a}{C_0}^3+252\mathrm{j}k_{10,1}{\tilde{\Omega }} a^6{C_0}^5 +5k_{5,2}{\tilde{\Omega }}^2 a^6{C_0}+28k_{7,2}{\tilde{\Omega }}^2 a^7\bar{a}{C_0}+35k_{7,2} {\tilde{\Omega }}^2 a^6{C_0}^3+117k_{9,2}{\tilde{\Omega }}^2 a^8\bar{a}^2{C_0}+336k_{9,2} {\tilde{\Omega }}^2 a^7\bar{a}{C_0}^3+126k_{9,2}{\tilde{\Omega }}^2 a^6{C_0}^5-4\mathrm{j}k_{4,3} {\tilde{\Omega }}^3 a^6{C_0}-12\mathrm{j}k_{6,3}{\tilde{\Omega }}^3 a^7\bar{a}{C_0}-20\mathrm{j} k_{6,3}{\tilde{\Omega }}^3 a^6{C_0}^3-24\mathrm{j}k_{8,3}{\tilde{\Omega }}^3 a^8\bar{a}^2{C_0} -112\mathrm{j}k_{8,3}{\tilde{\Omega }}^3 a^7\bar{a}{C_0}^3-56\mathrm{j}k_{8,3}{\tilde{\Omega }}^3 a^6{C_0}^5-3k_{3,4}{\tilde{\Omega }}^4 a^6\bar{a}{C_0}-10k_{5,4}{\tilde{\Omega }}^4 a^6{C_0}^3 +21k_{7,4}{\tilde{\Omega }}^4 a^8\bar{a}^2{C_0}-21k_{7,4}{\tilde{\Omega }}^4 a^6{C_0}^5 +2\mathrm{j}k_{2,5}{\tilde{\Omega }}^5 a^6{C_0}-8\mathrm{j}k_{4,5}{\tilde{\Omega }}^5 a^7 \bar{a}{C_0}+4\mathrm{j}k_{4,5}{\tilde{\Omega }}^5 a^6{C_0}^3-30\mathrm{j}k_{6,5}{\tilde{\Omega }}^5 a^8\bar{a}^2{C_0}-40\mathrm{j}k_{6,5}{\tilde{\Omega }}^5 a^7\bar{a}{C_0}^3+6\mathrm{j}k_{6,5}{\tilde{\Omega }}^5 a^6\bar{a}^6{C_0}^5+k_{1,6}{\tilde{\Omega }}^6 a^6{C_0}-12k_{3,6}{\tilde{\Omega }}^6 a^7\bar{a}{C_0}+k_{5,6}{\tilde{\Omega }}^6 a^6{C_0}^3-15k_{5,6}{\tilde{\Omega }}^6 a^8\bar{a}^2-40k_{5,6}{\tilde{\Omega }}^6 a^7\bar{a}{C_0}^3+k_{5,6}{\tilde{\Omega }}^6 a^6{C_0}^5+12\mathrm{j}k_{2,7}{\tilde{\Omega }}^7 a^7\bar{a}{C_0}-12\mathrm{j}k_{4,7}{\tilde{\Omega }}^7 a^8\bar{a}^2{C_0}+24\mathrm{j}k_{4,7}\tilde{\Omega }^7 a^7\bar{a}{C_0}^3+8k_{1,8}\tilde{\Omega }^8 a^7\bar{a}{C_0}-39k_{3,8}{\tilde{\Omega }}^8 a^8\bar{a}^2{C_0}+8k_{3,8}{\tilde{\Omega }}^8 a^7\bar{a}{C_0}^3+54\mathrm{j}k_{2,9}{\tilde{\Omega }}^9 a^8\bar{a}^2{C_0}+45k_{1,10}{\tilde{\Omega }}^{10} a^8\bar{a}^2{C_0} ,\)

\(b_{13}=-k_{7,0}a^7-9k_{9,0}a^8\bar{a}-36k_{9,0}a^7{C_0}^2-55k_{11,0}a^9\bar{a}^2 -495k_{11,0}a^8\bar{a}{C_0}^2-330k_{11,0}a^7{C_0}^4+\mathrm{j}k_{6,1}{\tilde{\Omega }} a^7+7\mathrm{j}k_{8,1}{\tilde{\Omega }} a^8\bar{a}+28\mathrm{j}k_{8,1}{\tilde{\Omega }} a^7{C_0}^2+35\mathrm{j}k_{10,1}{\tilde{\Omega }} a^9\bar{a}^2+315\mathrm{j}k_{10,1}{\tilde{\Omega }} a^8\bar{a}{C_0}^2+210\mathrm{j}k_{10,1}{\tilde{\Omega }} a^7{C_0}^4+k_{5,2}{\tilde{\Omega }}^2a^7+5k_{7,2}{\tilde{\Omega }}^2 a^8\bar{a}+210k_{7,2}{\tilde{\Omega }}^2 a^7{C_0}^2+19k_{9,2}{\tilde{\Omega }}^2 a^9\bar{a}^2+180k_{9,2}{\tilde{\Omega }}^2 a^8\bar{a}{C_0}^2+126k_{9,2}{\tilde{\Omega }}^2 a^7{C_0}^4-\mathrm{j}k_{4,3}{\tilde{\Omega }}^3 a^7-3\mathrm{j}k_{6,3}{\tilde{\Omega }}^3 a^8 \bar{a}-15\mathrm{j}k_{6,3}{\tilde{\Omega }}^3 a^7 {C_0}^2-7\mathrm{j}k_{8,3}{\tilde{\Omega }}^3 a^9 \bar{a}^2-84\mathrm{j}k_{8,3}{\tilde{\Omega }}^3 a^8 \bar{a}{C_0}^2-70\mathrm{j}k_{8,3}{\tilde{\Omega }}^3 a^7 {C_0}^4-k_{3,4}{\tilde{\Omega }}^4 a^7-k_{5,4}{\tilde{\Omega }}^4 a^8\bar{a}-10k_{5,4}{\tilde{\Omega }}^4 a^7{C_0}^2+k_{7,4}{\tilde{\Omega }}^4 a^9\bar{a}^2-21k_{7,4}{\tilde{\Omega }}^4 a^8\bar{a}{C_0}^2-35k_{7,4}{\tilde{\Omega }}^4 a^7{C_0}^4+\mathrm{j}k_{2,5}{\tilde{\Omega }}^5 a^7 -\mathrm{j}k_{4,5}{\tilde{\Omega }}^5 a^8 \bar{a} +6\mathrm{j}k_{4,5}{\tilde{\Omega }}^5 a^7 {C_0}^2-5\mathrm{j}k_{6,5}{\tilde{\Omega }}^5 a^9 \bar{a}^2-15\mathrm{j}k_{6,5}{\tilde{\Omega }}^5 a^8 \bar{a} {C_0}^2+15\mathrm{j}k_{6,5}{\tilde{\Omega }}^5 a^7 {C_0}^4+k_{1,6}{\tilde{\Omega }}^6 a^7-3k_{3,6}{\tilde{\Omega }}^6 a^8 \bar{a}+3k_{3,6}{\tilde{\Omega }}^6 a^7 {C_0}^2+5k_{5,6}{\tilde{\Omega }}^6 a^9 \bar{a}^2-30k_{5,6}{\tilde{\Omega }}^6 a^8 \bar{a} {C_0}^2+5k_{5,6}{\tilde{\Omega }}^6 a^7 {C_0}^4-\mathrm{j}k_{0,7}{\tilde{\Omega }}^7 a^7+5\mathrm{j}k_{2,7}{\tilde{\Omega }}^7 a^8 \bar{a}-\mathrm{j}k_{2,7}{\tilde{\Omega }}^7 a^7{C_0}^2+\mathrm{j}k_{4,7}{\tilde{\Omega }}^7 a^9 \bar{a}^2+30\mathrm{j}k_{4,7}{\tilde{\Omega }}^7 a^8 \bar{a} {C_0}^2-\mathrm{j}k_{4,7}{\tilde{\Omega }}^7 a^7 {C_0}^4+7k_{1,8}{\tilde{\Omega }}^8 a^8 \bar{a}-7k_{3,8}{\tilde{\Omega }}^8 a^9 \bar{a}^2+21k_{3,8}{\tilde{\Omega }}^8 a^8 \bar{a} {C_0}^2-9\mathrm{j}k_{0,9}{\tilde{\Omega }}^9 a^8 \bar{a}+19\mathrm{j}k_{2,9}{\tilde{\Omega }}^9 a^9 \bar{a}^2-9\mathrm{j}k_{2,9}{\tilde{\Omega }}^9 a^8 \bar{a} {C_0}^2+35k_{1,10}{\tilde{\Omega }}^{10} a^9 \bar{a}^2-55\mathrm{j}k_{0,11}{\tilde{\Omega }}^{11} a^9 \bar{a}^2 ,\)

\(b_{14}=-9k_{9,0}a^8{C_0}-110k_{11,0}a^8\bar{a}{C_0}-165k_{11,0}a^8{C_0}^3+8\mathrm{j}k_{8,1}{\tilde{\Omega }} a^8{C_0}+80\mathrm{j}k_{10,1}{\tilde{\Omega }} a^9 \bar{a}{C_0}+120\mathrm{j}k_{10,1}{\tilde{\Omega }} a^8 {C_0}^3+7k_{7,2}{\tilde{\Omega }}^2 a^8{C_0}+54k_{9,2}{\tilde{\Omega }}^2 a^9 \bar{a}{C_0}+84k_{9,2}{\tilde{\Omega }}^2 a^8 {C_0}^3-6\mathrm{j}k_{6,3}{\tilde{\Omega }}^3 a^8 {C_0}-32\mathrm{j}k_{8,3}{\tilde{\Omega }}^3 a^9 \bar{a}{C_0}-56\mathrm{j}k_{8,3}{\tilde{\Omega }}^3 a^8 {C_0}^3-5k_{5,4}{\tilde{\Omega }}^4 a^8{C_0}-14k_{7,4}{\tilde{\Omega }}^4 a^9 \bar{a}{C_0}-35k_{7,4}{\tilde{\Omega }}^4 a^8{C_0}^3+4\mathrm{j}k_{4,5}{\tilde{\Omega }}^5 a^8{C_0}+20\mathrm{j}k_{6,5}{\tilde{\Omega }}^5 a^8{C_0}^3+3k_{3,6}{\tilde{\Omega }}^6 a^8{C_0}-10k_{5,6}{\tilde{\Omega }}^6 a^9 \bar{a}{C_0}+10k_{5,6}{\tilde{\Omega }}^6 a^8 {C_0}^3-2\mathrm{j}k_{2,7}{\tilde{\Omega }}^7 a^8{C_0}+16\mathrm{j}k_{4,7}{\tilde{\Omega }}^7 a^9\bar{a}{C_0}-4\mathrm{j}k_{4,7}{\tilde{\Omega }}^7 a^8{C_0}^3-k_{1,8}{\tilde{\Omega }}^8 a^8{C_0}+18k_{3,8}{\tilde{\Omega }}^8 a^9\bar{a}{C_0}-k_{3,8}{\tilde{\Omega }}^8 a^8{C_0}^3-16\mathrm{j}k_{2,9}{\tilde{\Omega }}^9 a^9\bar{a}{C_0}-10k_{1,10}{\tilde{\Omega }}^{10} a^9\bar{a}{C_0} ,\)

\(b_{15}=-k_{9,0}a^9-11k_{11,0}a^{10}\bar{a}-55k_{11,0}a^9{C_0}^2+\mathrm{j}k_{8,1}{\tilde{\Omega }} a^9+9\mathrm{j}k_{10,1}{\tilde{\Omega }} a^{10} \bar{a}+45\mathrm{j}k_{10,1}{\tilde{\Omega }} a^9{C_0}^2+k_{7,2}{\tilde{\Omega }}^2 a^9+7k_{9,2}{\tilde{\Omega }}^2 a^{10}\bar{a}+36k_{9,2}{\tilde{\Omega }}^2 a^9{C_0}^2-\mathrm{j}k_{6,3}{\tilde{\Omega }}^3 a^9-5\mathrm{j}k_{8,3}{\tilde{\Omega }}^3 a^{10} \bar{a}-28\mathrm{j}k_{8,3}{\tilde{\Omega }}^3 a^9{C_0}^2-k_{5,4}{\tilde{\Omega }}^4 a^9-3k_{7,4}{\tilde{\Omega }}^4 a^{10} \bar{a}-21k_{7,4}{\tilde{\Omega }}^4 a^9 {C_0}^2+\mathrm{j}k_{4,5}{\tilde{\Omega }}^5 a^9+\mathrm{j}k_{6,5}{\tilde{\Omega }}^5 a^{10}\bar{a}+15\mathrm{j}k_{6,5}{\tilde{\Omega }}^5 a^9{C_0}^2+k_{3,6}{\tilde{\Omega }}^6 a^9-k_{5,6}{\tilde{\Omega }}^6 a^{10}\bar{a}+k_{5,6}{\tilde{\Omega }}^6 a^9{C_0}^2-\mathrm{j}k_{2,7}{\tilde{\Omega }}^7 a^9+3k_{4,7}{\tilde{\Omega }}^7 a^{10}\bar{a}-6\mathrm{j}k_{4,7}{\tilde{\Omega }}^7 a^9{C_0}^2-k_{1,8}{\tilde{\Omega }}^8 a^9+5k_{3,8}{\tilde{\Omega }}^8 a^{10}\bar{a}-3k_{3,8}{\tilde{\mathrm{\Omega }}}^8 a^9{C_0}^2+\mathrm{j}k_{0,9}{\tilde{\Omega }}^9 a^9-7\mathrm{j}k_{2,9}{\tilde{\Omega }}^9 a^{10}\bar{a}+\mathrm{j}k_{2,9}{\tilde{\Omega }}^9 a^9{C_0}^2-9k_{1,10}{\tilde{\Omega }}^{10} a^{10}\bar{a}+11\mathrm{j}k_{0,11}{\tilde{\Omega }}^{11} a^{10}\bar{a} ,\)

\(b_{16}=-11k_{11,0}a^{10}{C_0}+10\mathrm{j}k_{10,1}{\tilde{\Omega }} a^{10}{C_0}+9k_{9,2}{\tilde{\Omega }}^2 a^{10}{C_0}-8\mathrm{j}k_{8,3}{\tilde{\Omega }}^3 a^{10}{C_0}^3-7k_{7,4}{\tilde{\Omega }}^4 a^{10}{C_0}+6\mathrm{j}k_{6,5}{\tilde{\Omega }}^5 a^{10}{C_0}+5k_{5,6}{\tilde{\Omega }}^6 a^{10}{C_0}-4\mathrm{j}k_{4,7}{\tilde{\Omega }}^7 a^{10}{C_0}-3k_{3,8}\tilde{\Omega }^8 a^{10}{C_0}+2\mathrm{j}k_{2,9}\tilde{\Omega }^9 a^{10}{C_0}+k_{1,10}\tilde{\Omega }^{10} a^{10}{C_0} ,\)

\(b_{17}=-k_{11,0}a^{11}+\mathrm{j}k_{10,1}{\tilde{\Omega }} a^{11}+k_{9,2}{\tilde{\Omega }}^2 a^{11}-\mathrm{j}k_{8,3}{\tilde{\Omega }}^3 a^{11}-k_{7,4}{\tilde{\Omega }}^4 a^{11}+\mathrm{j}k_{6,5}{\tilde{\Omega }}^5 a^{11}+k_{5,6}{\tilde{\Omega }}^6 a^{11}-\mathrm{j}k_{4,7}{\tilde{\Omega }}^7 a^{11}-k_{3,8}{\tilde{\Omega }}^8 a^{11}+\mathrm{j}k_{2,9}{\tilde{\Omega }}^9 a^{11}+\mathrm{j}k_{1,10}{\tilde{\Omega }}^{10} a^{11}-\mathrm{j}k_{0,11}{\tilde{\Omega }}^{11}a^{11} .\)

Appendix C

In Eq. (20),

\(\tilde{F_f}\left( \xi _1,\xi _2\right) = k_{1,0}\xi _2 + k_{3,0}{{\xi _2}^3} + k_{5,0}{{\xi _2}^5} + k_{7,0}{{\xi _2}^7} + k_{9,0}{{\xi _2}^9} + k_{11,0}{{\xi _2}^{11}} + k_{0,1}\xi _1 + k_{2,1}{{\xi _2}^2}\xi _1 + k_{4,1}{{\xi _2}^4}\xi _1 + k_{6,1}{{\xi _2}^6}\xi _1 + k_{8,1}{{\xi _2}^8}\xi _1 + k_{10,1}{{\xi _2}^{10}}\xi _1 + k_{1,2}{\xi _2}{{\xi _1}^2} + k_{3,2}{{\xi _2}^3}{{\xi _1}^2} + k_{5,2}{{\xi _2}^5}{{\xi _1}^2} + k_{7,2}{{\xi _2}^7}{{\xi _1}^2}+ k_{9,2}{{\xi _2}^9}{{\xi _1}^2} + k_{0,3}{{\xi _1}^3} + k_{2,3}{{\xi _2}^2}{{\xi _1}^3} + k_{4,3}{{\xi _2}^4}{{\xi _1}^3} + k_{6,3}{{\xi _2}^6}{{\xi _1}^3} + k_{8,3}{{\xi _2}^8}{{\xi _1}^3} + k_{1,4}{\xi _2}{{\xi _1}^4} + k_{3,4}{{\xi _2}^3}{{\xi _1}^4} + k_{5,4}{{\xi _2}^5}{{\xi _1}^4} + k_{7,4}{{\xi _2}^7}{{\xi _1}^4} + k_{0,5}{{\xi _1}^5} + k_{2,5}{{\xi _2}^2}{{\xi _1}^5} + k_{4,5}{{\xi _2}^4}{{\xi _1}^5} + k_{6,5}{{\xi _2}^6}{{\xi _1}^5} + k_{1,6}{\xi _2}{{\xi _1}^6} + k_{3,6}{{\xi _2}^3}{{\xi _1}^6} + k_{5,6}{{\xi _2}^5}{{\xi _1}^6} + k_{0,7}{{\xi _1}^7} + k_{2,7}{{\xi _2}^2}{{\xi _1}^7} + k_{4,7}{{\xi _2}^4}{{\xi _1}^7} + k_{1,8}{\xi _2}{{\xi _1}^8} + k_{3,8}{{\xi _2}^3}{{\xi _1}^8} + k_{0,9}{{\xi _1}^9} + k_{2,9}{{\xi _2}^2}{{\xi _1}^9} + k_{1,10}{\xi _2}{{\xi _1}^{10}} + k_{0,11}{{\xi _1}^{11}} .\)

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Zhang, X., Sun, Z., Zhao, L. et al. Analysis of supercritical pitchfork bifurcation in active magnetic bearing-rotor system with current saturation. Nonlinear Dyn (2021). https://doi.org/10.1007/s11071-021-06220-w

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Keywords

  • Active magnetic bearing
  • Current saturation
  • Bifurcation
  • Static equilibrium
  • Method of multiple scales