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Theoretical and experimental investigations of the crossover phenomenon in micromachined arch resonator: part II—simultaneous 1:1 and 2:1 internal resonances

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We investigate in a silicon micromachined arch beam the activation of a one-to-one internal resonance between the first symmetric and first antisymmetric modes simultaneously with the activation of a two-to-one internal resonance between these modes and the second symmetric mode. The arch is excited electrically, using an antisymmetric partial electrode to activate both modes of vibrations, and tuned electrothermally via Joule’s heating. Theoretically, we explore the dynamics of the beam using the Galerkin and multiple timescales methods. The simulation results are shown to have good agreement with the experimental data. The results show the merging of both modes at crossing, after which the first antisymmetric mode exchanges the nonlinear behavior with the first symmetric mode. The nonlinear behavior of the arch beam is demonstrated and analyzed experimentally and theoretically as experiencing the simultaneous 2:1 and 1:1 internal resonances.

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We acknowledge the financial support from King Abdullah University of Science and Technology (KAUST).

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Correspondence to Mohammad I. Younis.

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Appendix A. Definition of coefficients used in Eqs. (10.1)–(10.3)

To simplify the expressions, we use the following definition to represent the integrals:

$$\begin{aligned}&\left\langle {A,B} \right\rangle =\mathop {\int }_0^1 {A(x)B(x)} \;\mathrm{d}x \end{aligned}$$
$$\begin{aligned}&\mu =\frac{c}{2} \end{aligned}$$
$$\begin{aligned}&F_{m_1 } =\alpha _2 V_\mathrm{AC} V_{\mathrm{DC}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _m \left( x \right) }{\left( {w_s \left( x \right) -1} \right) ^{2}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned}&F_{n_1 } =\alpha _2 V_\mathrm{AC} V_{\mathrm{DC}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _n \left( x \right) }{\left( {w_s \left( x \right) -1} \right) ^{2}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned}&F_{k_1 } =\frac{1}{4}\alpha _2 V_\mathrm{AC}^2 \left( {\mathop {\int }\limits _0^1 \frac{\phi _k \left( x \right) }{\left( {w_s \left( x \right) -1} \right) ^{2}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned}&R_{m_1 } =2\alpha _1 \left( \varGamma \left( {\phi _k ,\phi _m } \right) \left\langle {\phi _m ,w_s ^{\prime \prime }} \right\rangle \right. \nonumber \\&\qquad \quad \left. +\,\varGamma \left( {w_s ,\phi _m } \right) \left\langle {\phi _m ,\phi _k ^{\prime \prime }} \right\rangle +\varGamma \left( {w_s ,\phi _k } \right) \left\langle {\phi _m ,\phi _m ^{\prime \prime }} \right\rangle \right) \nonumber \\&\qquad \quad +\,6\alpha _2 V_{\mathrm{DCEff}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _k \phi _m ^{2}}{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned}&R_{m_2 } =2\alpha _1 \left( \varGamma \left( {\phi _k ,\phi _n } \right) \left\langle {\phi _m ,w_s ^{\prime \prime }} \right\rangle \right. \nonumber \\&\qquad \quad \left. +\,\varGamma \left( {w_s ,\phi _n } \right) \left\langle {\phi _m ,\phi _k ^{\prime \prime }} \right\rangle +\varGamma \left( {w_s ,\phi _k } \right) \left\langle {\phi _m ,\phi _n ^{\prime \prime }} \right\rangle \right) \nonumber \\&\qquad \quad +\,6V_{\mathrm{DCEff}} \alpha _2 \left( {\mathop {\int }\limits _0^1 \frac{\phi _k \phi _m \phi _n }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned}&R_{n_1 } =2\alpha _1 \left( \varGamma \left( {\phi _k ,\phi _m } \right) \left\langle {\phi _n ,w_s ^{\prime \prime }} \right\rangle \right. \nonumber \\&\qquad \quad \left. +\,\varGamma \left( {w_s ,\phi _m } \right) \left\langle {\phi _n ,\phi _k ^{\prime \prime }} \right\rangle +\varGamma \left( {w_s ,\phi _k } \right) \left\langle {\phi _n ,\phi _m ^{\prime \prime }} \right\rangle \right) \nonumber \\&\qquad \quad +\,6\alpha _2 V_{\mathrm{DCEff}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _k \phi _m \phi _n }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned}&R_{n_2 } =2\alpha _1 \left( \varGamma \left( {\phi _k ,\phi _n } \right) \left\langle {\phi _n ,w_s ^{\prime \prime }} \right\rangle \right. \nonumber \\&\qquad \quad \left. +\,\varGamma \left( {w_s ,\phi _n } \right) \left\langle {\phi _n ,\phi _k ^{\prime \prime }} \right\rangle +\varGamma \left( {w_s ,\phi _k } \right) \left\langle {\phi _n ,\phi _n ^{\prime \prime }} \right\rangle \right) \nonumber \\&\qquad \quad +\,6\alpha _2 V_{\mathrm{DCEff}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _k \phi _n ^{2}}{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned}&R_{k_1 } =\alpha _1 \left( \varGamma \left( {\phi _m ,\phi _m } \right) \left\langle {\phi _k ,w_s ^{\prime \prime }} \right\rangle \right. \nonumber \\&\qquad \quad \left. +\,2\varGamma \left( {w_s ,\phi _m } \right) \left\langle {\phi _k ,\phi _m ^{\prime \prime }} \right\rangle \right) \nonumber \\&\qquad \quad +\,3\alpha _2 V_{\mathrm{DCEff}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _k \phi _m ^{2}}{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned}&R_{k_2 } =2\alpha _1 \left( \varGamma \left( {\phi _m ,\phi _n } \right) \left\langle {\phi _k ,w_s ^{\prime \prime }} \right\rangle \right. \nonumber \\&\qquad \quad \left. +\,\varGamma \left( {w_s ,\phi _n } \right) \left\langle {\phi _k ,\phi _m ^{\prime \prime }} \right\rangle +\varGamma \left( {w_s ,\phi _m } \right) \left\langle {\phi _k ,\phi _n ^{\prime \prime }} \right\rangle \right) \nonumber \\&\qquad \quad +\,\alpha _2 6V_{\mathrm{DCEff}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _k \phi _m \phi _n }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned}&R_{k_3 } =\alpha _1 \left( \varGamma \left( {\phi _n ,\phi _n } \right) \left\langle {\phi _k ,w_s ^{\prime \prime }} \right\rangle \right. \nonumber \\&\qquad \quad \left. +\,2\varGamma \left( {w_s ,\phi _n } \right) \left\langle {\phi _k ,\phi _n ^{\prime \prime }} \right\rangle \right) \nonumber \\&\qquad \quad +\,3\alpha _2 V_{\mathrm{DCEff}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _k \phi _n ^{2}}{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \end{aligned}$$

Appendix B. Galerkin procedure

Here we explain the Galerkin procedure used to find the particular solutions by solving the boundary value problem in Eqs. (12.1)–(12.7) defined by

$$\begin{aligned} H[\omega ]= & {} H^{iv} - N H^{\prime {\prime }} - 2\alpha _{1}{{w}_{s}}^{\prime {\prime }}\left( \mathop \int \limits _{0}^{1}w_{s}^{\prime } H^{\prime } \hbox {d}x\right) \nonumber \\&-\frac{2{\alpha }_{2}V_{\mathrm{DCeff}}^{2}}{(1-w_{s})^{3}} H-\omega _{i}^{2}H \end{aligned}$$

We express the function H as a superposition of five mode shapes \(f_{i}(x)\) given by

$$\begin{aligned} H=\sum _{i=1}^{n=5} {q_i f_i (x)} \end{aligned}$$

where \(q_{i}\) are coefficients to be solved for in the Galerkin procedure. The \(f_{i}(x)\) are the mode shapes of the straight clamped–clamped beam. Then, the equation is reduced via orthogonality of the mode shapes to a set of five algebraic equations that are solved to obtain the value of coefficients.

Appendix C. Definition of coefficient used in Eqs. (14.1)–(14.3)

To simplify the expressions, we use the definition in (C.1) to represent the integrals in (C.2)–(C.34).

$$\begin{aligned} \langle A,B \rangle= & {} \mathop {\int }\limits _{0}^{1} A(x)B(x)\mathrm{d}x \end{aligned}$$
$$\begin{aligned} F_{mk}= & {} -\alpha _2 V_\mathrm{AC} V_{\mathrm{DC}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _k \phi _m }{\left( {1-w_s } \right) ^{3}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned} F_{m_2 }= & {} -\frac{1}{2}\alpha _2 V_\mathrm{AC}^2 \left( {\mathop {\int }\limits _0^1 \frac{\phi _m ^{2}}{\left( {1-w_s } \right) ^{3}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned} F_{mn_1 }= & {} -\frac{1}{2}\alpha _2 V_\mathrm{AC}^2 \left( {\mathop {\int }\limits _0^1 \frac{\phi _m \phi _n }{\left( {1-w_s } \right) ^{3}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned} F_{nk}= & {} \alpha _2 V_\mathrm{AC} V_{\mathrm{DC}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _k \phi _n }{\left( {1-w_s } \right) ^{3}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned} F_{n_2 }= & {} \frac{1}{2}\alpha _2 V_\mathrm{AC}^2 \left( {\mathop {\int }\limits _0^1 \frac{\phi _n ^{2}}{\left( {1-w_s } \right) ^{3}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned} K_{kmk_1 }= & {} 2\alpha _1 \left( \varGamma \left( {\phi _k ,\psi _{km} } \right) \left\langle {w_s ^{\prime \prime },\phi _m } \right\rangle \right. \nonumber \\&+\,\varGamma \left( {\phi _m ,\psi _{kk} } \right) \left\langle {w_s ^{\prime \prime },\phi _m } \right\rangle \nonumber \\&\left. +\varGamma \left( {w_s ,\psi _{km} } \right) \left\langle {\phi _k ^{\prime \prime },\phi _m } \right\rangle \right) \nonumber \\&+\,2\alpha _1 \left( 2\varGamma \left( {\phi _k ,\phi _m } \right) \left\langle {\phi _k ^{\prime \prime },\phi _m } \right\rangle \right. \nonumber \\&+\,\varGamma \left( {w_s ,\psi _{kk} } \right) \left\langle {\phi _m ^{\prime \prime },\phi _m } \right\rangle \nonumber \\&\left. +\varGamma \left( {\phi _k ,\phi _k } \right) \left\langle {\phi _m ^{\prime \prime },\phi _m } \right\rangle \right) \nonumber \\&+\,2\alpha _1 \left( \varGamma \left( {w_s ,\phi _k } \right) \left\langle {\psi _{km} ^{\prime \prime },\phi _m } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _m } \right) \left\langle {\psi _{kk} ^{\prime \prime },\phi _m } \right\rangle \right) \nonumber \\&+\,6\alpha _2 V_{\mathrm{DCEff}} \left( \left( {\mathop {\int }\limits _0^1 \frac{\phi _m \left( x \right) ^{2}\psi _{kk} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \right. \nonumber \\&\left. +\,\left( {\mathop {\int }\limits _0^1 \frac{\phi _k \phi _m \psi _{km} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \right) \end{aligned}$$
$$\begin{aligned} K_{knk_1 }= & {} 2\alpha _1 \left( \varGamma \left( {\phi _k ,\psi _{kn} } \right) \left\langle {w_s ^{\prime \prime },\phi _m } \right\rangle \right. \nonumber \\&+\,\varGamma \left( {\phi _n ,\psi _{kk} } \right) \left\langle {w_s ^{\prime \prime },\phi _m } \right\rangle \nonumber \\&\left. +\varGamma \left( {w_s ,\psi _{kn} } \right) \left\langle {\phi _k ^{\prime \prime },\phi _m } \right\rangle \right) \nonumber \\&+\,2\alpha _1 \left( 2\varGamma \left( {\phi _k ,\phi _n } \right) \left\langle {\phi _k ^{\prime \prime },\phi _m } \right\rangle \right. \nonumber \\&+\,\varGamma \left( {w_s ,\psi _{kk} } \right) \left\langle {\phi _n ^{\prime \prime },\phi _m } \right\rangle \nonumber \\&\left. +\varGamma \left( {\phi _k ,\phi _k } \right) \left\langle {\phi _n ^{\prime \prime },\phi _m } \right\rangle \right) \nonumber \\&+\,2\alpha _1 \left( \varGamma \left( {w_s ,\phi _n } \right) \left\langle {\psi _{kk} ^{\prime \prime },\phi _m } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _k } \right) \left\langle {\psi _{kn} ^{\prime \prime },\phi _m } \right\rangle \right) \nonumber \\&+\,6\alpha _2 V_{\mathrm{DCEff}} \left( \left( {\mathop {\int }\limits _0^1 \frac{\phi _m \phi _n \psi _{kk} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \right. \nonumber \\&\left. +\left( {\mathop {\int }\limits _0^1 \frac{\phi _k \phi _m \psi _{kn} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \right) \end{aligned}$$
$$\begin{aligned} K_{mmm_1 }= & {} 2\alpha _1 \varGamma \left( {\phi _m ,\psi _{mm} } \right) \left\langle {w_s ^{\prime \prime },\phi _m } \right\rangle \nonumber \\&+\,2\alpha _1 \varGamma \left( {w_s ,\psi _{mm} } \right) \left\langle {\phi _m ^{\prime \prime },\phi _m } \right\rangle \nonumber \\&+\,3\alpha _1 \varGamma \left( {\phi _m ,\phi _m } \right) \left\langle {\phi _m ^{\prime \prime },\phi _m } \right\rangle \nonumber \\&+\,2\alpha _1 \varGamma \left( {w_s ,\phi _m } \right) \left\langle {\psi _{mm} ^{\prime \prime },\phi _m \left( x \right) } \right\rangle \nonumber \\&+6\alpha _2 V_{\mathrm{DCEff}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _m ^{2}\psi _{mm} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned} K_{mnm_1 }= & {} 2\alpha _1 \left( \varGamma \left( {\phi _m ,\psi _{mn} } \right) \left\langle {w_s ^{\prime \prime },\phi _m } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {\phi _n ,\psi _{mm} } \right) \left\langle {w_s ^{\prime \prime },\phi _m } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\psi _{mn} } \right) \left\langle {\phi _m ^{\prime \prime },\phi _m } \right\rangle \right) \nonumber \\&+\,2\alpha _1 \left( \varGamma \left( {w_s ,\psi _{mm} } \right) \left\langle {\phi _n ^{\prime \prime },\phi _m } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {\phi _m ,\phi _m } \right) \left\langle {\phi _n ^{\prime \prime },\phi _m } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _n } \right) \left\langle {\psi _{mm} ^{\prime \prime },\phi _m } \right\rangle \right) \nonumber \\&+\,4\alpha _1 \varGamma \left( {\phi _m ,\phi _n } \right) \left\langle {\phi _m ^{\prime \prime },\phi _m } \right\rangle \nonumber \\&+2\alpha _1 \varGamma \left( {w_s ,\phi _m } \right) \left\langle {\psi _{mn} ^{\prime \prime },\phi _m } \right\rangle \nonumber \\&+6\alpha _2 V_{\mathrm{DCEff}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _m \phi _n \psi _{mm} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \nonumber \\&+\,6\alpha _2 V_{\mathrm{DCEff}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _m ^{2}\psi _{mn} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned} K_{nnm_1 }= & {} 2\alpha _1 \left( \varGamma \left( {\phi _n ,\psi _{mn} } \right) \left\langle {w_s ^{\prime \prime },\phi _m } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\psi _{mn} } \right) \left\langle {\phi _n ^{\prime \prime },\phi _m } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {\phi _m ,\phi _n } \right) \left\langle {\phi _n ^{\prime \prime },\phi _m } \right\rangle \right) \nonumber \\&+\,\alpha _1 \varGamma \left( {\phi _n ,\phi _n } \right) \left\langle {\phi _m ^{\prime \prime },\phi _m } \right\rangle \nonumber \\&+2\alpha _1 \varGamma \left( {w_s ,\phi _n } \right) \left\langle {\psi _{mn} ^{\prime \prime },\phi _m } \right\rangle \nonumber \\&+6\alpha _2 V_{\mathrm{DCEff}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _m \phi _n \psi _{mn} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned} K_{km_1 }= & {} 2\alpha _1 \left( \varGamma \left( {\phi _k ,\phi _m } \right) \left\langle {w_s ^{\prime \prime },\phi _m } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _m } \right) \left\langle {\phi _k ^{\prime \prime }\phi _m } \right\rangle +\varGamma \left( {w_s ,\phi _k } \right) \left\langle {\phi _m ^{\prime \prime }\phi _m } \right\rangle \right) \nonumber \\&+\,6\alpha _2 V_{\mathrm{DCEff}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _k \phi _m ^{2}}{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned} K_{mmn_1 }= & {} 2\alpha _1 \left( \varGamma \left( {\phi _m ,\psi _{mn} } \right) \left\langle {w_s ^{\prime \prime },\phi _m } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\psi _{mn} } \right) \left\langle {\phi _m ^{\prime \prime },\phi _m } \right\rangle \right. \nonumber \\&\left. +\varGamma \left( {\phi _m ,\phi _n } \right) \left\langle {\phi _m ^{\prime \prime },\phi _m } \right\rangle \right) \nonumber \\&+\,\alpha _1 \varGamma \left( {\phi _m ,\phi _m } \right) \left\langle {\phi _n ^{\prime \prime },\phi _m } \right\rangle \nonumber \\&+2\alpha _1 \varGamma \left( {w_s ,\phi _m } \right) \left\langle {\psi _{mn} ^{\prime \prime },\phi _m } \right\rangle \nonumber \\&+\,6\alpha _2 V_{\mathrm{DCEff}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _m ^{2}\psi _{mn} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned} K_{mnn_1 }= & {} 2\alpha _1 \left( \varGamma \left( {\phi _m ,\psi _{nn} } \right) \left\langle {w_s ^{\prime \prime },\phi _m } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {\phi _n ,\psi _{mn} } \right) \left\langle {w_s ^{\prime \prime },\phi _m } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\psi _{nn} } \right) \left\langle {\phi _m ^{\prime \prime },\phi _m } \right\rangle \right) \nonumber \\&+\,2\alpha _1 \left( \varGamma \left( {\phi _n ,\phi _n } \right) \left\langle {\phi _m ^{\prime \prime },\phi _m } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\psi _{mn} } \right) \left\langle {\phi _n ^{\prime \prime },\phi _m } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _n } \right) \left\langle {\psi _{mn} ^{\prime \prime },\phi _m } \right\rangle \right) \nonumber \\&+\,2\alpha _1 \left( \varGamma \left( {w_s ,\phi _m } \right) \left\langle {\psi _{nn} ^{\prime \prime },\phi _m } \right\rangle \right. \nonumber \\&\left. +\,2\varGamma \left( {\phi _m ,\phi _n } \right) \left\langle {\phi _n ^{\prime \prime },\phi _m } \right\rangle \right) \nonumber \\&+\,6\alpha _2 V_{\mathrm{DCEff}} \left( \left( {\mathop {\int }\limits _0^1 \frac{\phi _m \phi _n \psi _{mn} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \right. \nonumber \\&\left. +\,\left( {\mathop {\int }\limits _0^1 \frac{\phi _m ^{2}\psi _{nn} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \right) \end{aligned}$$
$$\begin{aligned} K_{nnn_1 }= & {} 2\alpha _1 \varGamma \left( {\phi _n ,\psi _{nn} } \right) \left\langle {w_s ^{\prime \prime },\phi _m } \right\rangle \nonumber \\&+2\alpha _1 \varGamma \left( {w_s ,\psi _{nn} } \right) \left\langle {\phi _n ^{\prime \prime },\phi _m } \right\rangle \nonumber \\&+2\alpha _1 \varGamma \left( {w_s ,\phi _n } \right) \left\langle {\psi _{nn} ^{\prime \prime },\phi _m } \right\rangle \nonumber \\&+\,3\alpha _1 \varGamma \left( {\phi _n ,\phi _n } \right) \left\langle {\phi _n ^{\prime \prime },\phi _m } \right\rangle \nonumber \\&+6\alpha _2 V_{\mathrm{DCEff}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _m \phi _n \psi _{nn} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned} K_{km_2 }= & {} 2\alpha _1 \left( \varGamma \left( {\phi _k ,\phi _m } \right) \left\langle {w_s ^{\prime \prime },\phi _m } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _m } \right) \left\langle {\phi _k ^{\prime \prime },\phi _m } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _k } \right) \left\langle {\phi _m ^{\prime \prime }\phi _m } \right\rangle \right) \nonumber \\&+\,6\alpha _2 V_{\mathrm{DCEff}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _k \phi _m ^{2}}{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned} K_{kn_1 }= & {} 2\alpha _1 \left( \varGamma \left( {\phi _k ,\phi _n } \right) \left\langle {w_s ^{\prime \prime },\phi _m } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _n } \right) \left\langle {\phi _k ^{\prime \prime },\phi _m } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _k } \right) \left\langle {\phi _n ^{\prime \prime },\phi _m } \right\rangle \right) \nonumber \\&+\,6\alpha _2 V_{\mathrm{DCEff}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _k \phi _m \phi _n }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned} S_{kmk_1 }= & {} 2\alpha _1 \left( \varGamma \left( {\phi _k ,\psi _{km} } \right) \left\langle {w_s ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {\phi _m ,\psi _{kk} } \right) \left\langle {w_s ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\psi _{km} } \right) \left\langle {\phi _k ^{\prime \prime },\phi _n } \right\rangle \right) \nonumber \\&+\,2\alpha _1 \left( \varGamma \left( {w_s ,\psi _{kk} } \right) \left\langle {\phi _m ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&+\,\varGamma \left( {\phi _k ,\phi _k } \right) \left\langle {\phi _m ^{\prime \prime },\phi _n } \right\rangle \nonumber \\&\left. +\varGamma \left( {w_s ,\phi _m } \right) \left\langle {\psi _{kk} ^{\prime \prime },\phi _n } \right\rangle \right) \nonumber \\&+\,2\alpha _1 \left( \varGamma \left( {w_s ,\phi _k } \right) \left\langle {\psi _{km} ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,2\varGamma \left( {\phi _k ,\phi _m } \right) \left\langle {\phi _k ^{\prime \prime },\phi _n } \right\rangle \right) \nonumber \\&+\,6\alpha _2 V_{\mathrm{DCEff}} \left( \left( {\mathop {\int }\limits _0^1 \frac{\phi _m \phi _n \psi _{kk} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \right. \nonumber \\&\left. +\,\left( {\mathop {\int }\limits _0^1 \frac{\phi _k \phi _n \psi _{km} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \right) \end{aligned}$$
$$\begin{aligned} S_{knk_1 }= & {} 2\alpha _1 \left( \varGamma \left( {\phi _k ,\psi _{kn} } \right) \left\langle {w_s ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {\phi _n ,\psi _{kk} } \right) \left\langle {w_s ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\psi _{kn} } \right) \left\langle {\phi _k ^{\prime \prime },\phi _n } \right\rangle \right) \nonumber \\&+\,2\alpha _1 \left( \varGamma \left( {w_s ,\psi _{kk} } \right) \left\langle {\phi _n ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {\phi _k ,\phi _k } \right) \left\langle {\phi _n ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _n } \right) \left\langle {\psi _{kk} ^{\prime \prime },\phi _n } \right\rangle \right) \nonumber \\&+\,2\alpha _1 \left( \varGamma \left( {w_s ,\phi _k } \right) \left\langle {\psi _{kn} ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,2\varGamma \left( {\phi _k ,\phi _n } \right) \left\langle {\phi _k ^{\prime \prime },\phi _n } \right\rangle \right) \nonumber \\&+\,6\alpha _2 V_{\mathrm{DCEff}} \left( \left( {\mathop {\int }\limits _0^1 \frac{\phi _n ^{2}\psi _{kk} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \right. \nonumber \\&\left. +\,\left( {\mathop {\int }\limits _0^1 \frac{\phi _k \phi _n \psi _{kn} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \right) \end{aligned}$$
$$\begin{aligned} S_{mmm_1 }= & {} 2\alpha _1 \left( \varGamma \left( {\phi _m ,\psi _{mm} } \right) \left\langle {w_s ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\psi _{mm} } \right) \left\langle {\phi _m ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _m } \right) \left\langle {\psi _{mm} ^{\prime \prime },\phi _n } \right\rangle \right) \nonumber \\&+\,3\alpha _1 \varGamma \left( {\phi _m ,\phi _m } \right) \left\langle {\phi _m ^{\prime \prime },\phi _n } \right\rangle \nonumber \\&+6\alpha _2 V_{\mathrm{DCEff}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _m \phi _n \psi _{mm} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned} S_{mnm_1 }= & {} 2\alpha _1 \left( \varGamma \left( {\phi _m ,\psi _{mn} } \right) \left\langle {w_s ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {\phi _n ,\psi _{mm} } \right) \left\langle {w_s ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\psi _{mn} } \right) \left\langle {\phi _m ^{\prime \prime },\phi _n } \right\rangle \right) \nonumber \\&+\,2\alpha _1 \left( \varGamma \left( {w_s ,\psi _{mm} } \right) \left\langle {\phi _n ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {\phi _m ,\phi _m } \right) \left\langle {\phi _n ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _n } \right) \left\langle {\psi _{mm} ^{\prime \prime },\phi _n } \right\rangle \right) \nonumber \\&+\,2\alpha _1 \left( \varGamma \left( {w_s ,\phi _m } \right) \left\langle {\psi _{mn} ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,2\varGamma \left( {\phi _m ,\phi _n } \right) \left\langle {\phi _m ^{\prime \prime },\phi _n } \right\rangle \right) \nonumber \\&+\,6\alpha _2 V_{\mathrm{DCEff}} \left( \left( {\mathop {\int }\limits _0^1 \frac{\phi _n ^{2}\psi _{mm} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \right. \nonumber \\&\left. +\,\left( {\mathop {\int }\limits _0^1 \frac{\phi _m \phi _n \psi _{mn} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \right) \end{aligned}$$
$$\begin{aligned} S_{nnm_1 }= & {} 2\alpha _1 \left( \varGamma \left( {\phi _n ,\psi _{mn} } \right) \left\langle {w_s ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\psi _{mn} } \right) \left\langle {\phi _n ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _n } \right) \left\langle {\psi _{mn} ^{\prime \prime },\phi _n } \right\rangle \right) \nonumber \\&+\,\alpha _1 \left( 2\varGamma \left( {\phi _m ,\phi _n } \right) \left\langle {\phi _n ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {\phi _n ,\phi _n } \right) \left\langle {\phi _m ^{\prime \prime },\phi _n } \right\rangle \right) \nonumber \\&+6\alpha _2 V_{\mathrm{DCEff}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _n ^{2}\psi _{mn} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned} S_{km_1 }= & {} 2\alpha _1 \left( \varGamma \left( {\phi _k ,\phi _m } \right) \left\langle {w_s ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _m } \right) \left\langle {\phi _k ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _k } \right) \left\langle {\phi _m ^{\prime \prime },\phi _n } \right\rangle \right) \nonumber \\&+\,6\alpha _2 V_{\mathrm{DCEff}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _k \phi _m \phi _n }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned} S_{mmn_1 }= & {} 2\alpha _1 \left( \varGamma \left( {\phi _m ,\psi _{mn} } \right) \left\langle {w_s ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\psi _{mn} } \right) \left\langle {\phi _m ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {\phi _m ,\phi _n } \right) \left\langle {\phi _m ^{\prime \prime },\phi _n } \right\rangle \right) \nonumber \\&+\,\alpha _1 \left( \varGamma \left( {\phi _m ,\phi _m } \right) \left\langle {\phi _n ^{\prime \prime }\phi _n } \right\rangle \right. \nonumber \\&\left. +\,2\varGamma \left( {w_s ,\phi _m } \right) \left\langle {\psi _{mn} ^{\prime \prime },\phi _n } \right\rangle \right) \nonumber \\&+6\alpha _2 V_{\mathrm{DCEff}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _m \phi _n \psi _{mn} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned}&S_{mnn_1 } =2\alpha _1 \left( \varGamma \left( {\phi _m ,\psi _{nn} } \right) \left\langle {w_s ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {\phi _n ,\psi _{mn} } \right) \left\langle {w_s ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\psi _{nn} } \right) \left\langle {\phi _m ^{\prime \prime },\phi _n } \right\rangle \right) \nonumber \\&+\,2\alpha _1 \left( \varGamma \left( {\phi _n ,\phi _n } \right) \left\langle {\phi _m ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\psi _{mn} } \right) \left\langle {\phi _n ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _n } \right) \left\langle {\psi _{mn} ^{\prime \prime },\phi _n } \right\rangle \right) \nonumber \\&+\,2\alpha _1 \left( \varGamma \left( {w_s ,\phi _m } \right) \left\langle {\psi _{nn} ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,2\varGamma \left( {\phi _m ,\phi _n } \right) \left\langle {\phi _n ^{\prime \prime },\phi _n } \right\rangle \right) \nonumber \\&+\,6\alpha _2 V_{\mathrm{DCEff}} \left( \left( {\mathop {\int }\limits _0^1 \frac{\phi _n ^{2}\psi _{mn} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \right. \nonumber \\&\left. +\,\left( {\mathop {\int }\limits _0^1 \frac{\phi _m \phi _n \psi _{nn} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \right) \end{aligned}$$
$$\begin{aligned} S_{nnn_1 }= & {} 2\alpha _1 \left( \varGamma \left( {\phi _n ,\psi _{nn} } \right) \left\langle {w_s ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\psi _{nn} } \right) \left\langle {\phi _n ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _n } \right) \left\langle {\psi _{nn} ^{\prime \prime },\phi _n } \right\rangle \right) \nonumber \\&+\,3\alpha _1 \varGamma \left( {\phi _n ,\phi _n } \right) \left\langle {\phi _n ^{\prime \prime },\phi _n } \right\rangle \nonumber \\&+6\alpha _2 V_{\mathrm{DCEff}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _n ^{2}\psi _{nn} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned} S_{kn_1 }= & {} 2\alpha _1 \left( \varGamma \left( {\phi _k ,\phi _n } \right) \left\langle {w_s ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _n } \right) \left\langle {\phi _k ^{\prime \prime },\phi _n } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _k } \right) \left\langle {\phi _n ^{\prime \prime },\phi _n } \right\rangle \right) \nonumber \\&+\,6\alpha _2 V_{\mathrm{DCEff}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _k \phi _n ^{2}}{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned} T_{mm}= & {} 2\alpha _1 \left( \varGamma \left( {\phi _m ,\phi _m } \right) \left\langle {w_s ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&\left. +\,2\varGamma \left( {w_s ,\phi _m } \right) \left\langle {\phi _m ^{\prime \prime }\phi _k } \right\rangle \right) \nonumber \\&+6\alpha _2 V_{\mathrm{DCEff}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _k \phi _m ^{2}}{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned} T_{nm}= & {} 2\alpha _1 \left( \varGamma \left( {\phi _m ,\phi _n } \right) \left\langle {w_s ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _n } \right) \left\langle {\phi _m ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _m } \right) \left\langle {\phi _n ^{\prime \prime }\phi _k } \right\rangle \right) \nonumber \\&+\,6\alpha _2 V_{\mathrm{DCEff}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _k \phi _m \phi _n }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned} T_{nn}= & {} 2\alpha _1 \left( \varGamma \left( {\phi _n ,\phi _n } \right) \left\langle {w_s ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&\left. +\,2\varGamma \left( {w_s ,\phi _n } \right) \left\langle {\phi _n ^{\prime \prime },\phi _k } \right\rangle \right) \nonumber \\&+6\alpha _2 V_{\mathrm{DCEff}} \left( {\mathop {\int }\limits _0^1 \frac{\phi _k \phi _n ^{2}}{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \end{aligned}$$
$$\begin{aligned} T_{kkk}= & {} 2\alpha _1 \left( \varGamma \left( {\phi _k ,\psi _{kk} } \right) \left\langle {w_s ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {\phi _k ,\psi _{k_2 } } \right) \left\langle {w_s ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\psi _{kk} } \right) \left\langle {\phi _k ^{\prime \prime },\phi _k } \right\rangle \right) \nonumber \\&+\,2\alpha _1 \left( \varGamma \left( {w_s ,\psi _{k_2 } } \right) \left\langle {\phi _k ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _k } \right) \left\langle {\psi _{kk} ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _k } \right) \left\langle {\psi _{k_2 } ^{\prime \prime },\phi _k } \right\rangle \right) \nonumber \\&+\,3\alpha _1 \varGamma \left( {\phi _k ,\phi _k } \right) \left\langle {\phi _k ^{\prime \prime },\phi _k } \right\rangle \nonumber \\&+6\alpha _2 V_{\mathrm{DCEff}} \left( \left( {\mathop {\int }\limits _0^1 \frac{\phi _k \left( x \right) ^{2}\psi _{kk} \left( x \right) }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \right. \nonumber \\&\left. +\,\left( {\mathop {\int }\limits _0^1 \frac{\phi _k \left( x \right) ^{2}\psi _{k_2 } \left( x \right) }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \right) \end{aligned}$$
$$\begin{aligned} T_{kmm}= & {} 2\alpha _1 \left( \varGamma \left( {\phi _k ,\psi _{mm} } \right) \left\langle {w_s ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {\phi _m ,\psi _{km} } \right) \left\langle {w_s ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\psi _{mm} } \right) \left\langle {\phi _k ^{\prime \prime },\phi _k } \right\rangle \right) \nonumber \\&+\,2\alpha _1 \left( \varGamma \left( {\phi _m ,\phi _m } \right) \left\langle {\phi _k ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&+\,\varGamma \left( {w_s ,\psi _{km} } \right) \left\langle {\phi _m ^{\prime \prime },\phi _k } \right\rangle \nonumber \\&\left. +\varGamma \left( {w_s ,\phi _k } \right) \left\langle {\psi _{mm} ^{\prime \prime },\phi _k } \right\rangle \right) \nonumber \\&+\,2\alpha _1 \left( \varGamma \left( {w_s ,\phi _m } \right) \left\langle {\psi _{km} ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&\left. +\,2\varGamma \left( {\phi _k ,\phi _m } \right) \left\langle {\phi _m ^{\prime \prime },\phi _k } \right\rangle \right) \nonumber \\&+\,6\alpha _2 V_{\mathrm{DCEff}} \left( \left( {\mathop {\int }\limits _0^1 \frac{\phi _k \phi _m \psi _{km} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \right. \nonumber \\&\left. +\,\left( {\mathop {\int }\limits _0^1 \frac{\phi _k ^{2}\psi _{mm} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \right) \end{aligned}$$
$$\begin{aligned} T_{knm}= & {} 2\alpha _1 \left( \varGamma \left( {\phi _k ,\psi _{mn} } \right) \left\langle {w_s ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {\phi _m ,\psi _{kn} } \right) \left\langle {w_s ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\psi _{mn} } \right) \left\langle {\phi _k ^{\prime \prime },\phi _k } \right\rangle \right) \nonumber \\&+\,2\alpha _1 \left( \varGamma \left( {\phi _m ,\phi _n } \right) \left\langle {\phi _k ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\psi _{kn} } \right) \left\langle {\phi _m ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {\phi _k ,\phi _n } \right) \left\langle {\phi _m ^{\prime \prime },\phi _k } \right\rangle \right) \nonumber \\&+\,2\alpha _1 \left( \varGamma \left( {\phi _k ,\phi _m } \right) \left\langle {\phi _n ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&+\varGamma \left( {w_s ,\phi _m } \right) \left\langle {\psi _{kn} ^{\prime \prime },\phi _k } \right\rangle \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _k } \right) \left\langle {\psi _{mn} ^{\prime \prime },\phi _k } \right\rangle \right) \nonumber \\&+\,6\alpha _2 V_{\mathrm{DCEff}} \left( \left( {\mathop {\int }\limits _0^1 \frac{\phi _k \phi _m \psi _{kn} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \right. \nonumber \\&\left. +\,\left( {\mathop {\int }\limits _0^1 \frac{\phi _k ^{2}\psi _{mn} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \right) \end{aligned}$$
$$\begin{aligned} T_{kmn}= & {} 2\alpha _1 \left( \varGamma \left( {\phi _k ,\psi _{mn} } \right) \left\langle {w_s ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {\phi _n ,\psi _{km} } \right) \left\langle {w_s ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\psi _{mn} } \right) \left\langle {\phi _k ^{\prime \prime },\phi _k } \right\rangle \right) \nonumber \\&+\,2\alpha _1 \left( \varGamma \left( {\phi _m ,\phi _n } \right) \left\langle {\phi _k ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {\phi _k ,\phi _n } \right) \left\langle {\phi _m ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\psi _{km} } \right) \left\langle {\phi _n ^{\prime \prime },\phi _k } \right\rangle \right) \nonumber \\&+\,2\alpha _1 \left( \varGamma \left( {\phi _k ,\phi _m } \right) \left\langle {\phi _n ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _n } \right) \left\langle {\psi _{km} ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _k } \right) \left\langle {\psi _{mn} ^{\prime \prime },\phi _k } \right\rangle \right) \nonumber \\&+\,6\alpha _2 V_{\mathrm{DCEff}} \left( \left( {\mathop {\int }\limits _0^1 \frac{\phi _k \phi _n \psi _{km} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \right. \nonumber \\&\left. +\,\left( {\mathop {\int }\limits _0^1 \frac{\phi _k ^{2}\psi _{mn} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \right) \end{aligned}$$
$$\begin{aligned} T_{knn}= & {} 2\alpha _1 \left( \varGamma \left( {\phi _k ,\psi _{nn} } \right) \left\langle {w_s ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {\phi _n ,\psi _{kn} } \right) \left\langle {w_s ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\psi _{nn} } \right) \left\langle {\phi _k ^{\prime \prime },\phi _k } \right\rangle \right) \nonumber \\&+\,2\alpha _1 \left( \varGamma \left( {\phi _n ,\phi _n } \right) \left\langle {\phi _k ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\psi _{kn} } \right) \left\langle {\phi _n ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&\left. +\,\varGamma \left( {w_s ,\phi _k } \right) \left\langle {\psi _{nn} ^{\prime \prime },\phi _k } \right\rangle \right) \nonumber \\&+\,2\alpha _1 \left( \varGamma \left( {w_s ,\phi _n } \right) \left\langle {\psi _{kn} ^{\prime \prime },\phi _k } \right\rangle \right. \nonumber \\&\left. +2\varGamma \left( {\phi _k ,\phi _n } \right) \left\langle {\phi _n ^{\prime \prime },\phi _k } \right\rangle \right) \nonumber \\&+\,6\alpha _2 V_{\mathrm{DCEff}} \left( \left( {\mathop {\int }\limits _0^1 \frac{\phi _k \phi _n \psi _{kn} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \right. \nonumber \\&\left. +\,\left( {\mathop {\int }\limits _0^1 \frac{\phi _k ^{2}\psi _{nn} }{\left( {1-w_s } \right) ^{4}}\hbox {d}x} \right) \right) \end{aligned}$$

Appendix D. Cartesian form of the modulation equations Eqs. (19.1)–(19.6)

To express the modulation equation in complex Cartesian form, we use the complex amplitude definition given by

$$\begin{aligned}&A_m =\frac{1}{2}\left( {p_m -\hbox {i }q_m } \right) e^{\mathrm {i}\lambda _1 t}\end{aligned}$$
$$\begin{aligned}&A_n =\frac{1}{2}\left( {p_n -\hbox {i }q_n } \right) e^{\mathrm {i}\lambda _2 t} \end{aligned}$$
$$\begin{aligned}&A_k =\frac{1}{2}\left( {p_k -\hbox {i }q_k } \right) e^{\mathrm {i}\lambda _3 t} \end{aligned}$$

Equations (D.1)–(D.3) are substituted in Eqs. (10.1)–(10.3) and Eqs. (14.1)–(14.3), and using Eqs. (16.1)–(16.3) with the method of reconstitution defined by Eq. (17) while separating the imaginary and real parts yields

$$\begin{aligned} \frac{\mathrm{d}p_m }{\mathrm{d}t}= & {} -\mu p_m \nonumber \\&+\,\left( {\frac{F_{k_1 } K_{km_1 } }{8\omega _k^2 \omega _m }-\frac{\mu ^{2}}{2\omega _m }-\sigma _1 -\frac{F_{m_2 } }{2\omega _m }} \right) q_m \nonumber \\&+\,\left( {\frac{F_{k_1 } K_{kn_1 } }{8\omega _k^2 \omega _m }-\frac{F_{mn_1 } }{2\omega _m }} \right) q_n +\,\frac{\mu F_{m_1 } }{2\omega _m^2 } \nonumber \\&+\,\left( {\frac{F_{mk} }{\omega _m }-\frac{F_{n_1 } K_{kn_1 } }{8\omega _m \omega _n^2 }-\frac{F_{m_1 } K_{km_1 } }{8\omega _m^3 }} \right) q_k \nonumber \\&-\left( {\frac{K_{mmn_1 } }{4\omega _m }+\frac{K_{kn_1 } R_{k_1 } }{16\omega _k^2 \omega _m }} \right) p_n q_m p_m \nonumber \\&+\,\left( {\frac{\mu K_{km_1 } }{8\omega _m^2 }-\frac{\mu K_{km_1 } }{8\omega _k \omega _m }+\frac{\mu R_{m_1 } }{8\omega _m^2 }} \right) \left( {p_k p_m +q_k q_m } \right) \nonumber \\&+\,\left( {\frac{\mu K_{kn_1 } }{8\omega _m \omega _n }-\frac{\mu K_{kn_1 } }{8\omega _k \omega _m }+\frac{\mu R_{m_2 } }{8\omega _m^2 }} \right) \left( {p_k p_n +q_k q_n } \right) \nonumber \\&-\left( {\frac{K_{mnn_1 } }{8\omega _m }+\frac{K_{nnm_1 } }{8\omega _m }+\frac{K_{kn_1 } R_{k_2 } }{32\omega _k^2 \omega _m }+\frac{K_{km_1 } R_{k_3 } }{32\omega _k^2 \omega _m }} \right) q_m q_n^2 \nonumber \\&-\left( {\frac{K_{nnm_1 } }{4\omega _m }+\frac{K_{km_1 } R_{k_3 } }{16\omega _k^2 \omega _m }} \right) p_n q_n p_m \nonumber \\&+\,\left( {\frac{K_{mmn_1 } }{8\omega _m }-\frac{K_{mnm_1 } }{8\omega _m }+\frac{K_{kn_1 } R_{k_1 } }{32\omega _k^2 \omega _m }-\frac{K_{km_1 } R_{k_2 } }{32\omega _k^2 \omega _m }} \right) \nonumber \\&\left( {q_n p_m^2 -q_m^2 q_n } \right) \nonumber \\&+\,\left( {\frac{K_{nnm_1 } }{8\omega _m }-\frac{K_{mnn_1 } }{8\omega _m }-\frac{K_{kn_1 } R_{k_2 } }{32\omega _k^2 \omega _m }+\frac{K_{km_1 } R_{k_3 } }{32\omega _k^2 \omega _m }} \right) p_n^2 q_m \nonumber \\&+\,\frac{R_{m_1 } }{4\omega _m }\left( {p_k q_m -q_k p_m } \right) \nonumber \\&-\left( {q_m p_k^2 +q_k^2 q_m } \right) \left( {\frac{K_{kmk_1 } }{8\omega _m }+\frac{K_{kn_1 } R_{n_1 } }{32\omega _m \omega _n^2 }+\frac{K_{km_1 } R_{m_1 } }{32\omega _m^3 }} \right) \nonumber \\&-\left( {\frac{K_{mmm_1 } }{8\omega _m }+\frac{K_{km_1 } R_{k_1 } }{32\omega _k^2 \omega _m }} \right) \left( {q_m^3 +p_m^2 q_m } \right) \nonumber \\&-\left( {q_n p_k^2 +q_k^2 q_n } \right) \left( {\frac{K_{knk_1 } }{8\omega _m }+\frac{K_{kn_1 } R_{n_2 } }{32\omega _m \omega _n^2 }+\frac{K_{km_1 } R_{m_2 } }{32\omega _m^3 }} \right) \nonumber \\&+\,\frac{R_{m_2 } }{4\omega _m }\left( {p_k q_n -p_n q_k } \right) \end{aligned}$$
$$\begin{aligned} \frac{\mathrm{d}q_m }{\mathrm{d}t}= & {} -\mu q_m +\left( {\frac{\mu ^{2}}{2\omega _m }+\sigma _1 -\frac{F_{m_2 } }{2\omega _m }+\frac{F_{k_1 } K_{km_1 } }{8\omega _k^2 \omega _m }} \right) p_m \nonumber \\&+\,\left( {\frac{F_{k_1 } K_{kn_1 } }{8\omega _k^2 \omega _m }-\frac{F_{mn_1 } }{2\omega _m }} \right) p_n +\frac{F_{m_1 } }{\omega _m } \nonumber \\&+\,\left( {\frac{F_{n_1 } K_{kn_1 } }{8\omega _m \omega _n^2 }-\frac{F_{mk} }{\omega _m }+\frac{F_{m_1 } K_{km_1 } }{8\omega _m^3 }} \right) p_k \nonumber \\&+\,\frac{R_{m_1 } }{4\omega _m }\left( {p_k p_m +q_k q_m } \right) +\frac{R_{m_2 } }{4\omega _m }\left( {p_k p_n +q_k q_n } \right) \nonumber \\&+\,\left( {\frac{\mu K_{kn_1 } }{8\omega _k \omega _m }-\frac{\mu K_{kn_1 } }{8\omega _m \omega _n }-\frac{\mu R_{m_2 } }{8\omega _m^2 }} \right) \left( {p_k q_n -p_n q_k } \right) \nonumber \\&+\,\left( {\frac{\mu K_{km_1 } }{8\omega _k \omega _m }-\frac{\mu K_{km_1 } }{8\omega _m^2 }-\frac{\mu R_{m_1 } }{8\omega _m^2 }} \right) \left( {p_k q_m -q_k p_m } \right) \nonumber \\&+\,\left( {\frac{K_{nnm_1 } }{4\omega _m }+\frac{K_{km_1 } R_{k_3 } }{16\omega _k^2 \omega _m }} \right) p_n q_m q_n \nonumber \\&+\,\left( {\frac{K_{kn_1 } R_{k_2 } }{32\omega _k^2 \omega _m }+\frac{K_{km_1 } R_{k_3 } }{32\omega _k^2 \omega _m }} \right) \left( {p_n^2 p_m +q_n^2 p_m } \right) \nonumber \\&+\,\left( {\frac{K_{kn_1 } R_{k_1 } }{32\omega _k^2 \omega _m }+\frac{K_{mmn_1 } }{8\omega _m }} \right) \left( {p_n p_m^2 -p_n q_m^2 } \right) \nonumber \\&+\,\left( {\frac{K_{mnn_1 } }{8\omega _m }+\frac{K_{nnm_1 } }{8\omega _m }} \right) \left( {p_n^2 p_m -q_n^2 p_m } \right) \nonumber \\&+\,\left( {\frac{K_{mnm_1 } }{8\omega _m }\frac{K_{km_1 } R_{k_2 } }{32\omega _k^2 \omega _m }} \right) \left( {p_n p_m^2 +p_n q_m^2 } \right) \nonumber \\&+\,\left( {\frac{K_{mmm_1 } }{8\omega _m }+\frac{K_{km_1 } R_{k_1 } }{32\omega _k^2 \omega _m }} \right) \left( {p_m^3 +q_m^2 p_m } \right) \nonumber \\&+\,\left( {\frac{K_{mmn_1 } }{4\omega _m }+\frac{K_{kn_1 } R_{k_1 } }{16\omega _k^2 \omega _m }} \right) q_m q_n p_m \nonumber \\&+\,\left( {\frac{K_{nnn_1 } }{8\omega _m }+\frac{K_{kn_1 } R_{k_3 } }{32\omega _k^2 \omega _m }} \right) \left( {p_n^3 +q_n^2 p_n } \right) \nonumber \\&+\,\left( {\frac{K_{kmk_1 } }{8\omega _m }+\frac{K_{kn_1 } R_{n_1 } }{32\omega _m \omega _n^2 }+\frac{K_{km_1 } R_{m_1 } }{32\omega _m^3 }} \right) \nonumber \\&\left( {p_m p_k^2 +p_m q_k^2 } \right) \nonumber \\&+\,\left( {\frac{K_{knk_1 } }{8\omega _m }+\frac{K_{kn_1 } R_{n_2 } }{32\omega _m \omega _n^2 }+\frac{K_{km_1 } R_{m_2 } }{32\omega _m^3 }} \right) \nonumber \\&\left( {p_n p_k^2 +p_n q_k^2 } \right) \end{aligned}$$
$$\begin{aligned} \frac{\mathrm{d}p_n }{\mathrm{d}t}= & {} -\mu p_n \nonumber \\&+\,q_n \left( {\frac{F_{k_1 } S_{kn_1 } }{8\omega _k^2 \omega _n }-\frac{\mu ^{2}}{2\omega _n }-\sigma _1 +\sigma _2 -\frac{F_{n_2 } }{2\omega _n }} \right) \nonumber \\&+\,\left( {\frac{F_{k_1 } S_{km_1 } }{8\omega _k^2 \omega _n }-\frac{F_{mn_1 } }{2\omega _n }} \right) q_m \nonumber \\&+\,\left( {\frac{F_{nk} }{\omega _n }-\frac{F_{m_1 } S_{km_1 } }{8\omega _m^2 \omega _n }-\frac{F_{n_1 } S_{kn_1 } }{8\omega _n^3 }} \right) q_k \nonumber \\&+\,\left( {\frac{\mu R_{n_2 } }{8\omega _n^2 }-\frac{\mu S_{kn_1 } }{8\omega _k \omega _n }+\frac{\mu S_{kn_1 } }{8\omega _n^2 }} \right) \left( {p_k p_n +q_k q_n } \right) \nonumber \\&+\,\left( {\frac{\mu R_{n_1 } }{8\omega _n^2 }-\frac{\mu S_{km_1 } }{8\omega _k \omega _n }+\frac{\mu S_{km_1 } }{8\omega _m \omega _n }} \right) \left( {p_k p_m +q_k q_m } \right) \nonumber \\&+\,\frac{R_{n_1 } }{4\omega _n }\left( {p_k q_m -q_k p_m } \right) \nonumber \\&+\,\frac{R_{n_2 } }{4\omega _n }\left( {p_k q_n -p_n q_k } \right) \nonumber \\&+\,\left( {\frac{S_{mmn_1 } }{8\omega _n }-\frac{S_{mnm_1 } }{8\omega _n }-\frac{R_{k_2 } S_{km_1 } }{32\omega _k^2 \omega _n }++\frac{R_{k_1 } S_{kn_1 } }{32\omega _k^2 \omega _n }} \right) p_m^2 q_n \nonumber \\&-\left( {\frac{R_{k_1 } S_{km_1 } }{32\omega _k^2 \omega _n }+\frac{S_{mmm_1 } }{8\omega _n }} \right) \left( {q_m^3 +p_m^2 q_m } \right) \nonumber \\&-\left( {\frac{R_{k_1 } S_{kn_1 } }{16\omega _k^2 \omega _n }+\frac{S_{mmn_1 } }{4\omega _n }} \right) p_m p_n q_m \nonumber \\&-\left( {\frac{R_{k_3 } S_{kn_1 } }{32\omega _k^2 \omega _n }+\frac{S_{nnn_1 } }{8\omega _n }} \right) \left( {q_n^3 +p_n^2 q_n } \right) \nonumber \\&-\left( {\frac{R_{k_3 } S_{km_1 } }{16\omega _k^2 \omega _n }+\frac{S_{nnm_1 } }{4\omega _n }} \right) p_n q_n p_m \nonumber \\&+\,\left( {\frac{R_{k_2 } S_{km_1 } }{32\omega _k^2 \omega _n }+\frac{S_{mmn_1 } }{8\omega _n }+\frac{S_{mnm_1 } }{8\omega _n }+\frac{R_{k_1 } S_{kn_1 } }{32\omega _k^2 \omega _n }} \right) q_m^2 q_n \nonumber \\&-\left( {\frac{R_{k_3 } S_{km_1 } }{32\omega _k^2 \omega _n }+\frac{S_{mnn_1 } }{8\omega _n }+\frac{S_{nnm_1 } }{8\omega _n }+\frac{R_{k_2 } S_{kn_1 } }{32\omega _k^2 \omega _n }} \right) q_m q_n^2 \nonumber \\&+\,p_n^2 q_m \left( {\frac{R_{k_3 } S_{km_1 } }{32\omega _k^2 \omega _n }-\frac{S_{mnn_1 } }{8\omega _n }+\frac{S_{nnm_1 } }{8\omega _n }-\frac{R_{k_2 } S_{kn_1 } }{32\omega _k^2 \omega _n }} \right) \nonumber \\&-\left( {\frac{R_{m_1 } S_{km_1 } }{32\omega _m^2 \omega _n }+\frac{S_{kmk_1 } }{8\omega _n }+\frac{R_{n_1 } S_{kn_1 } }{32\omega _n^3 }} \right) \left( {q_m p_k^2 +q_k^2 q_m } \right) \nonumber \\&-\left( {\frac{R_{m_2 } S_{km_1 } }{32\omega _m^2 \omega _n }+\frac{S_{knk_1 } }{8\omega _n }+\frac{R_{n_2 } S_{kn_1 } }{32\omega _n^3 }} \right) \nonumber \\&\left( {q_n p_k^2 +q_k^2 q_n } \right) +\frac{\mu F_{n_1 } }{2\omega _n^2 } \end{aligned}$$
$$\begin{aligned} \frac{\mathrm{d}q_n }{\mathrm{d}t}= & {} -\mu q_n \nonumber \\&+\,p_n \left( {\frac{\mu ^{2}}{2\omega _n }+\sigma _1 -\sigma _2 -\frac{F_{n_2 } }{2\omega _n }+\frac{F_{k_1 } S_{kn_1 } }{8\omega _k^2 \omega _n }} \right) \nonumber \\&+\,\left( {\frac{F_{k_1 } S_{km_1 } }{8\omega _k^2 \omega _n }-\frac{F_{mn_1 } }{2\omega _n }} \right) p_m \nonumber \\&+\,\left( {\frac{F_{m_1 } S_{km_1 } }{8\omega _m^2 \omega _n }-\frac{F_{nk} }{\omega _n }+\frac{F_{n_1 } S_{kn_1 } }{8\omega _n^3 }} \right) p_k \nonumber \\&+\,\left( {\frac{\mu R_{n_2 } }{8\omega _n^2 }-\frac{\mu S_{kn_1 } }{8\omega _k \omega _n }+\frac{\mu S_{kn_1 } }{8\omega _n^2 }} \right) \left( {p_n q_k -p_k q_n } \right) \nonumber \\&+\,\left( {\frac{\mu R_{n_1 } }{8\omega _n^2 }-\frac{\mu S_{km_1 } }{8\omega _k \omega _n }+\frac{\mu S_{km_1 } }{8\omega _m \omega _n }} \right) \left( {q_k p_m -p_k q_m } \right) \nonumber \\&+\,\frac{R_{n_1 } }{4\omega _n }\left( {p_k p_m +q_k q_m } \right) \nonumber \\&+\,\frac{R_{n_2 } }{4\omega _n }\left( {p_k p_n +q_k q_n } \right) \nonumber \\&+\,\left( {\frac{R_{k_2 } S_{km_1 } }{32\omega _k^2 \omega _n }+\frac{S_{mnm_1 } }{8\omega _n }} \right) \left( {p_n p_m^2 +p_n q_m^2 } \right) \nonumber \\&+\,\left( {\frac{S_{mmn_1 } }{8\omega _n }+\frac{R_{k_1 } S_{kn_1 } }{32\omega _k^2 \omega _n }} \right) \left( {p_n p_m^2 -p_n q_m^2 } \right) \nonumber \\&+\,\left( {\frac{S_{mnn_1 } }{8\omega _n }+\frac{R_{k_2 } S_{kn_1 } }{32\omega _k^2 \omega _n }} \right) \left( {p_m p_n^2 +p_m q_n^2 } \right) \nonumber \\&+\,\left( {\frac{R_{k_3 } S_{km_1 } }{32\omega _k^2 \omega _n }+\frac{S_{nnm_1 } }{8\omega _n }} \right) \left( {p_m p_n^2 -p_m q_n^2 } \right) \nonumber \\&+\,\left( {\frac{R_{k_1 } S_{kn_1 } }{16\omega _k^2 \omega _n }+\frac{S_{mmn_1 } }{4\omega _n }} \right) p_m q_m q_n \nonumber \\&+\,\left( {\frac{R_{k_3 } S_{km_1 } }{16\omega _k^2 \omega _n }+\frac{S_{nnm_1 } }{4\omega _n }} \right) p_n q_m q_n \nonumber \\&+\,\left( {p_m^3 +q_m^2 p_m } \right) \left( {\frac{R_{k_1 } S_{km_1 } }{32\omega _k^2 \omega _n }+\frac{S_{mmm_1 } }{8\omega _n }} \right) \nonumber \\&+\,\left( {\frac{R_{k_3 } S_{kn_1 } }{32\omega _k^2 \omega _n }+\frac{S_{nnn_1 } }{8\omega _n }} \right) \left( {p_n^3 +q_n^2 p_n } \right) \nonumber \\&+\,\left( {\frac{R_{m_1 } S_{km_1 } }{32\omega _m^2 \omega _n }+\frac{S_{kmk_1 } }{8\omega _n }+\frac{R_{n_1 } S_{kn_1 } }{32\omega _n^3 }} \right) \nonumber \\&\left( {p_m p_k^2 +p_m q_k^2 } \right) \nonumber \\&+\,\left( {\frac{R_{m_2 } S_{km_1 } }{32\omega _m^2 \omega _n }+\frac{S_{knk_1 } }{8\omega _n }+\frac{R_{n_2 } S_{kn_1 } }{32\omega _n^3 }} \right) \nonumber \\&\left( {p_n p_k^2 +p_n q_k^2 } \right) +\frac{F_{n_1 } }{\omega _n } \end{aligned}$$
$$\begin{aligned} \frac{\mathrm{d}p_k }{\mathrm{d}t}= & {} -\mu p_k +\left( {2\sigma _2 -2\sigma _1 +\sigma _3 -\frac{\mu ^{2}}{2\omega _k }} \right) q_k \nonumber \\&+\,q_m \left( {\frac{F_{mk} }{\omega _k }-\frac{F_{m_1 } T_{mm} }{8\omega _k \omega _m^2 }-\frac{F_{n_1 } T_{nm} }{8\omega _k \omega _n^2 }} \right) \nonumber \\&+\,q_n \left( {\frac{F_{nk} }{\omega _k }-\frac{F_{m_1 } T_{nm} }{8\omega _k \omega _m^2 }-\frac{F_{n_1 } T_{nn} }{8\omega _k \omega _n^2 }} \right) \nonumber \\&-\left( {\frac{T_{kmm} }{8\omega _k }+\frac{R_{m_1 } T_{mm} }{32\omega _k \omega _m^2 }+\frac{R_{n_1 } T_{nm} }{32\omega _k \omega _n^2 }} \right) \nonumber \\&\left( {q_k p_m^2 +q_k q_m^2 } \right) \nonumber \\&+\,\left( {\frac{\mu R_{k_2 } }{8\omega _k^2 }-\frac{\mu T_{nm} }{8\omega _k \omega _m }-\frac{\mu T_{nm} }{8\omega _k \omega _n }} \right) \left( {p_m p_n -q_m q_n } \right) \nonumber \\&-\frac{R_{k_2 } }{4\omega _k }\left( {p_n q_m +p_m q_n } \right) -\frac{R_{k_1 } }{2\omega _k }p_m q_m \nonumber \\&+\,\left( {\frac{\mu R_{k_1 } }{8\omega _k^2 }-\frac{\mu T_{mm} }{8\omega _k \omega _m }} \right) \left( {p_m^2 -q_m^2 } \right) \nonumber \\&+\,\left( {\frac{\mu R_{k_3 } }{8\omega _k^2 }-\frac{\mu T_{nn} }{8\omega _k \omega _n }} \right) \left( {p_n^2 -q_n^2 } \right) \nonumber \\&-\frac{T_{kkk} }{8\omega _k }\left( {p_k^2 q_k +q_k^3 } \right) \nonumber \\&-\left( \frac{T_{kmn} }{8\omega _k }+\frac{T_{knm} }{8\omega _k }+\frac{R_{m_2 } T_{mm} }{32\omega _k \omega _m^2 }+\frac{R_{m_1 } T_{nm} }{32\omega _k \omega _m^2 }\right. \nonumber \\&\left. +\,\frac{R_{n_2 } T_{nm} }{32\omega _k \omega _n^2 }+\frac{R_{n_1 } T_{nn} }{32\omega _k \omega _n^2 } \right) \left( {q_k p_m p_n +q_k q_m q_n } \right) \nonumber \\&+\,\left( \frac{T_{kmn} }{8\omega _k }-\frac{T_{knm} }{8\omega _k }+\frac{R_{m_2 } T_{mm} }{32\omega _k \omega _m^2 }-\frac{R_{m_1 } T_{nm} }{32\omega _k \omega _m^2 }+\frac{R_{n_2 } T_{nm} }{32\omega _k \omega _n^2 }\right. \nonumber \\&\left. -\frac{R_{n_1 } T_{nn} }{32\omega _k \omega _n^2 } \right) \left( {p_k p_m q_n -p_k p_n q_m } \right) \nonumber \\&-\left( {\frac{T_{knn} }{8\omega _k }+\frac{R_{m_2 } T_{nm} }{32\omega _k \omega _m^2 }+\frac{R_{n_2 } T_{nn} }{32\omega _k \omega _n^2 }} \right) \left( {p_n^2 q_k +q_n^2 q_k } \right) \nonumber \\&-\frac{R_{k_3 } }{2\omega _k }p_n q_n +\frac{\mu F_{k_1 } }{2\omega _k^2 } \\ \frac{\mathrm{d}q_k }{\mathrm{d}t}= & {} -\mu q_k +\left( {\frac{\mu ^{2}}{2\omega _k }+2\sigma _1 -2\sigma _2 -\sigma _3 } \right) p_k \nonumber \\&+\,\left( {\frac{F_{m_1 } T_{mm} }{8\omega _k \omega _m^2 }+\frac{F_{n_1 } T_{nm} }{8\omega _k \omega _n^2 }-\frac{F_{mk} }{\omega _k }} \right) p_m \nonumber \end{aligned}$$
$$\begin{aligned}&+\,\left( {\frac{\mu R_{k_1 } }{4\omega _k^2 }-\frac{\mu T_{mm} }{4\omega _k \omega _m }} \right) p_m q_m \nonumber \\&+\,\left( {\frac{\mu R_{k_3 } }{4\omega _k^2 }-\frac{\mu T_{nn} }{4\omega _k \omega _n }} \right) p_n q_n +\frac{R_{k_2 } }{4\omega _k }\left( {p_m p_n -q_m q_n } \right) \nonumber \\&+\,\frac{R_{k_1 } }{4\omega _k }\left( {p_m^2 -q_m^2 } \right) +\frac{R_{k_3 } }{4\omega _k }\left( {p_n^2 -q_n^2 } \right) \nonumber \\&+\,\left( {\frac{\mu R_{k_2 } }{8\omega _k^2 }-\frac{\mu T_{nm} }{8\omega _k \omega _m }-\frac{\mu T_{nm} }{8\omega _k \omega _n }} \right) \left( {p_n q_m +p_m q_n } \right) \nonumber \\&+\,\left( {\frac{T_{kmm} }{8\omega _k }+\frac{R_{m_1 } T_{mm} }{32\omega _k \omega _m^2 }+\frac{R_{n_1 } T_{nm} }{32\omega _k \omega _n^2 }} \right) \left( {p_k p_m^2 +p_k q_m^2 } \right) \nonumber \\&+\,\left( {\frac{T_{knn} }{8\omega _k }+\frac{R_{m_2 } T_{nm} }{32\omega _k \omega _m^2 }+\frac{R_{n_2 } T_{nn} }{32\omega _k \omega _n^2 }} \right) \left( {p_k p_n^2 +p_k q_n^2 } \right) \nonumber \\&+\,\left( {\frac{T_{kmn} }{8\omega _k }+\frac{R_{m_2 } T_{mm} }{32\omega _k \omega _m^2 }+\frac{R_{n_2 } T_{nm} }{32\omega _k \omega _n^2 }} \right) \nonumber \\&\left( {\left( {p_k p_m p_n +p_k q_m q_n } \right) +\left( {p_m q_k q_n -p_n q_k q_m } \right) } \right) \nonumber \\&+\,\left( {\frac{T_{knm} }{8\omega _k }+\frac{R_{m_1 } T_{nm} }{32\omega _k \omega _m^2 }+\frac{R_{n_1 } T_{nn} }{32\omega _k \omega _n^2 }} \right) \nonumber \\&\left( {\left( {p_k p_m p_n +p_k q_m q_n } \right) -\left( {p_m q_k q_n -p_n q_k q_m } \right) } \right) \nonumber \\&+\,\frac{T_{kkk} }{8\omega _k }\left( {p_k^3 +q_k^2 p_k } \right) +\frac{F_{k_1 } }{\omega _k }\nonumber \\&+\,\left( {\frac{F_{m_1 } T_{nm} }{8\omega _k \omega _m^2 }+\frac{F_{n_1 } T_{nn} }{8\omega _k \omega _n^2 }-\frac{F_{nk} }{\omega _k }} \right) p_n\end{aligned}$$

where the values of \(\lambda _1 \), \(\lambda _2\), and \(\lambda _3 \) are \(\lambda _1 =\sigma _1\), \(\lambda _2 =\sigma _1 -\sigma _2\), and \(\lambda _3 =2\sigma _1 -2\sigma _2 -\sigma _3 \).

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Hajjaj, A.Z., Alfosail, F.K., Jaber, N. et al. Theoretical and experimental investigations of the crossover phenomenon in micromachined arch resonator: part II—simultaneous 1:1 and 2:1 internal resonances. Nonlinear Dyn 99, 407–432 (2020).

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