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Complex dynamics of a harmonically excited structure coupled with a nonlinear energy sink

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Abstract

Nonlinear behaviors are investigated for a structure coupled with a nonlinear energy sink. The structure is linear and subject to a harmonic excitation, modeled as a forced single-degree-of-freedom oscillator. The nonlinear energy sink is modeled as an oscillator consisting of a mass, a nonlinear spring, and a linear damper. Based on the numerical solutions, global bifurcation diagrams are presented to reveal the coexistence of periodic and chaotic motions for varying nonlinear energy sink mass and stiffness. Chaos is numerically identified via phase trajectories, power spectra, and Poincaré maps. Amplitude-frequency response curves are predicted by the method of harmonic balance for periodic steady-state responses. Their stabilities are analyzed. The Hopf bifurcation and the saddle-node bifurcation are determined. The investigation demonstrates that a nonlinear energy sink may create dynamic complexity.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grants 11402151 and 11572182).

Author information

Authors and Affiliations

  1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, 200072, China

    Jian Zang & Li-Qun Chen

  2. Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai, 200072, China

    Li-Qun Chen

  3. Department of Mechanics, Shanghai University, Shanghai, 200444, China

    Li-Qun Chen

Authors
  1. Jian Zang
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  2. Li-Qun Chen
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Corresponding author

Correspondence to Li-Qun Chen.

Appendices

Appendix A

1.1 The nonlinear algebraic equations of the harmonic balance up to order 5

$$\begin{aligned}&-3/4\beta a_{11}^{2}b_{32} +3/4\beta a_{11}^{2}b_{31} +3/2\beta a_{51}^{2}b_{11} -3/2\beta a_{51}^{2}b_{12}\\&\quad -\,9/4\beta b_{11} ^{2}b_{12} +9/4\beta b_{11} b_{12}^{2}-3/4\beta a_{11}^{2}b_{12}+3/4\beta a_{11}^{2}b_{11} \\&\quad +\,3/4\beta a_{12}^{2}b_{11} -3/4\beta a_{12}^{2}b_{12}-3/2\beta a_{52} ^{2}b_{12} +3/2\beta a_{52}^{2}b_{11}\\&\quad -\,3/4\beta b_{12}^{2}b_{31} +\,3/4\beta b_{11}^{2}b_{32} -3/4\beta b_{11}^{2}b_{31} +3/4\beta a_{12}^{2}b_{31}\\&\quad -\,3/4\beta a_{12}^{2}b_{32} -3/2\beta b_{12} b_{51}^{2}-3/2\beta b_{12} b_{52}^{2}-\gamma \left( \zeta \right) a_{11} \\&\quad -\,\gamma \zeta _0 a_{11} +3/2\beta b_{11} b_{52}^{2}+3/2\beta b_{11} b_{51}^{2}+3/2\beta b_{11} b_{32}^{2}\\&\quad +\,3/2\beta b_{11} b_{31} ^{2}-3/4\beta a_{32}^{2}b_{51} +3/4\beta a_{32}^{2}b_{52} \\&\quad -\,3/2\beta a_{32}^{2}b_{12} +3/4\beta a_{31}^{2}b_{52} +3/2\beta a_{32}^{2}b_{11} -3/2\beta a_{31}^{2}b_{12}\\&\quad -3/4\beta a_{31} ^{2}b_{51} +3/2\beta a_{31}^{2}b_{11} +3/2\beta a_{32} a_{51} b_{11} \\&\quad -\,3/2\beta a_{32} a_{51} b_{12} -3/2\beta a_{32} a_{52} b_{11} +3/2\beta a_{32} a_{52} b_{12}\\&\quad -\,3/2\beta a_{32} a_{51} b_{31} +3/2\beta a_{32} a_{51} b_{32} +3/2\beta a_{32} a_{52} b_{31} \\&\quad +\,3/4\beta b_{12}^{2}b_{32} -3/2\beta a_{32} a_{52} b_{32} +3/2\beta a_{31} a_{52} b_{32}\\&\quad -\,3/2\beta a_{31} a_{52} b_{12} -3/2\beta a_{31} a_{52} b_{31} -3/2\beta a_{31} a_{51} b_{32} \\&\quad +\,3/2\beta a_{31} a_{52} b_{11} -3/2\beta a_{31} a_{32} b_{52} +3\beta a_{31} a_{32} b_{12} \\&\quad +\,3/2\beta a_{31} a_{32} b_{51} -3/4\beta b_{32}^{2}b_{52} +3/2\beta a_{31} a_{51} b_{12} \\&\quad +\,3/2\beta a_{31} a_{51} b_{31} -3/2\beta a_{31} a_{51} b_{11} +3/4\beta b_{32}^{2}b_{51}\\&\quad -\,3/4\beta b_{31}^{2}b_{52} +3/4\beta b_{31}^{2}b_{51} -3/2\beta b_{12} b_{32}^{2}\\&\quad -\,3/2\beta b_{12} b_{31}^{2} -3\beta b_{11} b_{51} b_{52} +3\beta b_{12} b_{31} b_{32}\\&\quad +\,3/2\beta b_{12} b_{31} b_{51}-\,3/2\beta b_{12} b_{31} b_{52} -3/2\beta b_{12} b_{32} b_{51}\\&\quad +\,3/2\beta b_{12} b_{32} b_{52}+3\beta b_{12} b_{51} b_{52}-\,3/2\beta b_{31} b_{32} b_{51}\\&\quad +\,3/2\beta b_{31} b_{32} b_{52} -3/2\beta b_{11} b_{32} b_{52}-3/2\beta a_{11} a_{12} b_{11}\\&\quad +\,3/2\beta a_{11} a_{12} b_{12} -3/2\beta a_{11} a_{12} b_{31}+3/2\beta a_{11} a_{12} b_{32} \\&\quad -\,3/2\beta a_{11} a_{31} b_{11} +3/2\beta a_{11} a_{31} b_{12}+3/2\beta a_{11} a_{31} b_{51} \\&\quad -\,3/2\beta a_{11} a_{31} b_{52} +3/2\beta a_{11} a_{32} b_{11}-3/2\beta a_{11} a_{32} b_{12} \\&\quad -3\beta a_{51} a_{52} b_{11} +3\beta a_{51} a_{52} b_{12}+3/2\beta b_{11} b_{12} b_{31} \\&\quad -\,3/2\beta b_{11} b_{12} b_{32} +b_{11} +3/2\beta a_{12} a_{52} b_{32}-3/2\beta a_{12} a_{52} b_{31} \\&\quad +\,3/2\beta a_{12} a_{51} b_{31} -3/2\beta a_{12} a_{51} b_{32}+3/2\beta a_{12} a_{32} b_{51}\\&\quad -\,3/2\beta a_{12} a_{32} b_{52} +3/2\beta a_{12} a_{31} b_{52}-3/2\beta a_{12} a_{31} b_{51} \\&\quad +\,3/2\beta a_{11} a_{52} b_{31} -3/2\beta a_{11} a_{52} b_{32}+3/2\beta b_{11} b_{32} b_{51} \\&\quad -\,3/2\beta a_{11} a_{32} b_{51} +3/2\beta a_{11} a_{32} b_{52}-3/2\beta a_{11} a_{51} b_{31} \\&\quad +\,3/2\beta a_{11} a_{51} b_{32} -3/2\beta b_{11} b_{31} b_{51}+3/2\beta b_{11} b_{31} b_{52} \\&\quad +\,3/4\beta b_{11}^{3}-b_{11} \gamma ^{2}-3/4\beta b_{12}^{3}-3/2\beta a_{12} a_{32} b_{11} \\&\quad +\,3/2\beta a_{12} a_{32} b_{12} -3\beta b_{11} b_{31} b_{32}+3/2\beta a_{12} a_{31} b_{11} \\&\quad -\,3/2\beta a_{12} a_{31} b_{12} -3\beta a_{31} a_{32} b_{11} =0, \end{aligned}$$
$$\begin{aligned}&f-3/2\beta a_{11} a_{31} a_{52} -3/2\beta a_{11} a_{32} a_{51} +3/2\beta a_{52} b_{11} b_{31} \\&\quad -\,3/2\beta a_{52} b_{11} b_{32} -3/2\beta a_{52} b_{12} b_{31}+3/2\beta a_{11} a_{32} a_{52} \\&\quad +\,3\beta a_{12} b_{31} b_{32} +3/2\beta a_{11} a_{12} a_{32}-3/2\beta a_{32} b_{32} b_{52}\\&\quad -\,3/2\beta a_{51} b_{11} b_{31} +3/2\beta a_{51} b_{11} b_{32}+3/2\beta a_{51} b_{12} b_{31} \\&\quad -\,3/2\beta a_{51} b_{12} b_{32} +3/2\beta a_{51} b_{31} b_{32}+3/2\beta a_{52} b_{12} b_{32}\\&\quad \,-3/2\beta a_{52} b_{31} b_{32} -3/2\beta a_{11} a_{12} a_{31}-3\beta a_{11} a_{31} a_{32} \\&\quad +\,3/2\beta a_{12} b_{31} b_{52}-3/2\beta a_{12} b_{31} b_{51}+3\beta a_{12} b_{51} b_{52}\\&\quad +\,3/2\beta a_{12} b_{32} b_{51} -3/2\beta a_{12} b_{32} b_{52}+3/2\beta a_{11} a_{31} a_{51} \\&\quad -\,3/2\beta a_{31} a_{32} a_{51}+3/2\beta a_{31} a_{32} a_{52}+3/2\beta a_{31} b_{11} b_{12}\\&\quad +\,3/2\beta a_{31} b_{11} b_{51} -3/2\beta a_{31} b_{11} b_{52}-3/2\beta a_{31} b_{12} b_{51} \\&\quad +\,3/2\beta a_{31} b_{12} b_{52}+3/2\beta a_{31} b_{31} b_{51}-3/2\beta a_{31} b_{31} b_{52} \\&\quad -\,3/2\beta a_{31} b_{32} b_{51} +3/2\beta a_{31} b_{32} b_{52}-3/2\beta a_{32} b_{11} b_{12} \\&\quad -\,3/2\beta a_{32} b_{11} b_{51} +3/2\beta a_{32} b_{11} b_{52}+3/2\beta a_{32} b_{12} b_{51} \\&\quad -\,3/2\beta a_{32} b_{12} b_{52} -3/2\beta a_{32} b_{31} b_{51}+3/2\beta a_{32} b_{31} b_{52} \\&\quad -\,3\beta a_{11} a_{51} a_{52} -3/2\beta a_{11} b_{11} b_{12}+3/2\beta a_{11} b_{11} b_{31} \\&\quad -\,3/2\beta a_{11} b_{11} b_{32} -3/2\beta a_{11} b_{12} b_{31}+3/2\beta a_{11} b_{12} b_{32} \\&\quad +\,a_{11} +\gamma \zeta _0 b_{11}+\gamma \left( \zeta \right) b_{11}+3\beta a_{12} a_{31} a_{32}\\&\quad -\,3\beta a_{11} b_{31} b_{32} +3/2\beta a_{12} b_{11} b_{32} -3/2\beta a_{12} b_{11} b_{31} \\&\quad +\,3/2\beta a_{12} b_{12} b_{31} +3/2\beta a_{12} b_{11} b_{12} -3/2\beta a_{12} a_{32} a_{52} \\&\quad +\,3\beta a_{12} a_{51} a_{52} -3/2\beta a_{12} a_{31} a_{51} +3/2\beta a_{12} a_{31} a_{52} \\&\quad +\,3/2\beta a_{12} a_{32} a_{51} -3/2\beta a_{11} b_{32} b_{51} +3/2\beta a_{11} b_{32} b_{52}\\&\quad -\,3\beta a_{11} b_{51} b_{52} -3/4\beta a_{12}^{2}a_{32} +3/4\beta a_{12}^{2}a_{31} \\&\quad +\,3/4\beta a_{52} b_{31}^{2}+3/2\beta a_{11} b_{51}^{2}+3/2\beta a_{11} b_{52}^{2}+3/4\beta a_{52} b_{32}^{2}\\&\quad +\,3/2\beta a_{11} b_{31} b_{51} -3/2\beta a_{11} b_{31} b_{52}-3/2\beta a_{12} b_{12} b_{32} \\&\quad +\,3/2\beta a_{32} b_{32} b_{51} -3/2\beta a_{12} b_{51}^{2}-3/2\beta a_{12} b_{52}^{2}\\&\quad +\,3/4\beta a_{31}^{2}a_{51} -3/4\beta a_{31}^{2}a_{52} -3/4\beta a_{31} b_{11}^{2}-3/4\beta a_{31} b_{12}^{2} \\&\quad +\,3/4\beta a_{32}^{2}a_{51} -3/4\beta a_{32}^{2}a_{52}+3/4\beta a_{32} b_{11}^{2}+3/4\beta a_{32}b_{12}^{2} \\&\quad -\,3/4\beta a_{51} b_{31}^{2}-3/4\beta a_{51} b_{32}^{2}-3/2\beta a_{12} b_{31}^{2}-3/2\beta a_{12} b_{32}^{2}\\&\quad -\,3/2\beta a_{12} a_{51}^{2}-3/2\beta a_{12} a_{52}^{2}-3/4\beta a_{12}b_{11}^{2}-3/4\beta a_{12} b_{12}^{2}\\&\quad +\,3/4\beta a_{11}^{2}a_{31}+9/4\beta a_{11} a_{12}^{2}+3/2\beta a_{11} a_{31}^{2}-3/4\beta a_{11}^{2}a_{32}\\&\quad -\,9/4\beta a_{11}^{2}a_{12}+3/2\beta a_{11}a_{51}^{2}+\,3/2\beta a_{11} a_{52}^{2}+3/4\beta a_{11} b_{11}^{2}\\&\quad +3/4\beta a_{11} b_{12}^{2}+\,3/2\beta a_{11} b_{31}^{2}+3/2\beta a_{11} b_{32}^{2}-3/2\beta a_{12} a_{31}^{2}\\&\quad -3/2\beta a_{12}a_{32}^{2}+\,3/2\beta a_{11} a_{32}^{2}+3/4\beta a_{11}^{3}-3/4\beta a_{12}^{3} \\&\quad -\,a_{11} \gamma ^{2}=0, \end{aligned}$$
$$\begin{aligned}&-3/2\beta a_{51} b_{51} b_{52} +3/2\beta a_{11} b_{31} b_{32} -3/2\beta a_{32} b_{12} b_{32} \\&\quad -\,3/2\beta a_{32} b_{11} b_{12} +3/2\beta a_{11} a_{12} a_{32} +3/2\beta a_{32} b_{12} b_{31} \\&\quad -\,3/2\beta a_{11} a_{12} a_{31} +3/2\beta a_{31} b_{11} b_{12} -3\beta a_{51} b_{31} b_{32}\\&\quad +3\beta a_{52} b_{31} b_{32} -3/2\beta a_{11} a_{31} a_{32} -3/2\beta a_{12} b_{31} b_{32}\\&\quad -\,3\beta a_{31}a_{32} a_{51}+\,3\beta a_{31} a_{32} a_{52} -3/2\beta a_{11} b_{11} b_{31}\\&\quad +\,3/2\beta a_{11} b_{11} b_{32}+3/2\beta a_{11} b_{12} b_{31}-3/2\beta a_{11} b_{12} b_{32}\\&\quad +3/2\beta a_{12} a_{31} a_{32}+\,3/2\beta a_{12} b_{11} b_{31}-3/2\beta a_{12} b_{11} b_{32} \\&\quad -\,3/2\beta a_{12} b_{12} b_{31} +3/2\beta a_{32} b_{11} b_{32}+3/2\beta a_{12} b_{12} b_{32} \\&\quad -\,3/2\beta a_{32} b_{11} b_{31} +3\beta a_{52} b_{11} b_{12}+3/2\beta a_{51} b_{12}^{2} \\&\quad -\,3/2\beta a_{52} b_{11}^{2}-3/2\beta a_{52} b_{12}^{2}+3/2\beta a_{12}^{2}a_{51}\\&\quad +\,3/2\beta a_{11} ^{2}a_{51} -3/2\beta a_{11}^{2}a_{52} +a_{51} -3/2\beta a_{52} b_{31}^{2} \\&\quad -\,3/2\beta a_{52} b_{32}^{2}+3/4\beta a_{51} b_{51}^{2}+9/4\beta a_{51} a_{52}^{2}+3/4\beta a_{51} b_{52}^{2}\\&\quad -\,3/4\beta a_{52} b_{51}^{2}-3/4\beta a_{52} b_{52}^{2}+5\gamma \left( \zeta \right) b_{51} +5\gamma \zeta _0 b_{51} \\&\quad -\,9/4\beta a_{51}^{2}a_{52} -3/2\beta a_{12}^{2}a_{52} -3/4\beta a_{12}^{2}a_{32} +3/4\beta a_{12}^{2}a_{31} \\&\quad +\,3/2\beta a_{51} b_{11}^{2}-3\beta a_{51} b_{11} b_{12} -3\beta a_{11} a_{12} a_{51} \\&\quad +\,3\beta a_{11} a_{12} a_{52} +3/2\beta a_{52} b_{51} b_{52} +3/2\beta a_{31} b_{11} b_{31} \\&\quad -\,3/2\beta a_{31} b_{11} b_{32} -3/2\beta a_{31} b_{12} b_{31} +3/2\beta a_{31} b_{12} b_{32} \\&\quad +\,3/2\beta a_{31}^{2}a_{51} -3/2\beta a_{31}^{2}a_{52} -3/4\beta a_{31} b_{11}^{2}-3/4\beta a_{31} b_{12}^{2}\\&\quad +\,3/2\beta a_{32} ^{2}a_{51} -3/2\beta a_{32}^{2}a_{52}+3/4\beta a_{32} b_{11}^{2}\\&\quad +\,3/4\beta a_{32} b_{12}^{2}+3/2\beta a_{51} b_{31}^{2}+3/2\beta a_{51} b_{32}^{2}+3/4\beta a_{12} b_{31}^{2}\\&\quad +\,3/4\beta a_{12} b_{32}^{2}+3/4\beta a_{11}^{2}a_{31} +3/4\beta a_{11} a_{31}^{2}\\&\quad -\,3/4\beta a_{11}^{2}a_{32} -3/4\beta a_{11} b_{31}^{2}-3/4\beta a_{11} b_{32}^{2}-3/4\beta a_{12} a_{31}^{2}\\&\quad -\,3/4\beta a_{12} a_{32}^{2}+3/4\beta a_{11} a_{32}^{2}+3/4\beta a_{51}^{3}-3/4\beta a_{52}^{3} \\&\quad -\,25a_{51} \gamma ^{2}=0, \end{aligned}$$
$$\begin{aligned}&3/2\beta a_{52}^{2}b_{31} +3/2\beta a_{11} a_{52} b_{11} -3/2\beta a_{11} a_{52} b_{12} \\&\quad +\,3/2\beta a_{12} a_{51} b_{11} -3/2\beta a_{12} a_{51} b_{12} +3/2\beta b_{11} b_{12} b_{51} \\&\quad -\,3/2\beta b_{11} b_{12} b_{52} -3\beta b_{31} b_{51} b_{52} +3\beta b_{32} b_{51} b_{52} \\&\quad -\,3/2\beta a_{12} a_{52} b_{11} +3/2\beta a_{12} a_{52} b_{12} -3/2\beta a_{31} a_{32} b_{31} \\&\quad +\,3/4\beta a_{11} ^{2}b_{51}+3/4\beta a_{12}^{2}b_{51} -3/4\beta a_{12}^{2}b_{52} -3/4\beta a_{11}^{2}b_{52} \\&\quad +\,3/2\beta a_{51}^{2}b_{31} -3/2\beta a_{51} ^{2}b_{32} -3/2\beta a_{52}^{2}b_{32} -3/4\beta b_{11}^{2}b_{51}\\&\quad +\,3/2\beta a_{31} a_{32} b_{32} -3\beta a_{51} a_{52} b_{31} +3\beta a_{51} a_{52} b_{32} \\&\quad -\,3/2\beta a_{11} a_{12} b_{51} +3/2\beta a_{11} a_{12} b_{52} -3/2\beta a_{11} a_{51} b_{11} \\&\quad +\,3/2\beta a_{11} a_{51}b_{12}-3/2\beta a_{11}^{2}b_{32} +3/2\beta a_{11}^{2}b_{31} +3/4\beta b_{11}^{2}b_{12} \\&\quad -\,3/4\beta b_{11} b_{12}^{2}-3/4\beta a_{11} ^{2}b_{12} +3/4\beta a_{11}^{2}b_{11} +3/4\beta a_{12}^{2}b_{11}\\&\quad -\,3/4\beta a_{12}^{2}b_{12} +3/2\beta a_{31} a_{51} b_{11} -3/2\beta a_{31} a_{51} b_{12} \\&\quad +\,3/2\beta b_{12}^{2}b_{31} -3/2\beta b_{11} ^{2}b_{32} +3/2\beta b_{11}^{2}b_{31} +3/2\beta a_{12}^{2}b_{31}\\&\quad -\,3/2\beta a_{12}^{2}b_{32} -3\gamma \left( \zeta \right) a_{31} -3\gamma \zeta _0 a_{31}-3/2\beta b_{32} b_{52}^{2} \\&\quad +\,3/4\beta a_{31}^{2}b_{31} +3/4\beta a_{32}^{2}b_{31} -3/4\beta a_{32}^{2}b_{32}+3/2\beta b_{31} b_{51}^{2} \\&\quad +\,3/2\beta b_{31} b_{52}^{2}+3/4\beta b_{12}^{2}b_{52}+3/4\beta b_{11}^{2}b_{52}-3/4\beta b_{12}^{2}b_{51} \\&\quad -\,3/4\beta a_{31}^{2}b_{32} +9/4\beta b_{31} b_{32}^{2}-9/4\beta b_{31}^{2}b_{32}-3/2\beta b_{32} b_{51}^{2}\\&\quad -\,3/2\beta b_{12}^{2}b_{32}-3/2\beta a_{31} a_{52} b_{11}+3/2\beta a_{31}a_{52} b_{12} \\&\quad -\,3/2\beta a_{32} a_{51} b_{11}+\,3/2\beta a_{32} a_{51} b_{12} +3/2\beta a_{32} a_{52} b_{11}\\&\quad -\,3/2\beta a_{32} a_{52} b_{12}-3/2\beta b_{12} b_{31} b_{51} +3/2\beta b_{12} b_{31} b_{52}\\&\quad +\,3/2\beta b_{12} b_{32} b_{51}-3/2\beta b_{12} b_{32} b_{52} +3/2\beta b_{11} b_{32} b_{52} \\&\quad -\,3/2\beta a_{11} a_{12} b_{11}+3/2\beta a_{11} a_{12} b_{12} -3\beta a_{11} a_{12} b_{31} \\&\quad +\,3\beta a_{11} a_{12} b_{32}+3/2\beta a_{11} a_{31} b_{51}-3/2\beta a_{11} a_{31} b_{52} \\&\quad -\,3\beta b_{11} b_{12} b_{31} +3\beta b_{11} b_{12} b_{32}+3/2\beta b_{11} b_{31} b_{51} \\&\quad -\,3/2\beta b_{11}b_{31} b_{52} -3/2\beta b_{11} b_{32} b_{51}-3/2\beta a_{11} a_{32} b_{51} \\&\quad +3/2\beta a_{11} a_{32} b_{52} -3/2\beta a_{11} a_{51} b_{31} +3/2\beta a_{11} a_{51} b_{32} \\&\quad +\,3/2\beta a_{11} a_{52} b_{31} -3/2\beta a_{11} a_{52} b_{32} -3/2\beta a_{12} a_{31} b_{51} \\&\quad +\,3/2\beta a_{12} a_{31} b_{52} +3/2\beta a_{12} a_{32} b_{51} -3/2\beta a_{12} a_{32} b_{52} \\&\quad +\,3/2\beta a_{12} a_{51} b_{31} -3/2\beta a_{12} a_{51} b_{32} -3/2\beta a_{12} a_{52} b_{31} \\&\quad +\,3/2\beta a_{12} a_{52} b_{32} +b_{31} -3/4\beta b_{32} ^{3}+3/4\beta b_{31}^{3}\\&\quad -\,9b_{31} \gamma ^{2}+1/4\beta b_{12} ^{3}-1/4\beta b_{11}^{3}=0, \end{aligned}$$
$$\begin{aligned}&-3/2\beta a_{11} a_{31} a_{52} -3/2\beta a_{11} a_{32} a_{51} -3/2\beta a_{31} b_{12} b_{52} \\&\quad -\,3/2\beta a_{52} b_{11} b_{31} +3/2\beta a_{52} b_{11} b_{32} +3/2\beta a_{52} b_{12} b_{31} \\&\quad +\,3/2\beta a_{11} a_{32} a_{52} +3\beta a_{32} b_{11} b_{12} +3/2\beta a_{32} b_{11} b_{51} \\&\quad +\,3\beta a_{11} a_{12} a_{32} +3/2\beta a_{51} b_{11} b_{31} -3/2\beta a_{51} b_{11} b_{32}\\&\quad -\,3/2\beta a_{51} b_{12} b_{31}+3/2\beta a_{51} b_{12} b_{32}+3/2\beta a_{31} b_{11} b_{52} \\&\quad -\,3/2\beta a_{52} b_{12} b_{32}-3\beta a_{11} a_{12} a_{31} +3/2\beta a_{12} b_{31} b_{52} \\&\quad -\,3/2\beta a_{12} b_{31} b_{51}+3/2\beta a_{12} b_{32} b_{51}-3/2\beta a_{12} b_{32} b_{52} \\&\quad +\,3/2\beta a_{11} a_{31} a_{51}+3/2\beta a_{31} b_{12} b_{51}-\,3/2\beta a_{31} b_{11} b_{51} \\&\quad -\,3\beta a_{31} b_{11} b_{12}-3/2\beta a_{32} b_{11} b_{52}-3/2\beta a_{32} b_{12} b_{51} \\&\quad +\,3/2\beta a_{32} b_{12} b_{52} +3/2\beta a_{11} b_{11} b_{12} +3/2\beta a_{11} b_{31} b_{51} \\&\quad -\,3/2\beta a_{11} b_{31} b_{52} -3/2\beta a_{11} b_{32} b_{51} +3/2\beta a_{11} b_{32} b_{52}\\&\quad -\,3/2\beta a_{12} a_{31} a_{51}+3/2\beta a_{12} a_{31} a_{52} +3/2\beta a_{12} a_{32} a_{51} \\&\quad -\,3/2\beta a_{12} a_{32} a_{52}-3/2\beta a_{12} b_{11} b_{12} -3/2\beta a_{12} b_{11} b_{51} \\&\quad +\,3/2\beta a_{12} b_{11} b_{52}+3/2\beta a_{12} b_{12} b_{51}-3/2\beta a_{12} b_{12} b_{52} \\&\quad -\,3\beta a_{31} a_{51} a_{52}-3/2\beta a_{52} b_{11} b_{12}-3/2\beta a_{31} b_{31} b_{32}\\&\quad -\,3\beta a_{31} b_{51} b_{52} +3\beta a_{32} a_{51} a_{52}+\,3/2\beta a_{32} b_{31} b_{32}\\&\quad +\,3\beta a_{32} b_{51} b_{52} -3/4\beta a_{51} b_{12}^{2}+3/4\beta a_{52} b_{11}^{2}+3\gamma \zeta _0 b_{31}\\&\quad +\,3/4\beta a_{52} b_{12} ^{2}+3/4\beta a_{12}^{2}a_{51} +3\gamma \left( \zeta \right) b_{31}+3/4\beta a_{11}^{2}a_{51}\\&\quad -\,3/4\beta a_{11}^{2}a_{52} -3/2\beta a_{32} a_{51}^{2}-9/4\beta a_{31}^{2}a_{32} \\&\quad -\,3/4\beta a_{12}^{2}a_{52} +3/2\beta a_{11} b_{12} b_{52} +3/2\beta a_{51} b_{11} b_{12} \\&\quad -\,3/2\beta a_{11} a_{12}a_{51}+3/2\beta a_{11} a_{12} a_{52} +3/2\beta a_{11} b_{11} b_{51}\\&\quad -\,3/2\beta a_{11} b_{11} b_{52}-3/2\beta a_{11} b_{12} b_{51} -3/2\beta a_{32} b_{51}^{2}\\&\quad -\,3/2\beta a_{32} b_{52}^{2}-3/2\beta a_{32} a_{52}^{2}-3/4\beta a_{32} b_{31}^{2}-3/4\beta a_{32} b_{32}^{2}\\&\quad +\,3/4\beta a_{31} b_{31}^{2}+3/4\beta a_{31} b_{32}^{2}+3/2\beta a_{31} b_{52}^{2}+3/2\beta a_{31} a_{51}^{2}\\&\quad +\,9/4\beta a_{31} a_{32}^{2}+3/2\beta a_{31} b_{51}^{2}+3/2\beta a_{31} a_{52}^{2}-3/4\beta a_{51} b_{11}^{2} \\&\quad +\,a_{31}+3/2\beta a_{12}^{2}a_{31} -3/2\beta a_{12}^{2}a_{32} +3/2\beta a_{31} b_{11}^{2}\\&\quad +\,3/2\beta a_{31} b_{12}^{2}-3/2\beta a_{32} b_{11}^{2}-3/2\beta a_{32} b_{12}^{2}+3/4\beta a_{12} b_{11}^{2}\\&\quad +\,3/4\beta a_{12} b_{12}^{2}+3/2\beta a_{11} ^{2}a_{31} +3/4\beta a_{11} a_{12}^{2}-3/2\beta a_{11}^{2}a_{32}\\&\quad -\,3/4\beta a_{11}^{2}a_{12} -3/4\beta a_{11} b_{11}^{2}-3/4\beta a_{11} b_{12}^{2}+3/4\beta a_{31}^{3}\\&\quad -\,3/4\beta a_{32}^{3}-9a_{31} \gamma ^{2}+1/4\beta a_{11}^{3}-1/4\beta a_{12}^{3}=0, \end{aligned}$$
$$\begin{aligned}&-3\beta a_{11} a_{12} b_{51} +3\beta a_{11} a_{12} b_{52} -3\beta b_{11} b_{12} b_{51} \\&\quad +\,3\beta b_{11} b_{12} b_{52} +3/2\beta a_{11} a_{31} b_{31} +3/2\beta a_{11}^{2}b_{51} \\&\quad +\,9/4\beta b_{51} b_{52}^{2}-5\gamma \zeta _0 a_{51} +3/4\beta a_{51}^{2}b_{51} -3/4\beta a_{51}^{2}b_{52} \\&\quad +\,3/2\beta a_{51} a_{52} b_{52} -3/2\beta a_{12} a_{32} b_{32} \\&\quad -3/2\beta a_{51} a_{52} b_{51}-3/2\beta a_{12} a_{31}b_{31}+3/2\beta a_{12} a_{31} b_{32} \\&\quad +\,3/2\beta a_{12} a_{32} b_{31}-3/2\beta a_{11} a_{31} b_{32}-3/2\beta a_{11} a_{32} b_{31}\\&\quad +\,3/2\beta a_{11} a_{32} b_{32} -5\gamma \left( \zeta \right) a_{51}\\&\quad +\,3/2\beta a_{12}^{2}b_{51}+3/4\beta a_{52}^{2}b_{51} -3/4\beta a_{52}^{2}b_{52} -9/4\beta b_{51}^{2}b_{52} \\&\quad +\,3/2\beta b_{11}^{2}b_{51} -3/2\beta b_{11} ^{2}b_{52} +3/2\beta b_{12}^{2}b_{51} \\&\quad -\,3/2\beta a_{12}^{2}b_{52}-3/2\beta a_{11}^{2}b_{52} +3/2\beta b_{12} b_{31} b_{32} -3/4\beta a_{11}^{2}b_{32} \\&\quad +\,3/4\beta a_{11}^{2}b_{31} -3/2\beta a_{31} a_{32} b_{12} -3/4\beta b_{12}^{2}b_{31}\\&\quad +\,3/4\beta b_{11} ^{2}b_{32} -\,3\beta a_{31} a_{32} b_{51} -3/4\beta b_{11}^{2}b_{31} +3/4\beta a_{12}^{2}b_{31} \\&\quad -\,3/4\beta a_{12}^{2}b_{32} +3\beta a_{31} a_{32} b_{52} -3\beta b_{31} b_{32} b_{51} -3/2\beta b_{12}^{2}b_{52} \\&\quad -\,3/4\beta b_{12} b_{32}^{2}-3/4\beta b_{12} b_{31}^{2}+3/2\beta b_{32}^{2}b_{51}\\&\quad -\,3/2\beta b_{32}^{2}b_{52} +3/2\beta b_{31} ^{2}b_{51} -3/2\beta b_{31}^{2}b_{52} +3/4\beta b_{11} b_{32}^{2}\\&\quad +\,3/4\beta b_{11} b_{31}^{2}-3/2\beta a_{32}^{2}b_{52} +3/2\beta a_{31}^{2}b_{51} \\&\quad -\,3/4\beta a_{32}^{2}b_{11} +3/2\beta a_{32} ^{2}b_{51} +3/4\beta a_{32}^{2}b_{12} -3/2\beta a_{31}^{2}b_{52}\\&\quad -\,3/4\beta a_{31}^{2}b_{11} +3/4\beta a_{31}^{2}b_{12} +3/4\beta b_{12}^{2}b_{32} \\&\quad +\,3\beta b_{31} b_{32} b_{52} -3/2\beta a_{11} a_{12} b_{31} +3/2\beta a_{11} a_{12} b_{32} \\&\quad +\,3/2\beta a_{11} a_{31} b_{11} \\&\quad -\,3/2\beta a_{11} a_{31} b_{12} -3/2\beta a_{11} a_{32} b_{11} +3/2\beta a_{11} a_{32} b_{12} \\&\quad +\,3/2\beta b_{11} b_{12} b_{31} -3/2\beta b_{11} b_{12} b_{32} -3/2\beta b_{11} b_{31} b_{32} \\&\quad -\,3/2\beta a_{12} a_{31} b_{11} +3/2\beta a_{12} a_{31} b_{12} +3/2\beta a_{12} a_{32} b_{11} \\&\quad -\,3/2\beta a_{12} a_{32} b_{12} +3/2\beta a_{31} a_{32} b_{11} +b_{51} -3/4\beta b_{52}^{3} \\&\quad +\,3/4\beta b_{51}^{3}-25b_{51} \gamma ^{2}=0, \end{aligned}$$
$$\begin{aligned}&-b_{12} \gamma ^{2}-3/2\beta \lambda a_{12} a_{51} b_{31} +3/2\beta \lambda a_{12} a_{51} b_{32}\\&\quad +3/2\beta \lambda a_{12} a_{52} b_{31} -3/2\beta \lambda a_{12} a_{52} b_{32} +3\beta \lambda a_{31} a_{32} b_{11} \\&\quad -\,3\beta \lambda a_{31} a_{32} b_{12} -3/2\beta \lambda a_{31} a_{32} b_{51} +3/2\beta \lambda a_{31} a_{32} b_{52}\\&\quad +\,3/2\beta \lambda a_{31} a_{51} b_{11} -3/2\beta \lambda a_{31} a_{51} b_{12} -3/2\beta \lambda a_{31} a_{51} b_{31} \\&\quad +\,3/2\beta \lambda a_{31} a_{51} b_{32} -3/2\beta \lambda a_{31} a_{52} b_{11} +3/2\beta \lambda a_{31} a_{52} b_{12}\\&\quad +3/2\beta \lambda a_{31} a_{52} b_{31} -3/2\beta \lambda a_{31} a_{52} b_{32} -3/2\beta \lambda a_{32} a_{51} b_{11} \\&\quad +\,3/2\beta \lambda a_{32} a_{51} b_{12} -3/2\beta \lambda a_{32} a_{51} b_{32} +3/2\beta \lambda a_{32} a_{52} b_{11}\\&\quad -3/2\beta \lambda a_{32} a_{52} b_{12} -3/2\beta \lambda a_{32} a_{52} b_{31} +3/2\beta \lambda a_{32} a_{52} b_{32} \\&\quad +\,3\beta \lambda a_{51} a_{52} b_{11} -3\beta \lambda a_{51} a_{52} b_{12} -3/2\beta \lambda b_{11} b_{12} b_{31}\\&\quad +3/2\beta \lambda b_{11} b_{12} b_{32} +3\beta \lambda b_{11} b_{31} b_{32} +3/2\beta \lambda b_{11} b_{31} b_{51} \\&\quad -\,3/2\beta \lambda b_{11} b_{31} b_{52} -3/2\beta \lambda b_{11} b_{32} b_{51} +3/2\beta \lambda b_{11} b_{32} b_{52} \\&\quad +\,3\beta \lambda b_{11} b_{51} b_{52} -3\beta \lambda b_{12} b_{31} b_{32} -3/2\beta \lambda b_{12} b_{31} b_{51} \\&\quad +\,3/2\beta \lambda b_{12} b_{31} b_{52} +3/2\beta \lambda b_{12} b_{32} b_{51} -3/2\beta \lambda b_{12} b_{32} b_{52}\\&\quad -\,3\beta \lambda b_{12} b_{51} b_{52} +3/2\beta \lambda b_{31} b_{32} b_{51} -3/2\beta \lambda b_{31} b_{32} b_{52} \\&\quad -\,3/2\beta \lambda a_{11} a_{12} b_{32} +3/2\beta \lambda a_{32} a_{51} b_{31} +3/2\beta \lambda a_{11} a_{12} b_{11}\\&\quad -\,3/2\beta \lambda a_{11} a_{12} b_{12} +3/2\beta \lambda a_{11} a_{12} b_{31} +3/2\beta \lambda a_{11} a_{31} b_{11} \\&\quad -\,3/2\beta \lambda a_{11} a_{31} b_{12} -3/2\beta \lambda a_{11} a_{31} b_{51} +3/2\beta \lambda a_{11} a_{31} b_{52}\\&\quad -\,3/2\beta \lambda a_{11} a_{32} b_{11} +3/2\beta \lambda a_{11} a_{32} b_{12} +3/2\beta \lambda a_{11} a_{32} b_{51} \\&\quad -\,3/2\beta \lambda a_{11} a_{32} b_{52} +3/2\beta \lambda a_{11} a_{51} b_{31} -3/2\beta \lambda a_{11} a_{51} b_{32}\\&\quad -\,3/2\beta \lambda a_{11} a_{52} b_{31} +3/2\beta \lambda a_{11} a_{52} b_{32} -3/2\beta \lambda a_{12} a_{31} b_{11} \\&\quad +\,3/2\beta \lambda a_{12} a_{31} b_{12} +3/2\beta \lambda a_{12} a_{31} b_{51} -3/2\beta \lambda a_{12} a_{31} b_{52} \\&\quad +\,3/2\beta \lambda a_{12} a_{32} b_{11}-3/2\beta \lambda a_{12} a_{32} b_{12} -3/2\beta \lambda a_{12} a_{32} b_{51} \\&\quad +\,3/2\beta \lambda a_{12} a_{32} b_{52} -3/2\beta \lambda a_{52} ^{2}b_{11} +3/2\beta \lambda a_{52}^{2}b_{12} \\&\quad +\,9/4\beta \lambda b_{11}^{2}b_{12} +3/4\beta \lambda a_{12}^{2}b_{32} -3/2\beta \lambda a_{31}^{2}b_{11} \\&\quad +\,3/2\beta \lambda a_{31}^{2}b_{12} +3/4\beta \lambda a_{31} ^{2}b_{51} -3/4\beta \lambda a_{31}^{2}b_{52}\\&\quad -\,3/2\beta \lambda a_{51}^{2}b_{11} +3/2\beta \lambda a_{51}^{2}b_{12} +3/4\beta \lambda b_{12}^{2}b_{31} \\&\quad +\,3/2\beta \lambda b_{12} b_{31}^{2} -3/2\beta \lambda a_{32}^{2}b_{11} +3/2\beta \lambda a_{32} ^{2}b_{12}+3/4\beta \lambda a_{32}^{2}b_{51} \\&\quad -3/4\beta \lambda a_{32}^{2}b_{52} -3/2\beta \lambda b_{11} b_{31}^{2}+3/4\beta \lambda b_{32}^{2}b_{52} -3/2\beta \lambda b_{11} b_{32}^{2}\\&\quad +3/2\beta \lambda b_{12} b_{51}^{2}-9/4\beta \lambda b_{11} b_{12} ^{2}+3/2\beta \lambda b_{12} b_{32}^{2}\\&\quad -\,3/2\beta \lambda b_{11} b_{51}^{2}-3/2\beta \lambda b_{11} b_{52}^{2}-3/4\beta \lambda b_{12}^{2}b_{32}\\&\quad +\,3/4\beta \lambda b_{11}^{2}b_{31} \\&\quad -\,3/4\beta \lambda b_{11}^{2}b_{32} +3/2\beta \lambda b_{12} b_{52} ^{2}-3/4\beta \lambda b_{31}^{2}b_{51} \\&\quad +\,3/4\beta \lambda b_{31} ^{2}b_{52} -3/4\beta \lambda b_{32}^{2}b_{51} -3/4\beta \lambda a_{11}^{2}b_{11}\\&\quad +3/4\beta \lambda a_{11}^{2}b_{12} \\&\quad -\,3/4\beta \lambda a_{11}^{2}b_{31} +3/4\beta \lambda a_{11} ^{2}b_{32} -3/4\beta \lambda a_{12}^{2}b_{11}\\&\quad +3/4\beta \lambda a_{12}^{2}b_{12} -3/4\beta \lambda a_{12}^{2}b_{31} -3/4\beta \lambda b_{11}^{3}\\&\quad +\,3/4\beta \lambda b_{12}^{3} \\&\quad +\,\gamma \lambda \zeta a_{11} -\gamma \lambda \zeta a_{12} =0, \end{aligned}$$
$$\begin{aligned}&3/2\beta \lambda a_{12} a_{32}^{2}+3/2\beta \lambda a_{12} a_{51} ^{2}+3/2\beta \lambda a_{12} a_{52}^{2}\\&\quad +\,3/4\beta \lambda a_{12} b_{11}^{2}+3/4\beta \lambda a_{12} b_{12}^{2}-3/2\beta \lambda a_{11} b_{31}^{2} \\&\quad -\,3/4\beta \lambda a_{11} b_{12}^{2}-3/2\beta \lambda a_{11} a_{52} ^{2}-3/4\beta \lambda a_{11} b_{11}^{2}\\&\quad -\,3/2\beta \lambda a_{11} a_{32}^{2}-3/2\beta \lambda a_{11} a_{51}^{2}-9/4\beta \lambda a_{11} a_{12}^{2} \\&\quad -\,3/2\beta \lambda a_{11} a_{31}^{2}-3/4\beta \lambda a_{11} ^{2}a_{31} +3/4\beta \lambda a_{11}^{2}a_{32} \\&\quad -\,3/4\beta \lambda a_{31}^{2}a_{51} +3/4\beta \lambda a_{31}^{2}a_{52} +3/4\beta \lambda a_{31} b_{11}^{2} \\&\quad +\,9/4\beta \lambda a_{11}^{2}a_{12} -3/2\beta \lambda a_{11} b_{52} ^{2}-3/4\beta \lambda a_{12}^{2}a_{31} \\&\quad +\,3/2\beta \lambda a_{12} b_{31}^{2}+3/2\beta \lambda a_{12} b_{32}^{2}+3/2\beta \lambda a_{12} b_{51}^{2} \\&\quad +\,3/2\beta \lambda a_{12} b_{52}^{2}-3/2\beta \lambda a_{11} b_{32} ^{2}-3/2\beta \lambda a_{11} b_{51}^{2}-a_{12} \gamma ^{2}\\&\quad -\,3/4\beta \lambda a_{52} b_{32}^{2}+3/4\beta \lambda a_{51} b_{32}^{2} \\&\quad -\,3/4\beta \lambda a_{52} b_{31}^{2}+3/4\beta \lambda a_{31} b_{12} ^{2}-3/4\beta \lambda a_{32}^{2}a_{51} \\&\quad +\,3/4\beta \lambda a_{32} ^{2}a_{52} -3/4\beta \lambda a_{32} b_{11}^{2}-3/4\beta \lambda a_{32} b_{12}^{2} \\&\quad +\,3/4\beta \lambda a_{51} b_{31}^{2}+3/4\beta \lambda a_{12} ^{2}a_{32} +3/2\beta \lambda a_{12} a_{31}^{2}\\&\quad +\,3/2\beta \lambda a_{32} b_{31} b_{51} -3/2\beta \lambda a_{32} b_{31} b_{52} -3/2\beta \lambda a_{32} b_{32} b_{51} \\&\quad +\,3/2\beta \lambda a_{32} b_{32} b_{52} +3/2\beta \lambda a_{51} b_{11} b_{31} -3/2\beta \lambda a_{51} b_{11} b_{32}\\&\quad -\,3/2\beta \lambda a_{51} b_{12} b_{31} +3/2\beta \lambda a_{51} b_{12} b_{32} -3/2\beta \lambda a_{51} b_{31} b_{32} \\&\quad -\,3/2\beta \lambda a_{52} b_{11} b_{31} +3/2\beta \lambda a_{52} b_{11} b_{32} -\gamma \lambda \zeta b_{11} \\&\quad +\,\gamma \lambda \zeta b_{12} +3/4\beta \lambda a_{12}^{3}-3/4\beta \lambda a_{11} ^{3}+3/2\beta \lambda a_{52} b_{12} b_{31} \\&\quad -\,3/2\beta \lambda a_{52} b_{12} b_{32} +3/2\beta \lambda a_{52} b_{31} b_{32} +3/2\beta \lambda a_{11} a_{32} a_{51}\\&\quad -\,3/2\beta \lambda a_{11} a_{32} a_{52} +3\beta \lambda a_{11} a_{51} a_{52} +3/2\beta \lambda a_{11} b_{11} b_{12} \\&\quad -\,3/2\beta \lambda a_{11} b_{11} b_{31} +3/2\beta \lambda a_{11} b_{11} b_{32} +3/2\beta \lambda a_{11} b_{12} b_{31}\\&\quad -\,3/2\beta \lambda a_{11} b_{12} b_{32} +3\beta \lambda a_{11} b_{31} b_{32} -3/2\beta \lambda a_{11} b_{31} b_{51} \\&\quad +\,3/2\beta \lambda a_{11} b_{31} b_{52} +3/2\beta \lambda a_{11} b_{32} b_{51} -3/2\beta \lambda a_{11} b_{32} b_{52} \\&\quad +\,3\beta \lambda a_{11} b_{51} b_{52} -3\beta \lambda a_{12} a_{31} a_{32} +3/2\beta \lambda a_{12} a_{31} a_{51} \\&\quad -\,3/2\beta \lambda a_{12} a_{31} a_{52} -3/2\beta \lambda a_{12} a_{32} a_{51} +3/2\beta \lambda a_{12} a_{32} a_{52}\\&\quad -3\beta \lambda a_{12} a_{51} a_{52} -3/2\beta \lambda a_{12} b_{11} b_{12} +3/2\beta \lambda a_{12} b_{11} b_{31} \\&\quad -\,3/2\beta \lambda a_{12} b_{11} b_{32} -3/2\beta \lambda a_{12} b_{12} b_{31} +3/2\beta \lambda a_{12} b_{12} b_{32}\\&\quad +3/2\beta \lambda a_{12} b_{31} b_{51} -3/2\beta \lambda a_{12} b_{31} b_{52} -3/2\beta \lambda a_{12} b_{32} b_{51} \\&\quad +\,3/2\beta \lambda a_{12} b_{32} b_{52} -3\beta \lambda a_{12} b_{51} b_{52} +3/2\beta \lambda a_{31} a_{32} a_{51}\\&\quad -\,3/2\beta \lambda a_{31} a_{32} a_{52} -3/2\beta \lambda a_{31} b_{11} b_{12} -3/2\beta \lambda a_{31} b_{11} b_{51} \\&\quad +\,3/2\beta \lambda a_{31} b_{11} b_{52} +3/2\beta \lambda a_{31} b_{12} b_{51} -3/2\beta \lambda a_{31} b_{12} b_{52} \\&\quad -\,3/2\beta \lambda a_{31} b_{31} b_{51} +3/2\beta \lambda a_{31} b_{31} b_{52} +3/2\beta \lambda a_{31} b_{32} b_{51} \\&\quad -\,3/2\beta \lambda a_{31} b_{32} b_{52} +3/2\beta \lambda a_{32} b_{11} b_{12} +3/2\beta \lambda a_{32} b_{11} b_{51} \\&\quad -\,3/2\beta \lambda a_{32} b_{11} b_{52} -3/2\beta \lambda a_{32} b_{12} b_{51} +3/2\beta \lambda a_{32} b_{12} b_{52} \\&\quad -\,3\beta \lambda a_{12} b_{31} b_{32} +3/2\beta \lambda a_{11} a_{12} a_{31} -3/2\beta \lambda a_{11} a_{12} a_{32} \\&\quad +\,3\beta \lambda a_{11} a_{31} a_{32} -3/2\beta \lambda a_{11} a_{31} a_{51} +3/2\beta \lambda a_{11} a_{31} a_{52} =0, \end{aligned}$$
$$\begin{aligned}&-9b_{32} \gamma ^{2}-3/2\beta \lambda a_{12} a_{51} b_{31} +3/2\beta \lambda a_{12} a_{51} b_{32} \\&\quad +\,3/2\beta \lambda a_{12} a_{52} b_{31} -3/2\beta \lambda a_{12} a_{52} b_{32} -3/2\beta \lambda a_{31} a_{51} b_{11} \\&\quad +\,3/2\beta \lambda a_{31} a_{51} b_{12} +3/2\beta \lambda a_{31} a_{52} b_{11} -3/2\beta \lambda a_{31} a_{52} b_{12} \\&\quad +\,3/2\beta \lambda a_{32} a_{51} b_{11} -3/2\beta \lambda a_{32} a_{51} b_{12} -3/2\beta \lambda a_{32} a_{52} b_{11} \\&\quad +\,3/2\beta \lambda a_{32} a_{52} b_{12} +3\beta \lambda b_{11} b_{12} b_{31} -3\beta \lambda b_{11} b_{12} b_{32} \\&\quad -\,3/2\beta \lambda b_{11} b_{31} b_{51} +3/2\beta \lambda b_{11} b_{31} b_{52} +3/2\beta \lambda b_{11} b_{32} b_{51} \\&\quad -\,3/2\beta \lambda b_{11} b_{32} b_{52} +3/2\beta \lambda b_{12} b_{31} b_{51} -3/2\beta \lambda b_{12} b_{31} b_{52} \\&\quad -\,3/2\beta \lambda b_{12} b_{32} b_{51} +3/2\beta \lambda b_{12} b_{32} b_{52} -3\beta \lambda a_{11} a_{12} b_{32} \\&\quad +\,3/2\beta \lambda a_{11} a_{12} b_{11} -3/2\beta \lambda a_{11} a_{12} b_{12} +3\beta \lambda a_{11} a_{12} b_{31} \\&\quad -\,3/2\beta \lambda a_{11} a_{31} b_{51} +3/2\beta \lambda a_{11} a_{31} b_{52} +3/2\beta \lambda a_{11} a_{32} b_{51} \\&\quad -\,3/2\beta \lambda a_{11} a_{32} b_{52} +3/2\beta \lambda a_{11} a_{51} b_{31} -3/2\beta \lambda a_{11} a_{51} b_{32} \\&\quad -\,3/2\beta \lambda a_{11} a_{52} b_{31} +3/2\beta \lambda a_{11} a_{52} b_{32} +3/2\beta \lambda a_{12} a_{31} b_{51} \\&\quad -\,3/2\beta \lambda a_{12} a_{31} b_{52} -3/2\beta \lambda a_{12} a_{32} b_{51} +3/2\beta \lambda a_{12} a_{32} b_{52} \\&\quad +\,3/2\beta \lambda a_{12}^{2}b_{32} -3/4\beta \lambda b_{11}^{2}b_{12} -3/2\beta \lambda b_{11}^{2}b_{31} \\&\quad +\,3/2\beta \lambda b_{11}^{2}b_{32} +3/4\beta \lambda b_{11} b_{12} ^{2}-3/2\beta \lambda b_{12}^{2}b_{31} \\&\quad +\,3/2\beta \lambda b_{12} ^{2}b_{32} -9/4\beta \lambda b_{31} b_{32}^{2}-3/4\beta \lambda b_{11}^{2}b_{52} \\&\quad -\,3/2\beta \lambda b_{31} b_{51}^{2}-3/4\beta \lambda b_{12} ^{2}b_{52} +9/4\beta \lambda b_{31}^{2}b_{32} \\&\quad -\,3/2\beta \lambda a_{12}^{2}b_{31} +3/4\beta \lambda b_{11}^{2}b_{51} -3/2\beta \lambda a_{52}^{2}b_{31} \\&\quad +\,3/2\beta \lambda a_{52}^{2}b_{32} +3/4\beta \lambda b_{12} ^{2}b_{51} +3/4\beta \lambda a_{11}^{2}b_{52} \\&\quad -\,3/4\beta \lambda a_{12}^{2}b_{51} +3/4\beta \lambda a_{12}^{2}b_{52} -3/4\beta \lambda a_{11}^{2}b_{51} \\&\quad -\,3/4\beta \lambda a_{32}^{2}b_{31} -3/4\beta \lambda a_{31} ^{2}b_{31} +3/4\beta \lambda a_{31}^{2}b_{32} \\&\quad +\,3/2\beta \lambda a_{11}^{2}b_{32} -3/4\beta \lambda a_{12}^{2}b_{11} +3/4\beta \lambda a_{12}^{2}b_{12} \\&\quad -\,3/4\beta \lambda a_{11}^{2}b_{11} +3/4\beta \lambda a_{11} ^{2}b_{12} -3/2\beta \lambda a_{11}^{2}b_{31} \\&\quad +\,3/2\beta \lambda a_{51}^{2}b_{32} +3/4\beta \lambda a_{32}^{2}b_{32} -3/2\beta \lambda a_{51}^{2}b_{31} \\&\quad +\,1/4\beta \lambda b_{11}^{3}-1/4\beta \lambda b_{12}^{3}-3/4\beta \lambda b_{31}^{3}-3/2\beta \lambda b_{31} b_{52}^{2}\\&\quad +\,3/2\beta \lambda b_{32} b_{51}^{2}+3/2\beta \lambda b_{32} b_{52} ^{2}+3/2\beta \lambda b_{11} b_{12} b_{52} \\&\quad -\,3/2\beta \lambda b_{11} b_{12} b_{51} -3\beta \lambda a_{51} a_{52} b_{32} +3\beta \lambda a_{51} a_{52} b_{31} \\&\quad -\,3/2\beta \lambda a_{31} a_{32} b_{32} +3/2\beta \lambda a_{31} a_{32} b_{31} -3/2\beta \lambda a_{12} a_{52} b_{12} \\&\quad +\,3/2\beta \lambda a_{12} a_{52} b_{11} +3/2\beta \lambda a_{12} a_{51} b_{12} -3/2\beta \lambda a_{12} a_{51} b_{11} \\&\quad +\,3/2\beta \lambda a_{11} a_{52} b_{12} -3/2\beta \lambda a_{11} a_{52} b_{11} -3/2\beta \lambda a_{11} a_{51} b_{12} \\&\quad +\,3/2\beta \lambda a_{11} a_{51} b_{11} +3/2\beta \lambda a_{11} a_{12} b_{51} -3/2\beta \lambda a_{11} a_{12} b_{52} \\&\quad +\,3\beta \lambda b_{31} b_{51} b_{52} -3\beta \lambda b_{32} b_{51} b_{52} +3/4\beta \lambda b_{32}^{3}-3\gamma \lambda \zeta a_{32} \\&\quad +\,3\gamma \lambda \zeta a_{31} =0, \end{aligned}$$
$$\begin{aligned}&-3/4\beta \lambda a_{12} b_{11}^{2}-3/4\beta \lambda a_{12} b_{12} ^{2}+3/4\beta \lambda a_{11} b_{12}^{2}\\&\quad +\,3/4\beta \lambda a_{11} b_{11}^{2}-3/4\beta \lambda a_{11} a_{12}^{2}-3/2\beta \lambda a_{11}^{2}a_{31} \\&\quad +\,3/2\beta \lambda a_{11}^{2}a_{32} -3/2\beta \lambda a_{31} b_{11} ^{2}+3/4\beta \lambda a_{11}^{2}a_{12} \\&\quad -\,9a_{32} \gamma ^{2}-3/2\beta \lambda a_{31} b_{12}^{2}+3/2\beta \lambda a_{32} b_{11}^{2}+3/2\beta \lambda a_{32} b_{12}^{2} \\&\quad +\,3/2\beta \lambda a_{12}^{2}a_{32} -3/2\beta \lambda a_{12} ^{2}a_{31} +3/4\beta \lambda a_{51} b_{12}^{2}\\&\quad -\,3/4\beta \lambda a_{52} b_{11}^{2}-3/4\beta \lambda a_{52} b_{12}^{2}-3/4\beta \lambda a_{11}^{2}a_{51} \\&\quad +\,3/4\beta \lambda a_{11}^{2}a_{52} -3/4\beta \lambda a_{12} ^{2}a_{51} +3/4\beta \lambda a_{12}^{2}a_{52} \\&\quad -\,3/4\beta \lambda a_{31} b_{31}^{2}-3/2\beta \lambda a_{31} a_{52}^{2}-3/4\beta \lambda a_{31} b_{32}^{2} \\&\quad +\,9/4\beta \lambda a_{31}^{2}a_{32} -9/4\beta \lambda a_{31} a_{32} ^{2}-3/2\beta \lambda a_{31} a_{51}^{2}\\&\quad -\,3/2\beta \lambda a_{32} b_{31} b_{32} +3\beta \lambda a_{31} b_{51} b_{52} -3\beta \lambda a_{32} a_{51} a_{52} \\&\quad +\,3/2\beta \lambda a_{11} b_{11} b_{52} -3/4\beta \lambda a_{31} ^{3}+3/4\beta \lambda a_{32}^{3}\\&\quad -\,3\gamma \lambda \zeta b_{31} +3\gamma \lambda \zeta b_{32} -3\beta \lambda a_{32} b_{51} b_{52} -3/2\beta \lambda a_{11} a_{12} a_{52} \\&\quad -\,3/2\beta \lambda a_{11} b_{11} b_{51} +3/2\beta \lambda a_{11} b_{12} b_{51} -3/2\beta \lambda a_{51} b_{11} b_{12} \\&\quad +\,3/2\beta \lambda a_{11} a_{12} a_{51} +3/4\beta \lambda a_{51} b_{11} ^{2}+3/2\beta \lambda a_{32} a_{52}^{2} \\&\quad +\,3/4\beta \lambda a_{32} b_{31}^{2}+3/4\beta \lambda a_{32} b_{32} ^{2}+3/2\beta \lambda a_{32} b_{51}^{2}\\&\quad +\,3/2\beta \lambda a_{32} b_{52}^{2}+3/2\beta \lambda a_{31} b_{31} b_{32} -3/2\beta \lambda a_{31} b_{51}^{2} \\&\quad -\,3/2\beta \lambda a_{31} b_{52}^{2}+3/2\beta \lambda a_{32} a_{51} ^{2}+3/2\beta \lambda a_{52} b_{11} b_{12} \\&\quad -\,3/2\beta \lambda a_{11} b_{12} b_{52} +3/2\beta \lambda a_{12} b_{11} b_{51} -3/2\beta \lambda a_{12} b_{11} b_{52} \\&\quad -\,3/2\beta \lambda a_{12} b_{12} b_{51} +3/2\beta \lambda a_{12} b_{12} b_{52} +3\beta \lambda a_{31} a_{51} a_{52} \\&\quad -\,1/4\beta \lambda a_{11}^{3}+1/4\beta \lambda a_{12}^{3}-3/2\beta \lambda a_{51} b_{11} b_{31} \\&\quad +\,3/2\beta \lambda a_{51} b_{11} b_{32} +3/2\beta \lambda a_{51} b_{12} b_{31} -3/2\beta \lambda a_{51} b_{12} b_{32} \\&\quad +\,3/2\beta \lambda a_{52} b_{11} b_{31} -3/2\beta \lambda a_{52} b_{11} b_{32} -3/2\beta \lambda a_{52} b_{12} b_{31} \\&\quad +\,3/2\beta \lambda a_{52} b_{12} b_{32} +3/2\beta \lambda a_{11} a_{32} a_{51} -3/2\beta \lambda a_{11} a_{32} a_{52} \\&\quad -\,3/2\beta \lambda a_{11} b_{11} b_{12} -3/2\beta \lambda a_{11} b_{31} b_{51} +3/2\beta \lambda a_{11} b_{31} b_{52} \\&\quad +\,3/2\beta \lambda a_{11} b_{32} b_{51} -3/2\beta \lambda a_{11} b_{32} b_{52} +3/2\beta \lambda a_{12} a_{31} a_{51} \\&\quad -\,3/2\beta \lambda a_{12} a_{31} a_{52} -3/2\beta \lambda a_{12} a_{32} a_{51} +3/2\beta \lambda a_{12} a_{32} a_{52} \\&\quad +\,3/2\beta \lambda a_{12} b_{11} b_{12} +3/2\beta \lambda a_{12} b_{31} b_{51} -3/2\beta \lambda a_{12} b_{31} b_{52} \\&\quad -\,3/2\beta \lambda a_{12} b_{32} b_{51} +3/2\beta \lambda a_{12} b_{32} b_{52} +3\beta \lambda a_{31} b_{11} b_{12} \\&\quad +\,3/2\beta \lambda a_{31} b_{11} b_{51} -3/2\beta \lambda a_{31} b_{11} b_{52} -3/2\beta \lambda a_{31} b_{12} b_{51} \\&\quad +\,3/2\beta \lambda a_{31} b_{12} b_{52} -3\beta \lambda a_{32} b_{11} b_{12} -3/2\beta \lambda a_{32} b_{11} b_{51} \\&\quad +\,3/2\beta \lambda a_{32} b_{11} b_{52} +3/2\beta \lambda a_{32} b_{12} b_{51} -3/2\beta \lambda a_{32} b_{12} b_{52} \\&\quad +\,3\beta \lambda a_{11} a_{12} a_{31} -3\beta \lambda a_{11} a_{12} a_{32} -3/2\beta \lambda a_{11} a_{31} a_{51} \\&\quad +\,3/2\beta \lambda a_{11} a_{31} a_{52} =0, \end{aligned}$$
$$\begin{aligned}&-25b_{52} \gamma ^{2}-3/2\beta \lambda a_{31} a_{32} b_{11} +3/2\beta \lambda a_{31} a_{32} b_{12} \\&\quad +\,3\beta \lambda a_{31} a_{32} b_{51} -3\beta \lambda a_{31} a_{32} b_{52} -3/2\beta \lambda b_{11} b_{12} b_{31} \\&\quad +\,3/2\beta \lambda b_{11} b_{12} b_{32} +3/2\beta \lambda b_{11} b_{31} b_{32} -3/2\beta \lambda b_{12} b_{31} b_{32} \\&\quad +\,3\beta \lambda b_{31} b_{32} b_{51} -3\beta \lambda b_{31} b_{32} b_{52} -3/2\beta \lambda a_{11} a_{12} b_{32} \\&\quad +\,3/2\beta \lambda a_{11} a_{12} b_{31} -3/2\beta \lambda a_{11} a_{31} b_{11} +3/2\beta \lambda a_{11} a_{31} b_{12} \\&\quad +\,3/2\beta \lambda a_{11} a_{32} b_{11} -3/2\beta \lambda a_{11} a_{32} b_{12} \\&\quad +\,3/2\beta \lambda a_{12} a_{31} b_{11} -3/2\beta \lambda a_{12} a_{31} b_{12} -3/2\beta \lambda a_{12} a_{32} b_{11} \\&\quad +\,3/2\beta \lambda a_{12} a_{32} b_{12} -3/2\beta \lambda a_{11} a_{31} b_{31} \\&\quad +\,3/2\beta \lambda a_{11} a_{31} b_{32} -3/4\beta \lambda b_{11} b_{32}^{2}-3/4\beta \lambda b_{11} b_{31}^{2}\\&\quad +\,3/2\beta \lambda b_{32}^{2}b_{52} +3/4\beta \lambda a_{12}^{2}b_{32} -3/2\beta \lambda a_{32}^{2}b_{51} \\&\quad +\,3/2\beta \lambda a_{32}^{2}b_{52} -3/4\beta \lambda a_{31} ^{2}b_{12} -3/2\beta \lambda a_{31}^{2}b_{51} \\&\quad +\,3/2\beta \lambda a_{31}^{2}b_{52} +3/4\beta \lambda a_{32}^{2}b_{11} -3/4\beta \lambda a_{32}^{2}b_{12} \\&\quad +\,3/4\beta \lambda a_{31}^{2}b_{11} +3/4\beta \lambda b_{12} b_{31} ^{2}+3/4\beta \lambda b_{12} b_{32}^{2}\\&\quad +\,3/4\beta \lambda b_{12} ^{2}b_{31} -3/2\beta \lambda a_{51} a_{52} b_{52} +3/2\beta \lambda a_{51} a_{52} b_{51} \\&\quad -\,3/2\beta \lambda a_{12} a_{32} b_{31} +3/2\beta \lambda a_{12} a_{32} b_{32} +3/2\beta \lambda a_{12} a_{31} b_{31} \\&\quad -\,3/2\beta \lambda a_{12} a_{31} b_{32} +3/2\beta \lambda a_{11} a_{32} b_{31} \\&\quad -\,3/2\beta \lambda a_{11} a_{32} b_{32} -3/4\beta \lambda b_{12} ^{2}b_{32} +3/4\beta \lambda b_{11}^{2}b_{31} \\&\quad -\,3/4\beta \lambda b_{11}^{2}b_{32} -3/4\beta \lambda a_{51}^{2}b_{51} +3/4\beta \lambda a_{51}^{2}b_{52} \\&\quad -\,3/4\beta \lambda a_{52}^{2}b_{51} +3/4\beta \lambda a_{52} ^{2}b_{52} +3/2\beta \lambda b_{11}^{2}b_{52}\\&\quad +\,3/2\beta \lambda b_{12}^{2}b_{52} -3/4\beta \lambda a_{12}^{2}b_{31} +9/4\beta \lambda b_{51}^{2}b_{52} \\&\quad -\,9/4\beta \lambda b_{51} b_{52}^{2}-5\gamma \lambda \zeta a_{52} -3/2\beta \lambda b_{11}^{2}b_{51} \\&\quad -\,3\beta \lambda b_{11} b_{12} b_{52} +3\beta \lambda b_{11} b_{12} b_{51} +3\beta \lambda a_{11} a_{12} b_{51} \\&\quad -\,3\beta \lambda a_{11} a_{12} b_{52} \\&\quad -\,3/2\beta \lambda b_{12}^{2}b_{51} +3/2\beta \lambda a_{11} ^{2}b_{52} -3/2\beta \lambda a_{12}^{2}b_{51} \\&\quad +\,3/2\beta \lambda a_{12}^{2}b_{52} -3/2\beta \lambda a_{11}^{2}b_{51} +3/4\beta \lambda a_{11}^{2}b_{32} \\&\quad -\,3/4\beta \lambda a_{11}^{2}b_{31} -3/2\beta \lambda b_{31} ^{2}b_{51} +3/2\beta \lambda b_{31}^{2}b_{52} \\&\quad -\,3/2\beta \lambda b_{32}^{2}b_{51} +5\gamma \lambda \zeta a_{51} -3/4\beta \lambda b_{51}^{3}+3/4\beta \lambda b_{52}^{3}=0, \end{aligned}$$
$$\begin{aligned}&-3/4\beta \lambda a_{11}^{2}a_{31} +3/4\beta \lambda a_{11} ^{2}a_{32} +3/4\beta \lambda a_{31} b_{11}^{2}\nonumber \\&\quad +\,3/2\beta \lambda a_{52} b_{32}^{2}-3/2\beta \lambda a_{51} b_{32}^{2}+3/2\beta \lambda a_{52} b_{31}^{2} \nonumber \\&\quad +\,3/4\beta \lambda a_{31} b_{12}^{2}-3/2\beta \lambda a_{32} ^{2}a_{51} +3/2\beta \lambda a_{32}^{2}a_{52} \nonumber \\&\quad -\,3/4\beta \lambda a_{32} b_{11}^{2}-3/4\beta \lambda a_{32} b_{12}^{2}-3/2\beta \lambda a_{51} b_{31}^{2} \nonumber \\&\quad +\,3/4\beta \lambda a_{12}^{2}a_{32} +3/4\beta \lambda a_{12} a_{31} ^{2}-3/4\beta \lambda a_{12}^{2}a_{31} \nonumber \\&\quad -\,3/2\beta \lambda a_{51} b_{12}^{2}+3/2\beta \lambda a_{52} b_{11}^{2}+3/2\beta \lambda a_{52} b_{12}^{2} \nonumber \\&\quad +\,3/4\beta \lambda a_{11} b_{32}^{2}-3/4\beta \lambda a_{11} a_{32} ^{2}+3/4\beta \lambda a_{11} b_{31}^{2}\nonumber \\&\quad -\,25a_{52} \gamma ^{2}+3/4\beta \lambda a_{12} a_{32}^{2}-3/4\beta \lambda a_{12} b_{31}^{2} \nonumber \\&\quad -\,3/4\beta \lambda a_{12} b_{32}^{2}-3/2\beta \lambda a_{31} ^{2}a_{51} +3/2\beta \lambda a_{31}^{2}a_{52} \nonumber \\&\quad -\,3/4\beta \lambda a_{11} a_{31}^{2}-3/2\beta \lambda a_{52} b_{51} b_{52} -3/2\beta \lambda a_{32} b_{12} b_{31} \nonumber \\&\quad +\,3/2\beta \lambda a_{32} b_{12} b_{32} +3/2\beta \lambda a_{51} b_{51} b_{52} +3/2\beta \lambda a_{31} b_{12} b_{31} \nonumber \\&\quad -\,3/2\beta \lambda a_{31} b_{11} b_{31} -3/2\beta \lambda a_{31} b_{12} b_{32} +3/2\beta \lambda a_{32} b_{11} b_{31} \nonumber \\&\quad -\,3/2\beta \lambda a_{11}^{2}a_{51} +3/2\beta \lambda a_{11} ^{2}a_{52} -3/2\beta \lambda a_{12}^{2}a_{51} \nonumber \\&\quad +\,3/2\beta \lambda a_{12}^{2}a_{52} -5\gamma \lambda \zeta b_{51} +5\gamma \lambda \zeta b_{52} +9/4\beta \lambda a_{51}^{2}a_{52} \nonumber \\&\quad -\,9/4\beta \lambda a_{51} a_{52}^{2}+3/4\beta \lambda a_{52} ^{3}-3/4\beta \lambda a_{51}^{3}\nonumber \\&\quad -\,3/4\beta \lambda a_{51} b_{51} ^{2}-3/4\beta \lambda a_{51} b_{52}^{2}+3/4\beta \lambda a_{52} b_{51}^{2} \nonumber \\&\quad -\,3/2\beta \lambda a_{32} b_{11} b_{32} +3/2\beta \lambda a_{31} b_{11} b_{32} +3/4\beta \lambda a_{52} b_{52}^{2}\nonumber \\&\quad -\,3/2\beta \lambda a_{51} b_{11}^{2}+3\beta \lambda a_{51} b_{11} b_{12} -3\beta \lambda a_{52} b_{11} b_{12} \nonumber \\&\quad +\,3\beta \lambda a_{11} a_{12} a_{51} -3\beta \lambda a_{11} a_{12} a_{52} +3\beta \lambda a_{51} b_{31} b_{32} \nonumber \\&\quad -\,3\beta \lambda a_{52} b_{31} b_{32} +3/2\beta \lambda a_{11} b_{11} b_{31} -3/2\beta \lambda a_{11} b_{11} b_{32} \nonumber \\&\quad -\,3/2\beta \lambda a_{11} b_{12} b_{31} +3/2\beta \lambda a_{11} b_{12} b_{32} -3/2\beta \lambda a_{11} b_{31} b_{32} \nonumber \\&\quad -\,3/2\beta \lambda a_{12} a_{31} a_{32} -3/2\beta \lambda a_{12} b_{11} b_{31} +3/2\beta \lambda a_{12} b_{11} b_{32} \nonumber \\&\quad +\,3/2\beta \lambda a_{12} b_{12} b_{31} -3/2\beta \lambda a_{12} b_{12} b_{32} +3\beta \lambda a_{31} a_{32} a_{51} \nonumber \\&\quad -\,3\beta \lambda a_{31} a_{32} a_{52} -3/2\beta \lambda a_{31} b_{11} b_{12} +3/2\beta \lambda a_{32} b_{11} b_{12} \nonumber \\&\quad +\,3/2\beta \lambda a_{12} b_{31} b_{32} +3/2\beta \lambda a_{11} a_{12} a_{31} -3/2\beta \lambda a_{11} a_{12} a_{32}\nonumber \\&\quad +\,3/2\beta \lambda a_{11} a_{31} a_{32} =0. \end{aligned}$$
(A1)

Appendix B

1.1 The nonlinear differential-algebraic equations of the harmonic balance up to order 3

$$\begin{aligned}&-f-3/2b_{11} \left( \tau \right) a_{12} \left( \tau \right) b_{32} \left( \tau \right) \beta +3/2b_{11} \left( \tau \right) b_{12} \left( \tau \right) a_{32} \left( \tau \right) \beta \\&\quad -\,3a_{31} \left( \tau \right) a_{12} \left( \tau \right) a_{32} \left( \tau \right) \beta \\&\quad -\,3/2b_{31} \left( \tau \right) a_{12} \left( \tau \right) b_{12} \left( \tau \right) \beta -3b_{31} \left( \tau \right) a_{12} \left( \tau \right) b_{32} \left( \tau \right) \beta \\&\quad +\,3/2a_{12} \left( \tau \right) b_{12} \left( \tau \right) b_{32} \left( \tau \right) \beta \\&\quad -\,3/2a_{11} \left( \tau \right) b_{11} \left( \tau \right) b_{31} \left( \tau \right) \beta +3/2a_{11} \left( \tau \right) b_{11} \left( \tau \right) b_{12} \left( \tau \right) \beta \\&\quad +\,3/2a_{11} \left( \tau \right) b_{11} \left( \tau \right) b_{32} \left( \tau \right) \beta \\&\quad +\,3/2a_{11} \left( \tau \right) a_{31} \left( \tau \right) a_{12} \left( \tau \right) \beta +3a_{11} \left( \tau \right) a_{31} \left( \tau \right) a_{32} \left( \tau \right) \beta \\&\quad +\,3/2a_{11} \left( \tau \right) b_{31} \left( \tau \right) b_{12} \left( \tau \right) \beta \\&\quad +\,3a_{11} \left( \tau \right) b_{31} \left( \tau \right) b_{32} \left( \tau \right) \beta -3/2a_{11} \left( \tau \right) a_{12} \left( \tau \right) a_{32} \left( \tau \right) \beta \\&\quad -\,3/2a_{11} \left( \tau \right) b_{12} \left( \tau \right) b_{32} \left( \tau \right) \beta \\&\quad -\,3/2b_{11} \left( \tau \right) a_{31} \left( \tau \right) b_{12} \left( \tau \right) \beta +3/2b_{11} \left( \tau \right) b_{31} \left( \tau \right) a_{12} \left( \tau \right) \beta \\&\quad -\,3/2b_{11} \left( \tau \right) a_{12} \left( \tau \right) b_{12} \left( \tau \right) \beta \\&\quad -\,2\left( {\frac{\hbox {d}}{\hbox {d}\tau }b_{11} \left( \tau \right) } \right) \gamma -3/4\left( {a_{11} \left( \tau \right) } \right) ^{3}\beta -\left( {\frac{\hbox {d}}{\hbox {d}\tau }a_{11} \left( \tau \right) } \right) \zeta _0 \\&\quad +\,a_{11} \left( \tau \right) \gamma ^{2}+3/4\left( {a_{12} \left( \tau \right) } \right) ^{3}\beta \\&\quad -\,\left( {\frac{\hbox {d}}{\hbox {d}\tau }a_{11} \left( \tau \right) } \right) \left( \zeta \right) +3/2a_{12} \left( \tau \right) \left( {a_{32} \left( \tau \right) } \right) ^{2}\beta \\&\quad +\,3/2a_{12} \left( \tau \right) \left( {b_{32} \left( \tau \right) } \right) ^{2}\beta -3/4\left( {b_{12} \left( \tau \right) } \right) ^{2}a_{32} \left( \tau \right) \beta \\&\quad -\,\gamma b_{11} \left( \tau \right) \left( \zeta \right) -\gamma b_{11} \left( \tau \right) \zeta _0 -3/4\left( {a_{11} \left( \tau \right) } \right) ^{2}a_{31} \left( \tau \right) \beta \\&\quad -\,3/2a_{11} \left( \tau \right) \left( {a_{32} \left( \tau \right) } \right) ^{2}\beta -3/4a_{11} \left( \tau \right) \left( {b_{11} \left( \tau \right) } \right) ^{2}\beta \\&\quad +\,3/2\left( {a_{31} \left( \tau \right) } \right) ^{2}a_{12} \left( \tau \right) \beta +3/4\left( {b_{11} \left( \tau \right) } \right) ^{2}a_{12} \left( \tau \right) \beta \\&\quad +\,9/4\left( {a_{11} \left( \tau \right) } \right) ^{2}a_{12} \left( \tau \right) \beta -3/2a_{11} \left( \tau \right) \left( {b_{32} \left( \tau \right) } \right) ^{2}\beta \\&\quad +\,3/4\left( {b_{11} \left( \tau \right) } \right) ^{2}a_{31} \left( \tau \right) \beta -3/4\left( {b_{11} \left( \tau \right) } \right) ^{2}a_{32} \left( \tau \right) \beta \\&\quad -\,3/2a_{11} \left( \tau \right) \left( {a_{31} \left( \tau \right) } \right) ^{2}\beta -3/2a_{11} \left( \tau \right) \left( {b_{31} \left( \tau \right) } \right) ^{2}\beta \\&\quad -\,9/4a_{11} \left( \tau \right) \left( {a_{12} \left( \tau \right) } \right) ^{2}\beta -3/4a_{11} \left( \tau \right) \left( {b_{12} \left( \tau \right) } \right) ^{2}\beta \\&\quad +\,3/4a_{31} \left( \tau \right) \left( {b_{12} \left( \tau \right) } \right) ^{2}\beta +3/2\left( {b_{31} \left( \tau \right) } \right) ^{2}a_{12} \left( \tau \right) \beta \\&\quad +\,3/4\left( {a_{12} \left( \tau \right) } \right) ^{2}a_{32} \left( \tau \right) \beta +3/4a_{12} \left( \tau \right) \left( {b_{12} \left( \tau \right) } \right) ^{2}\beta \\&\quad -\,3/4a_{31} \left( \tau \right) \left( {a_{12} \left( \tau \right) } \right) ^{2}\beta +3/4\left( {a_{11} \left( \tau \right) } \right) ^{2}a_{32} \left( \tau \right) \beta \\&\quad -\,a_{11} \left( \tau \right) -\frac{\hbox {d}^{2}}{\hbox {d}\tau ^{2}}a_{11} \left( \tau \right) =0, \end{aligned}$$
$$\begin{aligned}&-3/4\left( {a_{11} \left( \tau \right) } \right) ^{2}b_{11} \left( \tau \right) \beta -3/4\left( {a_{11} \left( \tau \right) } \right) ^{2}b_{31} \left( \tau \right) \beta \\&\quad +\,3/4\left( {a_{11} \left( \tau \right) } \right) ^{2}b_{12} \left( \tau \right) \beta +3/4\left( {a_{11} \left( \tau \right) } \right) ^{2}b_{32} \left( \tau \right) \beta \\&\quad +\,3/4\left( {b_{11} \left( \tau \right) } \right) ^{2}b_{31} \left( \tau \right) \beta +9/4\left( {b_{11} \left( \tau \right) } \right) ^{2}b_{12} \left( \tau \right) \beta \\&\quad -\,3/4\left( {b_{11} \left( \tau \right) } \right) ^{2}b_{32} \left( \tau \right) \beta -3/2b_{11} \left( \tau \right) \left( {a_{31} \left( \tau \right) } \right) ^{2}\beta \\&\quad -\,3/2b_{11} \left( \tau \right) \left( {b_{31} \left( \tau \right) } \right) ^{2}\beta -3/4b_{11} \left( \tau \right) \left( {a_{12} \left( \tau \right) } \right) ^{2}\beta \\&\quad -\,9/4b_{11} \left( \tau \right) \left( {b_{12} \left( \tau \right) } \right) ^{2}\beta -3/2b_{11} \left( \tau \right) \left( {a_{32} \left( \tau \right) } \right) ^{2}\beta \\&\quad -\,3/2b_{11} \left( \tau \right) \left( {b_{32} \left( \tau \right) } \right) ^{2}\beta +3/2\left( {a_{31} \left( \tau \right) } \right) ^{2}b_{12} \left( \tau \right) \beta \\&\quad +\,3/2\left( {b_{31} \left( \tau \right) } \right) ^{2}b_{12} \left( \tau \right) \beta -3/4b_{31} \left( \tau \right) \left( {a_{12} \left( \tau \right) } \right) ^{2}\beta \\&\quad +\,3/4b_{31} \left( \tau \right) \left( {b_{12} \left( \tau \right) } \right) ^{2}\beta +3/4\left( {a_{12} \left( \tau \right) } \right) ^{2}b_{12} \left( \tau \right) \beta \\&\quad +\,3/4\left( {a_{12} \left( \tau \right) } \right) ^{2}b_{32} \left( \tau \right) \beta -3/4\left( {b_{12} \left( \tau \right) } \right) ^{2}b_{32} \left( \tau \right) \beta \\&\quad +\,3/2b_{12} \left( \tau \right) \left( {a_{32} \left( \tau \right) } \right) ^{2}\beta +3/2b_{12} \left( \tau \right) \left( {b_{32} \left( \tau \right) } \right) ^{2}\beta \\&\quad +\,\gamma a_{11} \left( \tau \right) \left( \zeta \right) +\gamma a_{11} \left( \tau \right) \zeta _0 +3/2a_{11} \left( \tau \right) b_{11} \left( \tau \right) a_{31} \left( \tau \right) \beta \\&\quad +\,3/2a_{11} \left( \tau \right) b_{11} \left( \tau \right) a_{12} \left( \tau \right) \beta -3/2a_{11} \left( \tau \right) b_{11} \left( \tau \right) a_{32} \left( \tau \right) \beta \\&\quad -\,3/2a_{11} \left( \tau \right) a_{31} \left( \tau \right) b_{12} \left( \tau \right) \beta \\&\quad +\,3/2a_{11} \left( \tau \right) b_{31} \left( \tau \right) a_{12} \left( \tau \right) \beta -3/2a_{11} \left( \tau \right) a_{12} \left( \tau \right) b_{12} \left( \tau \right) \beta \\&\quad -\,3/2a_{11} \left( \tau \right) a_{12} \left( \tau \right) b_{32} \left( \tau \right) \beta +3/2a_{11} \left( \tau \right) b_{12} \left( \tau \right) a_{32} \left( \tau \right) \beta \\&\quad -\,3/2b_{11} \left( \tau \right) a_{31} \left( \tau \right) a_{12} \left( \tau \right) \beta +3b_{11} \left( \tau \right) a_{31} \left( \tau \right) a_{32} \left( \tau \right) \beta \\&\quad -\,3/2b_{11} \left( \tau \right) b_{31} \left( \tau \right) b_{12} \left( \tau \right) \beta +3b_{11} \left( \tau \right) b_{31} \left( \tau \right) b_{32} \left( \tau \right) \beta \\&\quad +3/2b_{11} \left( \tau \right) a_{12} \left( \tau \right) a_{32} \left( \tau \right) \beta +3/2b_{11} \left( \tau \right) b_{12} \left( \tau \right) b_{32} \left( \tau \right) \beta \\&\quad +\,3/2a_{31} \left( \tau \right) a_{12} \left( \tau \right) b_{12} \left( \tau \right) \beta -3a_{31} \left( \tau \right) b_{12} \left( \tau \right) a_{32} \left( \tau \right) \beta \\&\quad -\,3b_{31} \left( \tau \right) b_{12} \left( \tau \right) b_{32} \left( \tau \right) \beta -3/2a_{12} \left( \tau \right) b_{12} \left( \tau \right) a_{32} \left( \tau \right) \beta \\&\quad +\,b_{11} \left( \tau \right) \gamma ^{2}-3/4\left( {b_{11} \left( \tau \right) } \right) ^{3}\beta -\left( {\frac{\hbox {d}}{\hbox {d}\tau }b_{11} \left( \tau \right) } \right) \zeta _0 \\&\quad +\,3/4\left( {b_{12} \left( \tau \right) } \right) ^{3}\beta -\left( {\frac{\hbox {d}}{\hbox {d}\tau }b_{11} \left( \tau \right) } \right) \left( \zeta \right) +2\left( {\frac{\hbox {d}}{\hbox {d}\tau }a_{11} \left( \tau \right) } \right) \gamma \\&\quad -\,b_{11} \left( \tau \right) -\frac{\hbox {d}^{2}}{\hbox {d}\tau ^{2}}b_{11} \left( \tau \right) =0, \end{aligned}$$
$$\begin{aligned}&-9/4a_{31} \left( \tau \right) \left( {a_{32} \left( \tau \right) } \right) ^{2}\beta -3/4a_{31} \left( \tau \right) \left( {b_{32} \left( \tau \right) } \right) ^{2}\beta \\&\quad +\,3/4\left( {b_{31} \left( \tau \right) } \right) ^{2}a_{32} \left( \tau \right) \beta +9/4\left( {a_{31} \left( \tau \right) } \right) ^{2}a_{32} \left( \tau \right) \beta \\&\quad -\,3/4a_{31} \left( \tau \right) \left( {b_{31} \left( \tau \right) } \right) ^{2}\beta -3\gamma b_{31} \left( \tau \right) \left( \zeta \right) \\&\quad -\,3\gamma b_{31} \left( \tau \right) \zeta _0 +3/4a_{32} \left( \tau \right) \left( {b_{32} \left( \tau \right) } \right) ^{2}\beta \\&\quad -\,3b_{11} \left( \tau \right) b_{12} \left( \tau \right) a_{32} \left( \tau \right) \beta \\&\quad -\,3/2a_{11} \left( \tau \right) b_{11} \left( \tau \right) b_{12} \left( \tau \right) \beta +3a_{11} \left( \tau \right) a_{31} \left( \tau \right) a_{12} \left( \tau \right) \beta \\&\quad -\,3a_{11} \left( \tau \right) a_{12} \left( \tau \right) a_{32} \left( \tau \right) \beta +3b_{11} \left( \tau \right) a_{31} \left( \tau \right) b_{12} \left( \tau \right) \beta \\&\quad +\,3/2b_{11} \left( \tau \right) a_{12} \left( \tau \right) b_{12} \left( \tau \right) \beta -\left( {\frac{\hbox {d}}{\hbox {d}\tau }a_{31} \left( \tau \right) } \right) \left( \zeta \right) \\&\quad -\,6\left( {\frac{\hbox {d}}{\hbox {d}\tau }b_{31} \left( \tau \right) } \right) \gamma -\left( {\frac{\hbox {d}}{\hbox {d}\tau }a_{31} \left( \tau \right) } \right) \zeta _0 +3/4\left( {a_{32} \left( \tau \right) } \right) ^{3}\beta \\&\quad +\,9a_{31} \left( \tau \right) \gamma ^{2}-3/4\left( {a_{31} \left( \tau \right) } \right) ^{3}\beta -1/4\left( {a_{11} \left( \tau \right) } \right) ^{3}\beta \\&\quad +\,1/4\left( {a_{12} \left( \tau \right) } \right) ^{3}\beta +3/2\left( {b_{12} \left( \tau \right) } \right) ^{2}a_{32} \left( \tau \right) \beta \\&\quad -\,3/2\left( {a_{11} \left( \tau \right) } \right) ^{2}a_{31} \left( \tau \right) \beta +3/4a_{11} \left( \tau \right) \left( {b_{11} \left( \tau \right) } \right) ^{2}\beta \\&\quad -\,3/4\left( {b_{11} \left( \tau \right) } \right) ^{2}a_{12} \left( \tau \right) \beta +3/4\left( {a_{11} \left( \tau \right) } \right) ^{2}a_{12} \left( \tau \right) \beta \\&\quad -\,3/2\left( {b_{11} \left( \tau \right) } \right) ^{2}a_{31} \left( \tau \right) \beta +3/2\left( {b_{11} \left( \tau \right) } \right) ^{2}a_{32} \left( \tau \right) \beta \\&\quad -\,3/4a_{11} \left( \tau \right) \left( {a_{12} \left( \tau \right) } \right) ^{2}\beta +3/4a_{11} \left( \tau \right) \left( {b_{12} \left( \tau \right) } \right) ^{2}\beta \\&\quad -\,3/2a_{31} \left( \tau \right) \left( {b_{12} \left( \tau \right) } \right) ^{2}\beta +3/2\left( {a_{12} \left( \tau \right) } \right) ^{2}a_{32} \left( \tau \right) \beta \\&\quad -\,3/4a_{12} \left( \tau \right) \left( {b_{12} \left( \tau \right) } \right) ^{2}\beta -3/2a_{31} \left( \tau \right) \left( {a_{12} \left( \tau \right) } \right) ^{2}\beta \\&\quad +\,3/2\left( {a_{11} \left( \tau \right) } \right) ^{2}a_{32} \left( \tau \right) \beta -a_{31} \left( \tau \right) \\&\quad +\,3/2a_{31} \left( \tau \right) b_{31} \left( \tau \right) b_{32} \left( \tau \right) \beta \\&\quad -\,3/2b_{31} \left( \tau \right) a_{32} \left( \tau \right) b_{32} \left( \tau \right) \beta -\frac{\hbox {d}^{2}}{\hbox {d}\tau ^{2}}a_{31} \left( \tau \right) =0, \end{aligned}$$
$$\begin{aligned}&-3/4\left( {a_{11} \left( \tau \right) } \right) ^{2}b_{11} \left( \tau \right) \beta -3/2\left( {a_{11} \left( \tau \right) } \right) ^{2}b_{31} \left( \tau \right) \beta \\&\quad +\,3/4\left( {a_{11} \left( \tau \right) } \right) ^{2}b_{12} \left( \tau \right) \beta +3/2\left( {a_{11} \left( \tau \right) } \right) ^{2}b_{32} \left( \tau \right) \beta \\&\quad -\,3/2\left( {b_{11} \left( \tau \right) } \right) ^{2}b_{31} \left( \tau \right) \beta -3/4\left( {b_{11} \left( \tau \right) } \right) ^{2}b_{12} \left( \tau \right) \beta \\&\quad +\,3/2\left( {b_{11} \left( \tau \right) } \right) ^{2}b_{32} \left( \tau \right) \beta -3/4b_{11} \left( \tau \right) \left( {a_{12} \left( \tau \right) } \right) ^{2}\beta \\&\quad +\,3/4b_{11} \left( \tau \right) \left( {b_{12} \left( \tau \right) } \right) ^{2}\beta -3/2b_{31} \left( \tau \right) \left( {a_{12} \left( \tau \right) } \right) ^{2}\beta \\&\quad -\,3/2b_{31} \left( \tau \right) \left( {b_{12} \left( \tau \right) } \right) ^{2}\beta +3/4\left( {a_{12} \left( \tau \right) } \right) ^{2}b_{12} \left( \tau \right) \beta \\&\quad +\,3/2\left( {a_{12} \left( \tau \right) } \right) ^{2}b_{32} \left( \tau \right) \beta +3/2\left( {b_{12} \left( \tau \right) } \right) ^{2}b_{32} \left( \tau \right) \beta \\&\quad +\,3/2a_{11} \left( \tau \right) b_{11} \left( \tau \right) a_{12} \left( \tau \right) \beta +3a_{11} \left( \tau \right) b_{31} \left( \tau \right) a_{12} \left( \tau \right) \beta \\&\quad -\,3/2a_{11} \left( \tau \right) a_{12} \left( \tau \right) b_{12} \left( \tau \right) \beta -3a_{11} \left( \tau \right) a_{12} \left( \tau \right) b_{32} \left( \tau \right) \beta \\&\quad +\,3b_{11} \left( \tau \right) b_{31} \left( \tau \right) b_{12} \left( \tau \right) \beta -3b_{11} \left( \tau \right) b_{12} \left( \tau \right) b_{32} \left( \tau \right) \beta \\&\quad -\,3/4b_{31} \left( \tau \right) \left( {a_{32} \left( \tau \right) } \right) ^{2}\beta -9/4b_{31} \left( \tau \right) \left( {b_{32} \left( \tau \right) } \right) ^{2}\beta \\&\quad +\,3/4\left( {a_{32} \left( \tau \right) } \right) ^{2}b_{32} \left( \tau \right) \beta +3\gamma a_{31} \left( \tau \right) \left( \zeta \right) +3\gamma a_{31} \left( \tau \right) \zeta _0 \\&\quad -\,3/4\left( {a_{31} \left( \tau \right) } \right) ^{2}b_{31} \left( \tau \right) \beta +3/4\left( {a_{31} \left( \tau \right) } \right) ^{2}b_{32} \left( \tau \right) \beta \\&\quad +\,9/4\left( {b_{31} \left( \tau \right) } \right) ^{2}b_{32} \left( \tau \right) \beta +9b_{31} \left( \tau \right) \gamma ^{2}-\left( {\frac{\hbox {d}}{\hbox {d}\tau }b_{31} \left( \tau \right) } \right) \zeta _0 \\&\quad -\,\left( {\frac{\hbox {d}}{\hbox {d}\tau }b_{31} \left( \tau \right) } \right) \left( \zeta \right) +6\left( {\frac{\hbox {d}}{\hbox {d}\tau }a_{31} \left( \tau \right) } \right) \gamma \\&\quad -\,3/4\left( {b_{31} \left( \tau \right) } \right) ^{3}\beta +3/4\left( {b_{32} \left( \tau \right) } \right) ^{3}\beta +1/4\left( {b_{11} \left( \tau \right) } \right) ^{3}\beta \\&\quad -\,1/4\left( {b_{12} \left( \tau \right) } \right) ^{3}\beta \\&\quad -\,b_{31} \left( \tau \right) +3/2a_{31} \left( \tau \right) b_{31} \left( \tau \right) a_{32} \left( \tau \right) \beta \\&\quad -\,3/2a_{31} \left( \tau \right) a_{32} \left( \tau \right) b_{32} \left( \tau \right) \beta -\frac{\hbox {d}^{2}}{\hbox {d}\tau ^{2}}b_{31} \left( \tau \right) =0, \end{aligned}$$
$$\begin{aligned}&-\gamma b_{12} \left( \tau \right) \lambda \zeta +3/4\left( {a_{11} \left( \tau \right) } \right) ^{3}\beta \lambda -3/4\left( {a_{12} \left( \tau \right) } \right) ^{3}\beta \lambda \\&\quad +\,\gamma b_{11} \left( \tau \right) \lambda \zeta -2\left( {\frac{\hbox {d}}{\hbox {d}\tau }b_{12} \left( \tau \right) } \right) \gamma +a_{12} \left( \tau \right) \gamma ^{2} \\&\quad +\,\left( {\frac{\hbox {d}}{\hbox {d}\tau }a_{11} \left( \tau \right) } \right) \lambda \zeta -\left( {\frac{\hbox {d}}{\hbox {d}\tau }a_{12} \left( \tau \right) } \right) \lambda \zeta \\&\quad +3/2b_{11} \left( \tau \right) a_{31} \left( \tau \right) b_{12} \left( \tau \right) \beta \lambda -3/2b_{11} \left( \tau \right) b_{31} \left( \tau \right) a_{12} \left( \tau \right) \beta \lambda \\&\quad +\,3/2b_{11} \left( \tau \right) a_{12} \left( \tau \right) b_{12} \left( \tau \right) \beta \lambda +3/2b_{11} \left( \tau \right) a_{12} \left( \tau \right) b_{32} \left( \tau \right) \beta \lambda \\&\quad -\,3/2b_{11} \left( \tau \right) b_{12} \left( \tau \right) a_{32} \left( \tau \right) \beta \lambda \\&\quad +\,3a_{31} \left( \tau \right) a_{12} \left( \tau \right) a_{32} \left( \tau \right) \beta \lambda +3/2b_{31} \left( \tau \right) a_{12} \left( \tau \right) b_{12} \left( \tau \right) \beta \lambda \\&\quad +3b_{31} \left( \tau \right) a_{12} \left( \tau \right) b_{32} \left( \tau \right) \beta \lambda \\&\quad -\,3/2a_{12} \left( \tau \right) b_{12} \left( \tau \right) b_{32} \left( \tau \right) \beta \lambda +3/2a_{11} \left( \tau \right) b_{11} \left( \tau \right) b_{31} \left( \tau \right) \beta \lambda \\&\quad -\,3/2a_{11} \left( \tau \right) b_{11} \left( \tau \right) b_{12} \left( \tau \right) \beta \lambda \\&\quad -\,3/2a_{11} \left( \tau \right) b_{11} \left( \tau \right) b_{32} \left( \tau \right) \beta \lambda -3/2a_{11} \left( \tau \right) a_{31} \left( \tau \right) a_{12} \left( \tau \right) \beta \lambda \\&\quad -\,3a_{11} \left( \tau \right) a_{31} \left( \tau \right) a_{32} \left( \tau \right) \beta \lambda -3/2a_{11} \left( \tau \right) b_{31} \left( \tau \right) b_{12} \left( \tau \right) \beta \lambda \\&\quad -\,3a_{11} \left( \tau \right) b_{31} \left( \tau \right) b_{32} \left( \tau \right) \beta \lambda +3/2a_{11} \left( \tau \right) a_{12} \left( \tau \right) a_{32} \left( \tau \right) \beta \lambda \\&\quad +\,3/2a_{11} \left( \tau \right) b_{12} \left( \tau \right) b_{32} \left( \tau \right) \beta \lambda +3/4\left( {a_{11} \left( \tau \right) } \right) ^{2}a_{31} \left( \tau \right) \beta \lambda \\&\quad -\,9/4\left( {a_{11} \left( \tau \right) } \right) ^{2}a_{12} \left( \tau \right) \beta \lambda \\&\quad -\,3/4\left( {a_{11} \left( \tau \right) } \right) ^{2}a_{32} \left( \tau \right) \beta \lambda +3/4a_{11} \left( \tau \right) \left( {b_{11} \left( \tau \right) } \right) ^{2}\beta \lambda \\&\quad +\,3/2a_{11} \left( \tau \right) \left( {a_{31} \left( \tau \right) } \right) ^{2}\beta \lambda +3/2a_{11} \left( \tau \right) \left( {b_{31} \left( \tau \right) } \right) ^{2}\beta \lambda \\&\quad +\,9/4a_{11} \left( \tau \right) \left( {a_{12} \left( \tau \right) } \right) ^{2}\beta \lambda +3/4a_{11} \left( \tau \right) \left( {b_{12} \left( \tau \right) } \right) ^{2}\beta \lambda \\&\quad +\,3/2a_{11} \left( \tau \right) \left( {a_{32} \left( \tau \right) } \right) ^{2}\beta \lambda +3/2a_{11} \left( \tau \right) \left( {b_{32} \left( \tau \right) } \right) ^{2}\beta \lambda \\&\quad -\,3/4\left( {b_{11} \left( \tau \right) } \right) ^{2}a_{31} \left( \tau \right) \beta \lambda -3/4\left( {b_{11} \left( \tau \right) } \right) ^{2}a_{12} \left( \tau \right) \beta \lambda \\&\quad +\,3/4\left( {b_{11} \left( \tau \right) } \right) ^{2}a_{32} \left( \tau \right) \beta \lambda -3/2\left( {a_{31} \left( \tau \right) } \right) ^{2}a_{12} \left( \tau \right) \beta \lambda \\&\quad +\,3/4a_{31} \left( \tau \right) \left( {a_{12} \left( \tau \right) } \right) ^{2}\beta \lambda -3/4a_{31} \left( \tau \right) \left( {b_{12} \left( \tau \right) } \right) ^{2}\beta \lambda \\&\quad -\,3/2\left( {b_{31} \left( \tau \right) } \right) ^{2}a_{12} \left( \tau \right) \beta \lambda -3/4\left( {a_{12} \left( \tau \right) } \right) ^{2}a_{32} \left( \tau \right) \beta \lambda \\&\quad -\,3/4a_{12} \left( \tau \right) \left( {b_{12} \left( \tau \right) } \right) ^{2}\beta \lambda -3/2a_{12} \left( \tau \right) \left( {a_{32} \left( \tau \right) } \right) ^{2}\beta \lambda \\&\quad -\,3/2a_{12} \left( \tau \right) \left( {b_{32} \left( \tau \right) } \right) ^{2}\beta \lambda +3/4\left( {b_{12} \left( \tau \right) } \right) ^{2}a_{32} \left( \tau \right) \beta \lambda \\&\quad -\,\frac{\hbox {d}^{2}}{\hbox {d}\tau ^{2}}a_{12} \left( \tau \right) =0, \end{aligned}$$
$$\begin{aligned}&-\gamma a_{11} \left( \tau \right) \lambda \zeta +3/4\left( {b_{11} \left( \tau \right) } \right) ^{3}\beta \lambda -3/4\left( {b_{12} \left( \tau \right) } \right) ^{3}\beta \lambda \\&\quad +\,\gamma a_{12} \left( \tau \right) \lambda \zeta -\left( {\frac{\hbox {d}}{\hbox {d}\tau }b_{12} \left( \tau \right) } \right) \lambda \zeta \\&\quad +\,\left( {\frac{\hbox {d}}{\hbox {d}\tau }b_{11} \left( \tau \right) } \right) \lambda \zeta +2\left( {\frac{\hbox {d}}{\hbox {d}\tau }a_{12} \left( \tau \right) } \right) \gamma \\&\quad +\,b_{12} \left( \tau \right) \gamma ^{2}+3/2a_{11} \left( \tau \right) a_{12} \left( \tau \right) b_{32} \left( \tau \right) \beta \lambda \\&\quad -\,3/2a_{11} \left( \tau \right) b_{12} \left( \tau \right) a_{32} \left( \tau \right) \beta \lambda +3/2b_{11} \left( \tau \right) a_{31} \left( \tau \right) a_{12} \left( \tau \right) \beta \lambda \\&\quad -\,3b_{11} \left( \tau \right) a_{31} \left( \tau \right) a_{32} \left( \tau \right) \beta \lambda \\&\quad +\,3/2b_{11} \left( \tau \right) b_{31} \left( \tau \right) b_{12} \left( \tau \right) \beta \lambda -3b_{11} \left( \tau \right) b_{31} \left( \tau \right) b_{32} \left( \tau \right) \beta \lambda \\&\quad -\,3/2b_{11} \left( \tau \right) a_{12} \left( \tau \right) a_{32} \left( \tau \right) \beta \lambda \\&\quad -\,3/2b_{11} \left( \tau \right) b_{12} \left( \tau \right) b_{32} \left( \tau \right) \beta \lambda -3/2a_{31} \left( \tau \right) a_{12} \left( \tau \right) b_{12} \left( \tau \right) \beta \lambda \\&\quad +\,3a_{31} \left( \tau \right) b_{12} \left( \tau \right) a_{32} \left( \tau \right) \beta \lambda \\&\quad +\,3b_{31} \left( \tau \right) b_{12} \left( \tau \right) b_{32} \left( \tau \right) \beta \lambda +3/2a_{12} \left( \tau \right) b_{12} \left( \tau \right) a_{32} \left( \tau \right) \beta \lambda \\&\quad -\,3/2a_{11} \left( \tau \right) b_{11} \left( \tau \right) a_{31} \left( \tau \right) \beta \lambda \\&\quad -\,3/2a_{11} \left( \tau \right) b_{11} \left( \tau \right) a_{12} \left( \tau \right) \beta \lambda +3/2a_{11} \left( \tau \right) b_{11} \left( \tau \right) a_{32} \left( \tau \right) \beta \lambda \\&\quad +\,3/2a_{11} \left( \tau \right) a_{31} \left( \tau \right) b_{12} \left( \tau \right) \beta \lambda \\&\quad -\,3/2a_{11} \left( \tau \right) b_{31} \left( \tau \right) a_{12} \left( \tau \right) \beta \lambda +3/2a_{11} \left( \tau \right) a_{12} \left( \tau \right) b_{12} \left( \tau \right) \beta \lambda \\&\quad -\,3/4\left( {a_{11} \left( \tau \right) } \right) ^{2}b_{32} \left( \tau \right) \beta \lambda \\&\quad -\,3/4\left( {b_{11} \left( \tau \right) } \right) ^{2}b_{31} \left( \tau \right) \beta \lambda -3/4b_{31} \left( \tau \right) \left( {b_{12} \left( \tau \right) } \right) ^{2}\beta \lambda \\&\quad -\,3/4\left( {a_{12} \left( \tau \right) } \right) ^{2}b_{12} \left( \tau \right) \beta \lambda \\&\quad -\,3/4\left( {a_{12} \left( \tau \right) } \right) ^{2}b_{32} \left( \tau \right) \beta \lambda +3/4\left( {b_{12} \left( \tau \right) } \right) ^{2}b_{32} \left( \tau \right) \beta \lambda \\&\quad -\,3/2b_{12} \left( \tau \right) \left( {a_{32} \left( \tau \right) } \right) ^{2}\beta \lambda \\&\quad -\,3/2b_{12} \left( \tau \right) \left( {b_{32} \left( \tau \right) } \right) ^{2}\beta \lambda -9/4\left( {b_{11} \left( \tau \right) } \right) ^{2}b_{12} \left( \tau \right) \beta \lambda \\&\quad +\,3/4\left( {b_{11} \left( \tau \right) } \right) ^{2}b_{32} \left( \tau \right) \beta \lambda \\&\quad +\,3/2b_{11} \left( \tau \right) \left( {a_{31} \left( \tau \right) } \right) ^{2}\beta \lambda +3/2b_{11} \left( \tau \right) \left( {b_{31} \left( \tau \right) } \right) ^{2}\beta \lambda \\&\quad +\,3/4b_{11}\left( \tau \right) \left( {a_{12} \left( \tau \right) } \right) ^{2}\beta \lambda \\&\quad +\,9/4b_{11} \left( \tau \right) \left( {b_{12} \left( \tau \right) } \right) ^{2}\beta \lambda +3/2b_{11} \left( \tau \right) \left( {a_{32} \left( \tau \right) } \right) ^{2}\beta \lambda \\&\quad +\,3/2b_{11} \left( \tau \right) \left( {b_{32} \left( \tau \right) } \right) ^{2}\beta \lambda \\&\quad -\,3/2\left( {a_{31} \left( \tau \right) } \right) ^{2}b_{12} \left( \tau \right) \beta \lambda -3/2\left( {b_{31} \left( \tau \right) } \right) ^{2}b_{12} \left( \tau \right) \beta \lambda \\&\quad +\,3/4b_{31} \left( \tau \right) \left( {a_{12} \left( \tau \right) } \right) ^{2}\beta \lambda \\&\quad +\,3/4\left( {a_{11} \left( \tau \right) } \right) ^{2}b_{11} \left( \tau \right) \beta \lambda +3/4\left( {a_{11} \left( \tau \right) } \right) ^{2}b_{31} \left( \tau \right) \beta \lambda \\&\quad -\,3/4\left( {a_{11} \left( \tau \right) } \right) ^{2}b_{12} \left( \tau \right) \beta \lambda \\&\quad -\,\frac{\hbox {d}^{2}}{\hbox {d}\tau ^{2}}b_{12} \left( \tau \right) =0, \end{aligned}$$
$$\begin{aligned}&3\gamma b_{31} \left( \tau \right) \lambda \zeta -3\gamma b_{32} \left( \tau \right) \lambda \zeta +3/4\left( {a_{31} \left( \tau \right) } \right) ^{3}\beta \lambda \\&\quad -\,3/4\left( {a_{32} \left( \tau \right) } \right) ^{3}\beta \lambda +1/4\left( {a_{11} \left( \tau \right) } \right) ^{3}\beta \lambda \\&\quad -\,1/4\left( {a_{12} \left( \tau \right) } \right) ^{3}\beta \lambda +\left( {\frac{\hbox {d}}{\hbox {d}\tau }a_{31} \left( \tau \right) } \right) \lambda \zeta \\&\quad -\,6\left( {\frac{\hbox {d}}{\hbox {d}\tau }b_{32} \left( \tau \right) } \right) \gamma +9a_{32} \left( \tau \right) \gamma ^{2}-\left( {\frac{\hbox {d}}{\hbox {d}\tau }a_{32} \left( \tau \right) } \right) \lambda \zeta \\&\quad -\,3/2a_{31} \left( \tau \right) b_{31} \left( \tau \right) b_{32} \left( \tau \right) \beta \lambda +3/2b_{31} \left( \tau \right) a_{32} \left( \tau \right) b_{32} \left( \tau \right) \beta \lambda \\&\quad -\,3/4\left( {b_{31} \left( \tau \right) } \right) ^{2}a_{32} \left( \tau \right) \beta \lambda \\&\quad -\,3/4a_{32} \left( \tau \right) \left( {b_{32} \left( \tau \right) } \right) ^{2}\beta \lambda -9/4\left( {a_{31} \left( \tau \right) } \right) ^{2}a_{32} \left( \tau \right) \beta \lambda \\&\quad +\,3/4a_{31} \left( \tau \right) \left( {b_{31} \left( \tau \right) } \right) ^{2}\beta \lambda \\&\quad +\,9/4a_{31} \left( \tau \right) \left( {a_{32} \left( \tau \right) } \right) ^{2}\beta \lambda +3/4a_{31} \left( \tau \right) \left( {b_{32} \left( \tau \right) } \right) ^{2}\beta \lambda \\&\quad -\,3b_{11}\left( \tau \right) a_{31} \left( \tau \right) b_{12} \left( \tau \right) \beta \lambda \\&\quad -\,3/2b_{11} \left( \tau \right) a_{12} \left( \tau \right) b_{12} \left( \tau \right) \beta \lambda +3b_{11} \left( \tau \right) b_{12} \left( \tau \right) a_{32} \left( \tau \right) \beta \lambda \\&\quad +\,3/2a_{11} \left( \tau \right) b_{11} \left( \tau \right) b_{12} \left( \tau \right) \beta \lambda \\&\quad -\,3a_{11} \left( \tau \right) a_{31} \left( \tau \right) a_{12} \left( \tau \right) \beta \lambda +3a_{11} \left( \tau \right) a_{12} \left( \tau \right) a_{32} \left( \tau \right) \beta \lambda \\&\quad +\,3/2\left( {a_{11} \left( \tau \right) } \right) ^{2}a_{31} \left( \tau \right) \beta \lambda \\&\quad -\,3/4\left( {a_{11} \left( \tau \right) } \right) ^{2}a_{12} \left( \tau \right) \beta \lambda -3/2\left( {a_{11} \left( \tau \right) } \right) ^{2}a_{32} \left( \tau \right) \beta \lambda \\&\quad -\,3/4a_{11} \left( \tau \right) \left( {b_{11} \left( \tau \right) } \right) ^{2}\beta \lambda \\&\quad +\,3/4a_{11} \left( \tau \right) \left( {a_{12} \left( \tau \right) } \right) ^{2}\beta \lambda -3/4a_{11} \left( \tau \right) \left( {b_{12} \left( \tau \right) } \right) ^{2}\beta \lambda \\&\quad +\,3/2\left( {b_{11} \left( \tau \right) } \right) ^{2}a_{31} \left( \tau \right) \beta \lambda \\&\quad +\,3/4\left( {b_{11} \left( \tau \right) } \right) ^{2}a_{12} \left( \tau \right) \beta \lambda -3/2\left( {b_{11} \left( \tau \right) } \right) ^{2}a_{32} \left( \tau \right) \beta \lambda \\&\quad +\,3/2a_{31} \left( \tau \right) \left( {a_{12} \left( \tau \right) } \right) ^{2}\beta \lambda \\&\quad +\,3/2a_{31} \left( \tau \right) \left( {b_{12} \left( \tau \right) } \right) ^{2}\beta \lambda -3/2\left( {a_{12} \left( \tau \right) } \right) ^{2}a_{32} \left( \tau \right) \beta \lambda \\&\quad +\,3/4a_{12} \left( \tau \right) \left( {b_{12} \left( \tau \right) } \right) ^{2}\beta \lambda \\&\quad -\,3/2\left( {b_{12} \left( \tau \right) } \right) ^{2}a_{32} \left( \tau \right) \beta \lambda -\frac{\hbox {d}^{2}}{\hbox {d}\tau ^{2}}a_{32} \left( \tau \right) =0, \end{aligned}$$
$$\begin{aligned}&-3\gamma a_{31} \left( \tau \right) \lambda \zeta +3\gamma a_{32} \left( \tau \right) \lambda \zeta +3/4\left( {b_{31} \left( \tau \right) } \right) ^{3}\beta \lambda \nonumber \\&\quad -\,3/4\left( {b_{32} \left( \tau \right) } \right) ^{3}\beta \lambda -1/4\left( {b_{11} \left( \tau \right) } \right) ^{3}\beta \lambda \nonumber \\&\quad +\,1/4\left( {b_{12} \left( \tau \right) } \right) ^{3}\beta \lambda -\left( {\frac{\hbox {d}}{\hbox {d}\tau }b_{32} \left( \tau \right) } \right) \lambda \zeta +9b_{32} \left( \tau \right) \gamma ^{2}\nonumber \\&\quad +\,6\left( {\frac{\hbox {d}}{\hbox {d}\tau }a_{32} \left( \tau \right) } \right) \gamma +\left( {\frac{\hbox {d}}{\hbox {d}\tau }b_{31} \left( \tau \right) } \right) \lambda \zeta \nonumber \\&\quad -\,3/2a_{31} \left( \tau \right) b_{31} \left( \tau \right) a_{32} \left( \tau \right) \beta \lambda +3/2a_{31} \left( \tau \right) a_{32} \left( \tau \right) b_{32} \left( \tau \right) \beta \lambda \nonumber \\&\quad +\,3/4\left( {a_{31} \left( \tau \right) } \right) ^{2}b_{31} \left( \tau \right) \beta \lambda -3/4\left( {a_{31} \left( \tau \right) } \right) ^{2}b_{32} \left( \tau \right) \beta \lambda \nonumber \\&\quad -\,9/4\left( {b_{31} \left( \tau \right) } \right) ^{2}b_{32} \left( \tau \right) \beta \lambda +3/4b_{31} \left( \tau \right) \left( {a_{32} \left( \tau \right) }\right) ^{2}\beta \lambda \nonumber \\&\quad +\,9/4b_{31} \left( \tau \right) \left( {b_{32} \left( \tau \right) } \right) ^{2}\beta \lambda -3/4\left( {a_{32} \left( \tau \right) } \right) ^{2}b_{32} \left( \tau \right) \beta \lambda \nonumber \\&\quad +\,3a_{11} \left( \tau \right) a_{12} \left( \tau \right) b_{32} \left( \tau \right) \beta \lambda \nonumber \\&\quad -\,3b_{11} \left( \tau \right) b_{31} \left( \tau \right) b_{12} \left( \tau \right) \beta \lambda +3b_{11} \left( \tau \right) b_{12} \left( \tau \right) b_{32} \left( \tau \right) \beta \lambda \nonumber \\&\quad -\,3/2a_{11} \left( \tau \right) b_{11} \left( \tau \right) a_{12} \left( \tau \right) \beta \lambda \nonumber \\&\quad -\,3a_{11} \left( \tau \right) b_{31} \left( \tau \right) a_{12} \left( \tau \right) \beta \lambda +3/2a_{11} \left( \tau \right) a_{12} \left( \tau \right) b_{12} \left( \tau \right) \beta \lambda \nonumber \\&\quad -\,3/2\left( {a_{11} \left( \tau \right) } \right) ^{2}b_{32} \left( \tau \right) \beta \lambda \nonumber \\&\quad +\,3/2\left( {b_{11} \left( \tau \right) } \right) ^{2}b_{31} \left( \tau \right) \beta \lambda +3/2b_{31} \left( \tau \right) \left( {b_{12} \left( \tau \right) } \right) ^{2}\beta \lambda \nonumber \\&\quad -\,3/4\left( {a_{12} \left( \tau \right) } \right) ^{2}b_{12} \left( \tau \right) \beta \lambda \nonumber \\&\quad -\,3/2\left( {a_{12} \left( \tau \right) } \right) ^{2}b_{32} \left( \tau \right) \beta \lambda -3/2\left( {b_{12} \left( \tau \right) } \right) ^{2}b_{32} \left( \tau \right) \beta \lambda \nonumber \\&\quad +\,3/4\left( {b_{11} \left( \tau \right) } \right) ^{2}b_{12} \left( \tau \right) \beta \lambda \nonumber \\&\quad -\,3/2\left( {b_{11} \left( \tau \right) } \right) ^{2}b_{32} \left( \tau \right) \beta \lambda +3/4b_{11} \left( \tau \right) \left( {a_{12} \left( \tau \right) } \right) ^{2}\beta \lambda \nonumber \\&\quad -\,3/4b_{11} \left( \tau \right) \left( {b_{12} \left( \tau \right) } \right) ^{2}\beta \lambda \nonumber \\&\quad +\,3/2b_{31} \left( \tau \right) \left( {a_{12} \left( \tau \right) } \right) ^{2}\beta \lambda +3/4\left( {a_{11} \left( \tau \right) } \right) ^{2}b_{11} \left( \tau \right) \beta \lambda \nonumber \\&\quad +\,3/2\left( {a_{11} \left( \tau \right) } \right) ^{2}b_{31} \left( \tau \right) \beta \lambda \nonumber \\&\quad -\,3/4\left( {a_{11} \left( \tau \right) } \right) ^{2}b_{12} \left( \tau \right) \beta \lambda -\frac{\hbox {d}^{2}}{\hbox {d}\tau ^{2}}\hbox {b}_{32} \left( \tau \right) =0. \end{aligned}$$
(B1)

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Zang, J., Chen, LQ. Complex dynamics of a harmonically excited structure coupled with a nonlinear energy sink. Acta Mech. Sin. 33, 801–822 (2017). https://doi.org/10.1007/s10409-017-0671-x

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