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Detection of different dynamics of two coupled oscillators including a time-dependent cubic nonlinearity

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Abstract

Vibratory energy channelling between a linear and a nonlinear oscillator is studied at different time scales. The nonlinear system possesses a time-dependent periodic restoring forcing function. Detection of fast and slow system dynamics leads to revealing different dynamical characteristics, namely slow invariant manifold, equilibrium and singular points. We show that the time-dependent nonlinearity produces a phase-dependent slow invariant manifold, frequency responses, and modifications concerning stability borders of its slow invariant manifold and singularities zones. The backbone curves of the system and also isola are detected; the latter should be taken into account carefully if the aim is system control.

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Acknowledgements

The authors would like to thank the following organizations for supporting this research: (i) The “Ministère de la transition écologique” and (ii) LABEX CELYA (ANR-10-LABX-0060) of the “Université de Lyon” within the program “Investissement d’Avenir”(ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

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  1. ENTPE, LTDS UMR CNRS 5513, Univ Lyon, 3 rue Maurice Audin, F-69518, Vaulx-en-Velin Cedex, France

    A. Labetoulle, A. Ture Savadkoohi & E. Gourdon

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  1. A. Labetoulle
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  2. A. Ture Savadkoohi
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  3. E. Gourdon
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Appendices

Treatment of \({\mathcal {E}}=0\)

$$\begin{aligned} {\mathcal {E}}= 0 \Rightarrow if_0 + [i(2\sigma - \gamma _0) +\xi _1]\phi _1 +i\gamma _0\phi _2 =0. \end{aligned}$$
(61)

Let us insert Eq. (22) in Eq. (61),

$$\begin{aligned}&if_0 + [i(2\sigma - \gamma _0) +\xi _1] \frac{\phi _2}{\gamma _0}\left[ -1+i\xi _2 + \gamma _0 + \frac{3}{4}k_0 |\phi _2 |^2 \right] +i\gamma _0\phi _2 =0, \end{aligned}$$
(62)
$$\begin{aligned}&if_0 + [i(2\sigma - \gamma _0) +\xi _1] \frac{N_2e^{i\delta _2}}{\gamma _0}\left[ -1+i\xi _2 + \gamma _0 + \frac{3}{4}k_0 N_2^2\right] +i\gamma _0N_2e^{i\delta _2} =0, \end{aligned}$$
(63)
$$\begin{aligned}&if_0 + \frac{N_2(\cos (\delta _2)+i\sin (\delta _2))}{\gamma _0} \left[ -\xi _2(2\sigma - \gamma _0) + i\left( 2\sigma - \gamma _0\right) \left( -1+\gamma _0+\frac{3}{4}k_0N_2^2\right) \right. \nonumber \\&\quad \left. + i\xi _1\xi _2 + \xi _1\left( -1+\gamma _0+\frac{3}{4}k_0N_2^2\right) \right] +i\gamma _0 N_2(\cos (\delta _2)+i\sin (\delta _2)) =0.\nonumber \\ \end{aligned}$$
(64)

Real part:

$$\begin{aligned}&\cos (\delta _2)\underbrace{\left[ -\xi _2(2\sigma - \gamma _0) +\xi _1\left( -1+\gamma _0+\frac{3}{4}k_0N_2^2\right) \right] }_{A_{11}} - \sin (\delta _2)\underbrace{\left[ \left( 2\sigma - \gamma _0\right) \left( -1+\gamma _0+\frac{3}{4}k_0N_2^2\right) +\xi _1\xi _2+\gamma _0^2 \right] }_{A_{12}}\nonumber \\&\qquad =0. \end{aligned}$$
(65)

Imaginary part:

$$\begin{aligned}&\cos (\delta _2)\underbrace{\left[ \left( 2\sigma - \gamma _0\right) \left( -1+\gamma _0+\frac{3}{4}k_0N_2^2\right) +\xi _1\xi _2+ \gamma _0^2 \right] }_{A_{12}}\nonumber \\&\quad +\sin (\delta _2)\underbrace{\left[ -\xi _2(2\sigma - \gamma _0) +\xi _1\left( -1+\gamma _0+\frac{3}{4}k_0N_2^2\right) \right] }_{A_{11}}=\underbrace{-\frac{f_0\gamma _0}{N_2}}_{C_2}, \end{aligned}$$
(66)
$$\begin{aligned}&\left\{ \begin{array}{l} A_{11}\cos (\delta _2) -A_{12}\sin (\delta _2)=0\\ A_{12}\cos (\delta _2) +A_{11}\sin (\delta _2)=C_2 \end{array} \right. , \end{aligned}$$
(67)
$$\begin{aligned}&A_{11}^2+A_{12}^2 = C_2^2, \end{aligned}$$
(68)
$$\begin{aligned}&\xi _2^2(2\sigma -\gamma _0)^2-2\xi _2\xi _1(2\sigma -\gamma _0)\left( -1+\gamma _0+\frac{3}{4}k_0N_2^2\right) +\xi _1^2\left( -1+\gamma _0+\frac{3}{4}k_0N_2^2\right) ^2\nonumber \\&\quad +\left( 2\sigma -\gamma _0\right) ^2\left( -1+\gamma _0+\frac{3}{4}k_0N_2^2\right) ^2\nonumber \\&\quad +2\left( 2\sigma -\gamma _0\right) \left( -1+\gamma _0+\frac{3}{4}k_0N_2^2\right) \left( \xi _1\xi _2+\gamma _0^2\right) +\left( \xi _1\xi _2+\gamma _0^2\right) ^2 =\frac{f_0^2\gamma _0^2}{N_2^2}. \end{aligned}$$
(69)

Let us consider \(X=N_2^2\):

$$\begin{aligned}&X^3\left( \frac{3}{4}k_0\right) ^2\left[ \xi _1^2 +(2\sigma -\gamma _0)^2\right] +X^2\frac{3}{2}k_0\left[ (-1+\gamma _0)\left[ \xi _1^2 +(2\sigma _-\gamma _0)^2 \right] + (2\sigma -\gamma _0)\gamma _0^2 \right] ,\nonumber \\&\qquad +X\left[ (-1+\gamma _0)^2\left[ \xi _1^2+2(\sigma -\gamma _0)^2 \right] + 2(-1+\gamma _0)(2\sigma -\gamma _0)\gamma _0^2\right. \nonumber \\&\qquad \left. + \left( \xi _1\xi _2+\gamma _0^2\right) ^2 +(2\sigma -\gamma _0)^2\xi _2^2\right] =f_0^2\gamma _0^2,\nonumber \\ \end{aligned}$$
(70)
$$\begin{aligned}&aX^3+bX^2+cX+d=0. \end{aligned}$$
(71)

Development of fast dynamics for the system with time-dependent nonlinearity

$$\begin{aligned} \gamma _0N_1e^{i\delta _1}&=\left( -1+\gamma _0+i\xi _2\right) N_2e^{i\delta _2}\nonumber \\&\quad -\frac{N_2^3}{4} \left( e^{3i\delta _2}\left( K_{2R}-iK_{2I}\right) \right. \left. +3 \left[ -K_{0}e^{i\delta _2}+\left( K_{2R}+iK_{2I}\right) e^{-i\delta _2} \right] -\left( K_{4R}+iK_{4I}\right) e^{-i3\delta _2}\right) .\nonumber \\ \end{aligned}$$
(72)

Real part:

$$\begin{aligned} \gamma _0\frac{N_1}{N_2}\cos (\delta _1-\delta _2)= \underbrace{\left( -1+\gamma _0\right) - \frac{N_2^2}{4} \left( 4K_{2R}\cos (2\delta _2)+4K_{2I}\sin (2\delta _2) \right. \left. -3K_{0}-K_{4R}\cos (4\delta _2)-K_{4I}\sin (4\delta _2)\right) }_{A}.\nonumber \\ \end{aligned}$$
(73)

Imaginary part:

$$\begin{aligned} \gamma _0\frac{N_1}{N_2}\sin (\delta _1-\delta _2)= \underbrace{\xi _2 - \frac{N_2^2}{4} \left( -2K_{2R}\sin (2\delta _2)+2K_{2I}\cos (2\delta _2) +K_{4R}\sin (4\delta _2)-K_{4I}\cos (4\delta _2)\right) }_{B}.\nonumber \\ \end{aligned}$$
(74)

So,

$$\begin{aligned} N_1&=\frac{N_2}{\gamma _0}\sqrt{A^2+B^2}, \end{aligned}$$
(75)
$$\begin{aligned} A^2+B^2&= \frac{1}{16}\left[ 16\xi _2^2+\left( -8 K_{4R}N_{2}^2\sin \left( 4\delta _{2}\right) +8 K_{4I}N_{2}^2\cos \left( 4\delta _{2}\right) \right. \right. \left. \left. +16K_{2R}N_{2}^2\sin \left( 2\delta _{2}\right) \right. \right. \nonumber \\&\quad \left. -16 K_{2I} N_{2}^2\cos \left( 2\delta _{2}\right) \right) \xi _2+\left( \left( -4K_{2R} K_{4R} -8 K_{2I} K_{4I}\right) N_{2}^4\sin \left( 2\delta _{2}\right) \right. \nonumber \\&\quad \left. \left. +\left( 4 K_{2I} K_{4R}-8K_{2R} K_{4I}\right) N_{2}^4\cos \left( 2\delta _{2}\right) \right. \right. \left. \left. +6K_{0} K_{4I}N_{2}^4\right. \right. \nonumber \\&\quad \left. +\left( 8 K_{4I}\gamma _{0}-8 K_{4I}\right) N_{2}^2\right) \sin \left( 4\delta _{2}\right) +\left( \left( 4 K_{2R} K_{4I} \right. \right. \nonumber \\&\quad \left. \left. \left. -8 K_{2I} K_{4R}\right) N_{2}^4\sin \left( 2\delta _{2}\right) +\left( -8 K_{2R}K_{4R}-4 K_{2I} K_{4I}\right) N_{2}^4\cos \left( 2 \delta _{2}\right) +6K_{0} K_{4R}N_{2}^4\right. \right. \nonumber \\&\quad \left. +\left( 8 K_{4R}\gamma _{0}-8 K_{4R}\right) N_{2}^2\right) \cos \left( 4\delta _{2}\right) \nonumber \\&\quad \left. +\left( 24 K_{2I} K_{2R}N_{2}^4\cos \left( 2\delta _{2}\right) -24K_{0} K_{2I}N_{2}^4+\left( 32K_{2I}-32 K_{2I}\gamma _{0}\right) N_{2}^2\right) \sin \left( 2\delta _{2}\right) \right. \nonumber \\&\quad \left. +\left( 12 K_{2R}^2 -12 K_{2I}^2 \right) N_{2}^4\cos ^2\left( 2\delta _{2}\right) +\left( \left( 32K_{2R}-32 K_{2R}\gamma _{0}\right) N_{2}^2 \right. \right. \nonumber \\&\quad \left. \left. -24K_{0}K_{2R}N_{2}^4\right) \cos \left( 2\delta _{2}\right) +\left( K_{4R}^2+ K_{4I}^2+4 K_{2R}^2+16K_{2I}^2+9K_{0}^2\right) N_{2}^4\right. \nonumber \\&\quad \left. +\left( 24K_{0}\gamma _{0}-24K_{0}\right) N_{2}^2+16\gamma _{0}^2-32\gamma _{0}+16\right] . \end{aligned}$$
(76)

Extrema of the SIM for the system with time-dependent nonlinearity

$$\begin{aligned} \frac{\partial N_1^2}{\partial N_2}&= \frac{-1}{8\gamma _{0}^2}\left[ \left( \left( 12 K_{2R} K_{4R}+24 K_{2I} K_{4I}\right) N_{2}^5\sin \left( 2\delta _{2}\right) \right. \right. \nonumber \\&\quad +\left( 24 K_{2R} K_{4I} -12 K_{2I} K_{4R}\right) N_{2}^5\cos \left( 2\delta _{2}\right) -18K_{0} K_{4I}N_{2}^5 \nonumber \\&\quad \left. \left. +\left( 16 K_{4R}\xi _{2} +\left( 16 -16\gamma _{0}\right) K_{4I}\right) N_{2}^3\right) \sin \left( 4\delta _{2}\right) +\left( \left( 24 K_{2I} K_{4R}-12 K_{2R} K_{4I}\right) N_{2}^5\sin \left( 2\delta _{2}\right) \right. \right. \nonumber \\&\quad \left. \left. +\left( 24 K_{2R} K_{4R}+12 K_{2I} K_{4I}\right) N_{2}^5\cos \left( 2\delta _{2}\right) -18K_{0} K_{4R}N_{2}^5+\left( \left( 16 -16\gamma _{0}\right) K_{4R}-16 K_{4I}\xi _{2}\right) N_{2}^3\right) \cos \left( 4\delta _{2}\right) \right. \nonumber \\&\quad \left. +\left( -72 K_{2I} K_{2R}N_{2}^5\cos \left( 2\delta _{2}\right) +72K_{0} K_{2I}N_{2}^5+\left( \left( 64\gamma _{0}-64\right) K_{2I}-32 K_{2R}\xi _{2}\right) N_{2}^3\right) \sin \left( 2\delta _{2}\right) +\left( 36 K_{2I}^2 \right. \right. \nonumber \\&\quad \left. \left. -36 K_{2R}^2\right) N_{2}^5\cos ^2\left( 2\delta _{2}\right) +\left( 72K_{0} K_{2R}N_{2}^5+\left( 32 K_{2I}\xi _{2}+\left( 64\gamma _{0} -64\right) K_{2R}\right) N_{2}^3\right) \cos \left( 2\delta _{2}\right) \right. \nonumber \\&\quad \left. +\left( -3 K_{4R}^2-3 K_{4I}^2-12 K_{2R}^2-48 K_{2I}^2 -27K_{0}^2\right) N_{2}^5\right. \nonumber \\&\quad \left. +\left( 48-48\gamma _{0}\right) K_{0}N_{2}^3+\left( -16\xi _{2}^2-16\gamma _{0}^2+32\gamma _{0}-16\right) N_{2}\right] =0, \end{aligned}$$
(77)
$$\begin{aligned}&\frac{\partial N_1^2}{\partial \delta _2} = \frac{1}{2 \gamma _{0}^2}\left[ \left( 3 K_{2I} K_{4R}N_{2}^6\sin \left( 2\delta _{2}\right) +3 K_{2R} K_{4R}N_{2}^6\cos \left( 2\delta _{2} \right) -3K_{0} K_{4R}N_{2}^6\right. \right. \nonumber \\&\quad \left. +\left( \left( 4-4\gamma _{0}\right) K_{4R}-4 K_{4I}\xi _{2}\right) N_{2}^4\right) \sin \left( 4\delta _{2}\right) \nonumber \\&\quad \left. +\left( -3 K_{2I} K_{4I} N_{2}^6\sin \left( 2\delta _{2}\right) -3 K_{2R} K_{4I}N_{2}^6\cos \left( 2\delta _{2}\right) +3K_{0} K_{4I}N_{2}^6 \right. \right. \nonumber \\&\quad \left. +\left( \left( 4\gamma _{0}-4\right) K_{4I}-4 K_{4R}\xi _{2}\right) N_{2}^4\right) \cos \left( 4\delta _{2}\right) +\left( \left( 6 K_{2I}^2 \right. \right. \nonumber \\&\quad \left. \left. \left. -6 K_{2R}^2\right) N_{2}^6\cos \left( 2\delta _{2}\right) +6K_{0} K_{2R}N_{2}^6+\left( 4 K_{2I}\xi _{2}+\left( 8\gamma _{0} -8\right) K_{2R}\right) N_{2}^4\right) \sin \left( 2\delta _{2}\right) \right. \nonumber \\&\quad \left. +12 K_{2I} K_{2R}N_{2}^6\cos ^2\left( 2\delta _{2}\right) \right. \nonumber \\&\quad \left. +\left( \left( 4 K_{2R}\xi _{2}+\left( 8 -8\gamma _{0}\right) K_{2I}\right) N_{2}^4-6K_{0} K_{2I}N_{2}^6\right) \cos \left( 2 \delta _{2}\right) -6 K_{2I} K_{2R}N_{2}^6\right] =0. \end{aligned}$$
(78)

Equations (77) and (78) read as polynomials of \( N_2 \):

$$\begin{aligned}&\frac{N_2}{8 \gamma _0^2}\left( 16-32\gamma _0+16\gamma _0^2+16\xi _2^2\right) + \frac{N_2^3}{8 \gamma _0^2}\left( 48K_0\left( -1+\gamma _0\right) + 64 K_{2R} \cos (2\delta _2) \right. \nonumber \\&\quad -64K_{2R}\gamma _0\cos (2\delta _2) - 32 K_{2I} \xi _2 \cos (2\delta _2) - 16 K_{4R} \cos (4\delta _2)+ 16 K_{4R} \gamma _0 \cos (4\delta _2) \nonumber \\&\quad \left. + 16 K_{4I} \xi _2 \cos (4\delta _2)+ 64 K_{2I} \sin (2\delta _2) - 64 K_{2I} \gamma _0 \sin (2\delta _2) \right. \nonumber \\&\quad \left. + 32 K_{2R} \xi _2 \sin (2\delta _2)- 16 K_{4I} \sin (4\delta _2) + 16 K_{4I} \gamma _0 \sin (4\delta _2) - 16 K_{4R} \xi _2 \sin (4\delta _2) \right) \nonumber \\&\quad + \frac{N_2^5}{8 \gamma _0^2}\left( 27 K_0^2 + 30 K_{2I}^2 + 30 K_{2R}^2 + 3 K_{4I}^2 + 3 K_{4R}^2 - 72 K_0 K_{2R} \cos (2\delta _2)-18 K_{2I} K_{4I} \cos (2\delta _2) \right. \nonumber \\&\quad \left. - 18 K_{2R} K_{4R} \cos (2\delta _2) - 18 K_{2I}^2 \cos (4\delta _2) + 18 K_{2R}^2 \cos (4\delta _2) +18 K_0 K_{4R} \cos (4\delta _2) + 6 K_{2I} K_{4I} \cos (6\delta _2) \right. \nonumber \\&\quad \left. - 6 K_{2R} K_{4R} \cos (6\delta _2) - 72 K_0 K_{2I} \sin (2\delta _2) -18 K_{2R} K_{4I} \sin (2\delta _2) + 18 K_{2I} K_{4R} \sin (2\delta _2)\right. \nonumber \\&\quad \left. + 36 K_{2I} K_{2R} \sin (4\delta _2) + 18 K_0 K_{4I} \sin (4\delta _2) - 6 K_{2R} K_{4I} \sin (6\delta _2) - 6 K_{2I} K_{4R} \sin (6\delta _2) \right) =0, \end{aligned}$$
(79)
$$\begin{aligned}&\frac{N_2^4}{8\gamma _0^2} \left( -32 K_{2I} \left( -1 + \gamma _0\right) \cos (2\delta _2) + 16 K_{2R} \xi _2 \cos (2\delta _2) - 16 K_{4I} \cos (4\delta _2)\right. \nonumber \\&\quad + 16 K_{4I} \gamma _0 \cos (4\delta _2) - 16 K_{4R} \xi _2 \cos (4\delta _2) \nonumber \\&\quad \left. + 32 K_{2R} \left( -1 + \gamma _0 \right) \sin (2\delta _2) +16 K_{2I} \xi _2 \sin (2\delta _2) + 16 K_{4R}\sin (4\delta _2) - 16 K_{4R} \gamma _0\sin (4\delta _2) - 16 K_{4I} \xi _2 \sin (4\delta _2)\right) \nonumber \\&\quad + \frac{N_2^6}{8\gamma _0^2} \left( -6 K_{2R} K_{4I}\cos (2\delta _2) + 6 K_{2I} \left( -4 K_{0} + K_{4R}\right) \cos (2\delta _2) + 24 K_{2I} K_{2R}\cos (4\delta _2) + 12 K_{0} K_{4I}\cos (4\delta _2) \right. \nonumber \\&\quad \left. -6 \left( K_{2R} K_{4I} + K_{2I} K_{4R}\right) \cos (6\delta _2) + 6 K_{2I} K_{4I}\sin (2\delta _2) + 6 K_{2R} \left( 4 K_{0} + K_{4R}\right) \sin (2\delta _2) \right. \nonumber \\&\quad \left. + 12 \left( K_{2I} - K_{2R}\right) \left( K_{2I} + K_{2R}\right) \sin (4\delta _2) - 12 K_{0} K_{4R}\sin (4\delta _2) - 6 \left( K_{2I} K_{4I} - K_{2R} K_{4R}\right) \sin (6\delta _2)\right) =0. \end{aligned}$$
(80)

Setting \(X=N_2^2\), then Eq. (79) can be written simplified:

$$\begin{aligned} \frac{\partial N_1^2}{\partial N_2} =0 \Rightarrow \alpha _4 X^2 + \alpha _2 X + \alpha _0 =0 \end{aligned}$$
(81)

with

$$\begin{aligned} \alpha _0&=16(1-2\gamma _0+\gamma _0^2+\xi _2^2), \end{aligned}$$
(82)
$$\begin{aligned} \alpha _2&=48K_0\left( -1+\gamma _0\right) + 64 K_{2R} \cos (2\delta _2) -64K_{2R}\gamma _0\cos (2\delta _2) - 32 K_{2I} \xi _2 \cos (2\delta _2) \nonumber \\&\quad - 16 K_{4R} \cos (4\delta _2) + 16 K_{4R} \gamma _0 \cos (4\delta _2) \nonumber \\&\quad + 16 K_{4I} \xi _2 \cos (4\delta _2)+ 64 K_{2I} \sin (2\delta _2) - 64 K_{2I} \gamma _0 \sin (2\delta _2) + 32 K_{2R} \xi _2 \sin (2\delta _2) \nonumber \\&\quad - 16 K_{4I} \sin (4\delta _2) + 16 K_{4I} \gamma _0 \sin (4\delta _2) - 16 K_{4R} \xi _2 \sin (4\delta _2), \end{aligned}$$
(83)
$$\begin{aligned} \alpha _4&= 27 K_0^2 + 30 K_{2I}^2 + 30 K_{2R}^2 + 3 K_{4I}^2 + 3 K_{4R}^2 - 72 K_0 K_{2R} \cos (2\delta _2) \nonumber \\&\quad -18 K_{2I} K_{4I} \cos (2\delta _2) - 18 K_{2R} K_{4R} \cos (2\delta _2)\nonumber \\&\quad - 18 K_{2I}^2 \cos (4\delta _2) + 18 K_{2R}^2 \cos (4\delta _2) +18 K_0 K_{4R} \cos (4\delta _2) \nonumber \\&\quad + 6 K_{2I} K_{4I} \cos (6\delta _2) - 6 K_{2R} K_{4R} \cos (6\delta _2) \nonumber \\&\quad - 72 K_0 K_{2I} \sin (2\delta _2) -18 K_{2R} K_{4I} \sin (2\delta _2) + 18 K_{2I} K_{4R} \sin (2\delta _2) \nonumber \\&\quad + 36 K_{2I} K_{2R} \sin (4\delta _2) + 18 K_0 K_{4I} \sin (4\delta _2) \nonumber \\&\quad - 6 K_{2R} K_{4I} \sin (6\delta _2) - 6 K_{2I} K_{4R} \sin (6\delta _2). \end{aligned}$$
(84)

Equation (80) can be written as:

$$\begin{aligned} \frac{\partial N_1^2}{\partial \delta _2} =0 \Rightarrow \beta _2 X + \beta _0 =0 \end{aligned}$$
(85)

with

$$\begin{aligned} \beta _0&= -32 K_{2I} \left( -1 + \gamma _0\right) \cos (2\delta _2) + 16 K_{2R} \xi _2 \cos (2\delta _2) - 16 K_{4I} \cos (4\delta _2) \nonumber \\&\quad + 16 K_{4I} \gamma _0 \cos (4\delta _2) - 16 K_{4R} \xi _2 \cos (4\delta _2) \nonumber \\&\quad + 32 K_{2R} \left( -1 + \gamma _0 \right) \sin (2\delta _2) +16 K_{2I} \xi _2 \sin (2\delta _2) + 16 K_{4R}\sin (4\delta _2)\nonumber \\&\quad - 16 K_{4R} \gamma _0\sin (4\delta _2) - 16 K_{4I} \xi _2 \sin (4\delta _2), \end{aligned}$$
(86)
$$\begin{aligned} \beta _2&=-6 K_{2R} K_{4I}\cos (2\delta _2) + 6 K_{2I} \left( -4 K_{0} + K_{4R}\right) \cos (2\delta _2) + 24 K_{2I} K_{2R}\cos (4\delta _2) + 12 K_{0} K_{4I}\cos (4\delta _2) \nonumber \\&\quad -6 \left( K_{2R} K_{4I} + K_{2I} K_{4R}\right) \cos (6\delta _2) + 6 K_{2I} K_{4I}\sin (2\delta _2) + 6 K_{2R} \left( 4 K_{0} + K_{4R}\right) \sin (2\delta _2) \nonumber \\&\quad + 12 \left( K_{2I} - K_{2R}\right) \left( K_{2I} + K_{2R}\right) \sin (4\delta _2) - 12 K_{0} K_{4R}\sin (4\delta _2) - 6 \left( K_{2I} K_{4I} - K_{2R} K_{4R}\right) \sin (6\delta _2).\nonumber \\ \end{aligned}$$
(87)

Development of the unstable zone of the SIM of the system with time-dependent nonlinearity

$$\begin{aligned} a_v&=\frac{9}{64}(3K_0^2-2K_{2I}^2-2K_{2R}^2-K_{4I}^2-K_{4R}^2+ 4(-K_0K_{2R}+K_{2I}K_{4I}+K_{2R}K_{4R})\cos (2\delta _2)\nonumber \\&\quad -2(K_{2I}^2- K_{2R}^2+K_0K_{4R})\cos (4\delta _2) -4(-K_{2R}K_{4I}+K_{2I}(K_0+K_{4R})) \sin (2\delta _2)\nonumber \\&\quad +2(2K_{2I}K_{2R}-K_0K_{4I})\sin (4\delta _2)), \end{aligned}$$
(88)
$$\begin{aligned} b_v&=\frac{3}{4}\left( -1+\gamma _0\right) \left( K_0-K_{2R}\cos (2\delta _2)-K_{2I}\sin (2 \delta _2)\right) , \end{aligned}$$
(89)
$$\begin{aligned} c_v&=\frac{1}{4}((-1+\gamma _0)^2+\xi 2^2). \end{aligned}$$
(90)

Development of \(\det ( {\mathbb {A}})\) (singular points of the system with time-dependent nonlinearity)

$$\begin{aligned} \det ({\mathbb {A}})&= \frac{\partial {{\mathscr {H}}}}{\partial \phi _2}\frac{\partial {{\mathscr {H}}}^*}{\partial \phi _2^*}-\frac{\partial {{\mathscr {H}}}^*}{\partial \phi _2}\frac{\partial {{\mathscr {H}}}}{\partial \phi _2^*}, \end{aligned}$$
(91)
$$\begin{aligned} \det ({\mathbb {A}})&= \frac{1}{4}\left[ \xi _2^2 + \left( -1+\gamma _0 -\frac{3}{4} \left( \phi _2^2K_2^*-2|\phi _2 |^2 K_0+\phi _2^{*2}k_2\right) \right) ^2 - \left( \frac{3}{4}\right) ^2\left( |\phi _2 |^4 - 2|\phi _2 |^2\phi _2^{*2}K_0K_2 \right. \right. \nonumber \\&\quad \left. \left. +\phi _2^{*4}K_0K_4-2|\phi _2 |^2\phi _2^2K_0K_2^*+4|\phi _2 |^4|K_2 |^2 \right. \right. \nonumber \\&\quad \left. \left. -2|\phi _2 |^2 \phi _2^{*2}K_2^*K_4+\phi _2^4K_0K_4^*-2|\phi _2 |^2\phi _2^2K_2K_4^*+|\phi _2 |^4|K_4 |^2 \right) \right] , \end{aligned}$$
(92)
$$\begin{aligned} \det ({\mathbb {A}})&=\frac{1}{4} \left[ \xi _2^2 + \left( -1+\gamma _0\right) ^2 - 3\left( -1+\gamma _0\right) N_2^2\left( K_{2R}\cos (2\delta _2) + K_{2I}\sin (2\delta _2)-K_0\right) \right. \nonumber \\&\quad +\frac{9}{4}N_2^4\left( K_{2R}\cos (2\delta _2) + K_{2I}\sin (2\delta _2) -K_0\right) ^2 \nonumber \\&\quad \left. -N_2^4\left( \frac{3}{4}\right) ^2\left( K_0^2-4K_0\left( K_{2R}\cos (2\delta _2) +K_{2I}\sin (2\delta _2)\right) + 2K_0 \left( K_{4R}\cos (4\delta _2) \right. \right. \right. \nonumber \\&\quad \left. \left. \left. + K_{4I}\sin (4\delta _2)\right) -4\left( K_{2R}K_{4R}\cos (2\delta _2)-K_{2I}K_{4R}\sin (2\delta _2) \right. \right. \right. \nonumber \\&\quad \left. \left. \left. +K_{4I}K_{2I}\cos (2\delta _2)+K_{2R}K_{4I}\sin (2\delta _2)\right) +4\left( K_{2R}^2+K_{2I}^2\right) + K_{4R}^2+K_{4I}^2 \right) \right] , \end{aligned}$$
(93)
$$\begin{aligned} \det ({\mathbb {A}})&= aN_2^4+bN_2^2+c. \end{aligned}$$
(94)

Let us consider \(X=N_2^2\):

$$\begin{aligned} \det ({\mathbb {A}}) = \frac{1}{4}\left( aX^2+bX+c\right) \end{aligned}$$
(95)

with

$$\begin{aligned} a&=\frac{-1}{64}\left[ 18K_0K_{4I}\sin (4\delta _2)+18K_0K_{4R}\cos (4\delta _2)+\left( -72K_{2I}K_{2r}\cos (2\delta _2)\right. \right. \nonumber \\&\quad \left. \left. +36K_{2I}K_{4R}-36K_{2R}K_{4I}+36K_0K_{2I}\right) \sin (2\delta _2)\right. \nonumber \\&\quad \left. +\left( 36K_{2I}^2-36K_{2R}^2\right) \cos (2\delta _2)^2 +\left( -36K_{2R}K_{4R}-36K_{2I}K_{4I} +36K_0K_{2R}\right) \cos (2\delta _2)+9K_{4R}^2\right. \nonumber \\&\quad \left. +9K_{4I}^2+36K_{2R}^2-27K_0^2\right] , \end{aligned}$$
(96)
$$\begin{aligned} b&=\frac{-3}{4}\left( -1+\gamma _0\right) \left( K_{2R}\cos (2\delta _2) + K_{2I}\sin (2\delta _2)-K_0\right) , \end{aligned}$$
(97)
$$\begin{aligned} c&=\frac{\xi _2^2}{4}\left( -1+\gamma _0\right) ^2. \end{aligned}$$
(98)

Development of equilibrium points of the system with time-dependent nonlinearity

$$\begin{aligned} if_0+\left( i\left( 2\sigma -\gamma _0\right) + \xi _1\right) \phi _1+i\gamma _0 \phi _2 =0. \end{aligned}$$
(99)

Inserting Eq. (45) in Eq. (99):

$$\begin{aligned}&if_0+\left( i\left( 2\sigma -\gamma _0\right) + \xi _1\right) \frac{1}{\gamma _0}\left( -1+\gamma _0 + i\xi _2\right) \phi _2 - \frac{1}{4}\left( \phi _2^3K_2^* -3|\phi _2 |^2 \phi _2K_0 + 3 |\phi _2 |^2\phi _2^*K_2-\phi _2^{*3}K_4 \right) \nonumber \\&\qquad +i\gamma _0 \phi _2 =0, \end{aligned}$$
(100)
$$\begin{aligned}&if_0+\left( i\left( 2\sigma -\gamma _0\right) + \xi _1\right) \frac{N_2e^{i\delta _2}}{\gamma _0}\left[ -1+\gamma _0+i\xi _2 -\frac{N_2^2}{4}\left( K_2^*e^{2i\delta 2}-3K_0+3e^{-2i\delta _2}K_2 - e^{-4i\delta _2}K_4\right) \right] \nonumber \\&\qquad + i\gamma _0 N_2e^{i\delta _2}=0. \end{aligned}$$
(101)

Real part:

$$\begin{aligned}&N_2 ((3 \gamma _0 K_{2I} N_2^2 - 6 K_{2I} N_2^2 \sigma + 4 \xi _1 - 4 \gamma _0 \xi _1 - 3 K_0N_2^2 \xi _1 + 3 K_{2R} N_2^2 \xi _1 - 4 \gamma _0 \xi _2 + 8 \sigma \xi _2) \cos (\delta _2) \nonumber \\&\quad + (-\gamma _0 K_{2I} N_2^2 - \gamma _0 K_{4I} N_2^2 + 2 K_{2I} N_2^2 \sigma + 2 K_{4I}N_2^2 \sigma + K_{2R} N_2^2 \xi _1 - K_{4R} N_2^2 \xi _1) \cos (3\delta _2)+ (4 \gamma _0 \nonumber \\&\quad - 3 \gamma _0 K_0N_2^2 - 3 \gamma _0 K_{2R} N_2^2 - 8\sigma + 8 \gamma _0 \sigma + 6 K_0N_2^2 \sigma + 6 K_{2R} N_2^2 \sigma + 3 K_{2I} N_2^2 \xi _1 + 4 \xi _1 \xi _2) \sin (\delta _2) \nonumber \\&\quad + (\gamma _0 K_{2R} N_2^2 + \gamma _0 K_{4R} N_2^2 - 2 K_{2R} N_2^2 \sigma - 2 K_{4R} N_2^2 \sigma + K_{2I} N_2^2 \xi _1 - K_{4I} N_2^2 \xi _1) \sin (3\delta _2))=0. \end{aligned}$$
(102)

Imaginary part:

$$\begin{aligned} \begin{aligned}&4 f_0 \gamma _0 + (4 \gamma _0 N_2 - 3 \gamma _0 K_0N_2^3 + 3 \gamma _0 K_{2R} N_2^3 - 8 N_2 \sigma + 8 \gamma _0 N_2 \sigma + 6 K_0N_2^3 \sigma - 6 K_{2R} N_2^3 \sigma \\&\quad - 3 K_{2I} N_2^3 \xi _1 + 4 N_2 \xi _1 \xi _2) \cos (\delta _2)+ (\gamma _0 K_{2R} N_2^3 - \gamma _0 K_{4R} N_2^3 - 2 K_{2R} N_2^3 \sigma + 2 K_{4R} N_2^3 \sigma + K_{2I} N2^3 \xi _1 \\&\quad + K_{4I} N_2^3 \xi _1) \cos (3\delta _2) + (3 \gamma _0 K_{2I} N_2^3 - 6 K_{2I} N_2^3 \sigma - 4 N_2 \xi _1 + 4 \gamma _0 N_2 \xi _1 + 3 K_0N_2^3 \xi _1 + 3 K_{2R} N_2^3 \xi _1 + 4 \gamma _0 N_2 \xi _2 \\&\quad - 8 N_2 \sigma \xi _2) \sin (\delta _2)+ (\gamma _0 K_{2I} N_2^3 - \gamma _0 K_{4I} N_2^3 - 2 K_{2I} N_2^3 \sigma + 2 K_{4I} N_2^3 \sigma - K_{2R} N_2^3 \xi _1 - K_{4R} N_2^3 \xi _1) \sin (3 \delta _2)=0. \end{aligned}\end{aligned}$$
(103)

Hence,

$$\begin{aligned} \left\{ \begin{array}{l} a_1\cos (\delta _2)+b_1\sin (\delta _2)=c_1\\ a_2\cos (\delta _2)+b_2\sin (\delta _2)=c_2 \end{array} \right. \end{aligned}$$
(104)

with

$$\begin{aligned} \left\{ \begin{array}{ll} a_1=&{} (3 \gamma _0 K_{2I} N_2^2 - 6 K_{2I} N_2^2 \sigma + 4 \xi _1 - 4 \gamma _0 \xi _1 - 3 K_0 N_2^2 \xi _1 + 3 K_{2R} N_2^2 \xi _1 - 4 \gamma _0 \xi _2 + 8 \sigma \xi _2)\\ b_1=&{}(4 \gamma _0 - 3 \gamma _0 K_0 N_2^2 - 3 \gamma _0 K_{2R} N_2^2 - 8 \sigma + 8 \gamma _0 \sigma + 6 K_0 N_2^2 \sigma + 6 K_{2R} N_2^2 \sigma + 3 K_{2I} N_2^2 \xi _1 + 4 \xi _1 \xi _2) \\ c_1=&{} -((-\gamma _0 K_{2I} N_2^2 - \gamma _0 K_{4I} N_2^2 + 2 K_{2I} N_2^2 \sigma + 2 K_{4I} N_2^2 \sigma + K_{2R} N_2^2 \xi _1 - K_{4R} N_2^2 \xi _1) \cos (3\delta _2)\\ &{} + (\gamma _0 K_{2R} N_2^2 + \gamma _0 K_{4R} N_2^2 - 2 K_{2R} N_2^2 \sigma - 2 K_{4R} N_2^2 \sigma + K_{2I} N_2^2 \xi _1 - K_{4I} N_2^2 \xi _1) \sin (3 \delta _2)),\\ a_2=&{}(4 \gamma _0 N_2 - 3 \gamma _0 K_0 N_2^3 + 3 \gamma _0 K_{2R} N_2^3 - 8 N_2 \sigma + 8 \gamma _0 N_2 \sigma + 6 K_0 N_2^3 \sigma - 6 K_{2R} N_2^3 \sigma \\ &{} - 3 K_{2I} N_2^3 \xi _1 + 4 N_2 \xi _1 \xi _2) \\ b_2=&{} (3 \gamma _0 K_{2I} N_2^3 - 6 K_{2I} N_2^3 \sigma - 4 N_2 \xi _1 + 4 \gamma _0 N_2 \xi _1 + 3 K_0 N_2^3 \xi _1 + 3 K_{2R} N_2^3 \xi _1 + 4 \gamma _0 N_2 \xi _2 - 8 N_2 \sigma \xi _2) \\ c_2=&{} -(4 f_0 \gamma _0 + (\gamma _0 K_{2R} N_2^3 - \gamma _0 K_{4R} N_2^3 - 2 K_{2R} N_2^3 \sigma + 2 K_{4R} N_2^3 \sigma + K_{2I} N_2^3 \xi _1 \\ &{}+K_{4I} N_2^3 \xi _1) \cos (3\delta _2)+ (\gamma _0 K_{2I} N_2^3 - \gamma _0 K_{4I} N_2^3 - 2 K_{2I} N_2^3 \sigma \\ &{}+ 2 K_{4I} N_2^3 \sigma - K_{2R} N_2^3 \xi _1 - K_{4R} N_2^3 \xi _1) \sin (3 \delta _2))\\ \end{array} \right. . \end{aligned}$$
(105)

So,

$$\begin{aligned} (b_2c1 - b_1 c_2)^2 + (a_2 c_1 - a_1 c_2)^2 - (-a_2b_1 + a_1b_2)^2=0. \end{aligned}$$
(106)

Equation (106) can be reorganized as a polynomial of \(N_2\):

$$\begin{aligned} p_{10}N_2^{10}+p_8N_2^8+p_7N_2^7+p_6N_2^6+p_5N_2^5+p_4N_2^4+p_3N_2^3+p_2N_2^2+p_0=0 \end{aligned}$$
(107)

with

$$\begin{aligned} p_{10}&= -9((\gamma _0-2\sigma )^2+\xi _1^2)^2 (9K_0^4+K_0(8K_{2I}K_{2R}K_{4I}-4K_{2I}^2 K_{4R}+4K_{2R}^2 K_{4R})\nonumber \\&\quad +(K_{2I}^2+K_{2R}^2)(8K_{2I}^2+8K_{2R}^2-K_{4I}^2-K_{4R}^2)-K_0^2 (19K_{2I}^2+19K_{2R}^2+K_{4I}^2+K_{4R}^2)\nonumber \\&\quad -2(K_0^2 (K_{2I}K_{4I}-K_{2R}K_{4R})+(K_{2I}^2+K_{2R}^2)(K_{2I}K_{4I}-K_{2R}K_{4R})+K_0(-3K_{2I}^2 \nonumber \\&\quad K_{2R}+K_{2R}^3-K_{2R}K_{4I}^2+2K_{2I}K_{4I}K_{4R}+K_{2R}K_{4R}^2))\cos (6 \delta _2)+2(K_0^2 (K_{2R}K_{4I}+K_{2I}K_{4R})\nonumber \\&\quad +(K_{2I}^2+K_{2R}^2)(K_{2R}K_{4I}+K_{2I}K_{4R})+K_0(K_{2I}^3-2K_{2R}K_{4I}K_{4R}\nonumber \\&\quad +K_{2I}(-3K_{2R}^2-K_{4I}^2+K_{4R}^2)))\sin (6\delta _2), \end{aligned}$$
(108)
$$\begin{aligned} p_8&=24((\gamma _0-2\sigma )^2+\xi _1^2)(\gamma _0(-(18K_0^3+4K_{2I}K_{2R}K_{4I}-2K_{2I}^2 K_{4R}+2K_{2R}^2 K_{4R}\nonumber \\&\quad -K_0(19K_{2I}^2+19K_{2R}^2+K_{4I}^2+K_{4R}^2))(4\sigma (1+\sigma )+\xi _1^2)-8(K_{2I}^2 K_{4I}-K_{2R}^2 K_{4I}\nonumber \\&\quad +2K_{2I}K_{2R}K_{4R})\sigma \xi _2)+(4\sigma ^2+\xi _1^2)(18K_0^3-K_0(19K_{2I}^2+19K_{2R}^2\nonumber \\&\quad +K_{4I}^2+K_{4R}^2)-2K_{2I}^2\nonumber \\&(K_{4R}-K_{4I}\xi _2)+2K_{2R}^2 (K_{4R}-K_{4I}\xi _2)+4K_{2I}K_{2R}(K_{4I}+K_{4R}\xi _2))+\gamma _0^2 (18K_0^3 (1+2\sigma )\nonumber \\&\quad -K_0(19K_{2I}^2+19K_{2R}^2+K_{4I}^2+K_{4R}^2)(1+2\sigma )+2K_{2R}^2 (K_{4R}+2K_{4R}\sigma -K_{4I}(\xi _1+\xi _2))\nonumber \\&\quad +2K_{2I}^2 (-K_{4R}(1+2\sigma )+K_{4I}(\xi _1+\xi _2))\nonumber \\&\quad +4K_{2I}K_{2R}(K_{4I}+2K_{4I}\sigma +K_{4R}(\xi _1+\xi _2)))\nonumber \\&\quad +(\gamma _0(-(3K_{2I}^2 K_{2R}-2K_{2I}K_{4I}(K_0+K_{4R})\nonumber \\&\quad -K_{2R}(K_{2R}^2-K_{4I}^2-2K_0K_{4R}+K_{4R}^2))(4\sigma (1+\sigma )+\xi _1^2)\nonumber \\&\quad -4(K_{2I}^3+2K_{2R}K_{4I}K_{4R}+K_{2I}(-3K_{2R}^2\nonumber \\&\quad +K_{4I}^2-K_{4R}^2))\sigma \xi _2)+(4\sigma ^2+\xi _1^2)(3K_{2I}^2 K_{2R}-2K_{2I}K_{4I}(K_0+K_{4R})\nonumber \\&\quad +K_{2I}^3 \xi _2+K_{2I}(-3K_{2R}^2+K_{4I}^2-K_{4R}^2)\xi _2+K_{2R}(-K_{2R}^2\nonumber \\&\quad +K_{4I}^2+2K_0K_{4R}-K_{4R}^2+2K_{4I}K_{4R}\xi _2))\nonumber \\&\quad +\gamma _0^2 (3K_{2I}^2 (K_{2R}+2K_{2R}\sigma )+K_{2I}^3 (\xi _1+\xi _2)\nonumber \\&\quad +K_{2R}(-K_{2R}^2 (1+2\sigma )+K_{4I}^2 (1+2\sigma )-K_{4R}(-2K_0+K_{4R})\nonumber \\&\quad (1+2\sigma )+2K_{4I}K_{4R}(\xi _1+\xi _2))-K_{2I}(2K_0(K_{4I}+2K_{4I}\sigma )\nonumber \\&\quad +2K_{4I}(K_{4R}+2K_{4R}\sigma )-K_{4I}^2 (\xi _1+\xi _2)\nonumber \\&\quad +(3K_{2R}^2+K_{4R}^2)(\xi _1+\xi _2))))\cos (6 \delta _2)+(-\gamma _0((K_{2I}^3+2K_{2R}K_{4I}(K_0-K_{4R})+K_{2I}(-3K_{2R}^2-K_{4I}^2\nonumber \\&\quad +2K_0K_{4R}+K_{4R}^2))(4\sigma (1+\sigma )+\xi _1^2)+4(-3K_{2I}^2 K_{2R}-2K_{2I}K_{4I}K_{4R}+K_{2R}(K_{2R}^2+K_{4I}^2-K_{4R}^2))\sigma \xi _2)\nonumber \\&\quad +\gamma _0^2 (K_{2I}^3 (1+2\sigma )-3K_{2I}^2 K_{2R}(\xi _1+\xi _2)-K_{2I}(K_{4I}^2 (1+2\sigma )-K_{4R}(2K_0+K_{4R})(1+2\sigma )+K_{2R}^2 (3+6\sigma )\nonumber \\&\quad +2K_{4I}K_{4R}(\xi _1+\xi _2))+K_{2R}(2K_0(K_{4I}+2K_{4I}\sigma )\nonumber \\&\quad -2K_{4I}(K_{4R}+2K_{4R}\sigma )+K_{4I}^2 (\xi _1+\xi _2)+(K_{2R}-K_{4R})\nonumber \\&\quad (K_{2R}+K_{4R})(\xi _1+\xi _2)))+(4\sigma ^2+\xi _1^2)(K_{2I}^3\nonumber \\&\quad +2K_{2R}K_{4I}(K_0-K_{4R})-3K_{2I}^2 K_{2R}\xi _2+K_{2R}(K_{2R}^2\nonumber \\&\quad +K_{4I}^2-K_{4R}^2)\xi _2+K_{2I}(-3K_{2R}^2+K_{4R}(2K_0+K_{4R})\nonumber \\&\quad -K_{4I}(K_{4I}+2K_{4R}\xi _2))))\sin (6\delta _2), \end{aligned}$$
(109)
$$\begin{aligned} p_7&=72f_0\gamma _0((\gamma _0-2\sigma )^2+\xi _1^2)(((K_0^2 (K_{2R}-K_{4R})+(K_{2I}^2+K_{2R}^2)(K_{2R}-K_{4R})\nonumber \\&\quad -2K_0(K_{2I}(K_{2I}+K_{4I})+K_{2R}(-K_{2R}+K_{4R})))(\gamma _0-2\sigma )+(K_0^2 (K_{2I}+K_{4I})\nonumber \\&\quad +(K_{2I}^2+K_{2R}^2)(K_{2I}+K_{4I})-2K_0K_{2R}(2K_{2I}+K_{4I})+2K_0K_{2I}K_{4R})\xi _1)\cos (3\delta _2)\nonumber \\&\quad +((K_0^2 (K_{2I}-K_{4I})+(K_{2I}^2+K_{2R}^2)(K_{2I}-K_{4I})-2K_0K_{2R}K_{4I}+2K_0K_{2I}(2K_{2R}+K_{4R}))(\gamma _0-2\sigma )\nonumber \\&\quad -(2K_0K_{2I}(K_{2I}-K_{4I})+K_0^2 (K_{2R}+K_{4R})-2K_0K_{2R}(K_{2R}+K_{4R})+(K_{2I}^2+K_{2R}^2)(K_{2R}\nonumber \\&\quad +K_{4R}))\xi _1)\sin (3\delta _2), \end{aligned}$$
(110)
$$\begin{aligned} p_6&= -16 (\gamma _0^2 (-(-54 K_0^2 + 19 K_{2I}^2 + 19 K_{2R}^2 + K_{4I}^2\nonumber \\&\quad + K_{4R}^2) (8 \sigma ^2 (3 + 2 \sigma (3 + \sigma )) + 2 (1 + 6 \sigma + 4 \sigma ^2) \xi _1^2 + \xi _1^4)\nonumber \\&\quad - 2 (-18 K_0^2 + 19 K_{2I}^2 + 19 K_{2R}^2 + K_{4I}^2 + K_{4R}^2) \xi _1 (4 \sigma ^2 + \xi _1^2) \xi _2 \nonumber \\&\quad - 2 (-18 K_0^2 + 19 K_{2I}^2 + 19 K_{2R}^2 + K_{4I}^2 + K_{4R}^2) \nonumber \\&\quad (12 \sigma ^2 + \xi _1^2) \xi _2^2) + (4 \sigma ^2 + \xi _1^2)^2 (-(19 K_{2I}^2 + 19 K_{2R}^2 + K_{4I}^2 \nonumber \\&\quad + K_{4R}^2) (1 + \xi _2^2) + 18 K_0^2 (3 + \xi _2^2)) \nonumber \\&\quad + 2 \gamma _0^3 ((19 K_{2I}^2 + 19 K_{2R}^2 + K_{4I}^2 + K_{4R}^2) ((1 + 2 \sigma ) (4 \sigma (1 + \sigma ) + \xi _1^2)\nonumber \\&\quad + 4 \sigma \xi _1 \xi _2 + 4 \sigma \xi _2^2) - 18 K_0^2 (3 (1 + 2 \sigma ) \nonumber \\&\quad (4 \sigma (1 + \sigma ) + \xi _1^2) + 4 \sigma \xi _1 \xi _2 + 4 \sigma \xi _2^2)) + \gamma _0^4 (-(19 K_{2I}^2\nonumber \\&\quad + 19 K_{2R}^2 + K_{4I}^2 + K_{4R}^2) (1 + 4 \sigma (1 + \sigma ) + (\xi _1 + \xi _2)^2)\nonumber \\&\quad + 18 K_0^2 (3 + 12 \sigma (1 + \sigma ) + (\xi _1 + \xi _2)^2)) - 2 \gamma _0 (4 \sigma ^2 + \xi _1^2) (-(19 K_{2I}^2 \nonumber \\&\quad + 19 K_{2R}^2 + K_{4I}^2 + K_{4R}^2) (\xi _1^2 + 4 \sigma (1 + \sigma + \xi _2^2))\nonumber \\&\quad + 18 K_0^2 (3 \xi _1^2 + 4 \sigma (3 + 3 \sigma + \xi _2^2))) - 2 ((\gamma _0 - 2 \sigma )^2 + \xi _1^2) ((4 \sigma ^2 \nonumber \\&\quad + \xi _1^2) (1 + \xi _2^2) + \gamma _0^2 (1 + 4 \sigma (1 + \sigma ) + (\xi _1 + \xi _2)^2)\nonumber \\&\quad - 2 \gamma _0 (\xi _1^2 + 2 \sigma (1 + 2 \sigma + \xi _2^2))) ((K_{2I} K_{4I} \nonumber \\&\quad - K_{2R} K_{4R}) \cos (6 \delta _2])- (K_{2R} K_{4I} + K_{2I} K_{4R}) \sin (6 \delta _2))), \end{aligned}$$
(111)
$$\begin{aligned} p_5&=192f_{0}\gamma _{0}((\gamma _0^3 (K_{2I}^2 (1+2\sigma )-K_0(K_{2R}-K_{4R})(1+2\sigma ) -K_{2R}(K_{2R}-K_{4R}+2K_{2R}\sigma \nonumber \\&\quad -2K_{4R}\sigma +K_{4I}(\xi _1+\xi _2))+K_{2I}(K_{4I}+2K_{4I}\sigma -(2K_{2R}-K_{4R})(\xi _1+\xi _2)))\nonumber \\&\quad -(4\sigma ^2+\xi _1^2)(K_0(-2K_{2R}\sigma +2K_{4R}\sigma +K_{4I}\xi _1)+K_{2I}^2 (2\sigma -\xi _1\xi _2)-K_{2R}(2K_{2R}\sigma \nonumber \\&\quad -2K_{4R}\sigma +K_{4I}\xi _1+2K_{4I}\sigma \xi _2-K_{2R}\xi _1\xi _2+K_{4R}\xi _1\xi _2)\nonumber \\&\quad +K_{2I}((K_0-2K_{2R}+K_{4R})\xi _1+2(-2K_{2R}+K_{4R})\sigma \xi _2\nonumber \\&\quad +K_{4I}(2\sigma -\xi _1\xi _2)))-\gamma _0^2 (K_0(-2(K_{2R}-K_{4R})\sigma (3+4\sigma )+K_{4I}\xi _1+2K_{4I}\sigma \xi _1+(-K_{2R} +K_{4R})\xi _1^2)\nonumber \\&\quad +K_{2I}^2 (6\sigma +8\sigma ^2-\xi _1\xi _2)-K_{2R}(2(K_{2R}-K_{4R})\sigma (3+4\sigma )+K_{4I}\xi _1+4K_{4I}\sigma \xi _1+6K_{4I}\sigma \xi _2\nonumber \\&\quad +(-K_{2R}+K_{4R})\xi _1\xi _2)+K_{2I}(K_0(\xi _1+2\sigma \xi _1)-(2K_{2R}\nonumber \\&\quad -K_{4R})(\xi _1+4\sigma \xi _1+6\sigma \xi _2)+K_{4I}(6\sigma +8\sigma ^2-\xi _1\xi _2)))\nonumber \\&\quad +\gamma _0(K_0(-4(K_{2R}-K_{4R})\sigma ^2 (3+2\sigma )+4K_{4I}\sigma (1+\sigma )\xi _1-(K_{2R}-K_{4R})(1+2\sigma )\xi _1^2 +K_{4I}\xi _1^3)\nonumber \\&\quad +K_{2I}^2 (\xi _1^2+2\sigma (6\sigma +4\sigma ^2+\xi _1(\xi _1-2\xi _2)))+K_{2I}((K_0-2K_{2R}\nonumber \\&\quad +K_{4R})\xi _1(4\sigma (1+\sigma )+\xi _1^2) +K_{4I}(\xi _1^2+2\sigma (6\sigma +4\sigma ^2+\xi _1(\xi _1-2\xi _2)))-(2K_{2R}\nonumber \\&\quad -K_{4R})(12\sigma ^2+\xi _1^2)\xi _2) -K_{2R}(K_{2R}(\xi _1^2+2\sigma (6\sigma +4\sigma ^2+\xi _1(\xi _1-2\xi _2)))\nonumber \\&\quad -K_{4R}(\xi _1^2+2\sigma (6\sigma +4\sigma ^2+\xi _1(\xi _1-2\xi _2)))\nonumber \\&\quad +K_{4I}(4\sigma \xi _1+\xi _1^2 (\xi _1+\xi _2)+4\sigma ^2 (\xi _1+3\xi _2)))))\cos (3 \delta _2)+(\gamma _0^3 (-K_0(K_{2I}-K_{4I})(1+2\sigma )-K_{2I}^2 (\xi _1+\xi _2)\nonumber \\&\quad -K_{2I}(K_{4R}+2(K_{2R}+2K_{2R}\sigma +K_{4R}\sigma )-K_{4I}(\xi _1+\xi _2))\nonumber \\&\quad +K_{2R}(K_{4I}+2K_{4I}\sigma +(K_{2R}+K_{4R})(\xi _1+\xi _2)))\nonumber \\&\quad +(4\sigma ^2+\xi _1^2)(2K_0(K_{2I}-K_{4I})\sigma +K_0(K_{2R}+K_{4R})\xi _1+K_{2I}^2 (\xi _1+2\sigma \xi _2)\nonumber \\&\quad +K_{2I}(4K_{2R}\sigma +2K_{4R}\sigma -K_{4I}\xi _{1}\nonumber \\&\quad -(2K_{4I}\sigma +(2K_{2R}+K_{4R})\xi _1)\xi _2)-K_{2R}((K_{2R}+K_{4R})(\xi _1+2\sigma \xi _2)+K_{4I}(2\sigma -\xi _1\xi _2)))\nonumber \\&\quad +\gamma _0^2 (K_0(2(K_{2I}-K_{4I})\sigma (3+4\sigma )+(K_{2R}+K_{4R})(1+2\sigma )\xi _1\nonumber \\&\quad +(K_{2I}-K_{4I})\xi _1^2)+K_{2I}^2 (\xi _1+4\sigma \xi _1+6\sigma \xi _2)\nonumber \\&\quad +K_{2I}(2(2K_{2R}+K_{4R})\sigma (3+4\sigma )-K_{4I}\xi _1-4K_{4I}\sigma \xi _1\nonumber \\&\quad -(6K_{4I}\sigma +(2K_{2R}+K_{4R})\xi _1)\xi _2)\nonumber \\&\quad -K_{2R}((K_{2R}+K_{4R})(\xi _1+4\sigma \xi _1+6\sigma \xi _2)+K_{4I}(6\sigma +8\sigma ^2-\xi _1\xi _2)))\nonumber \\&\quad +\gamma _0(K_0(-4(K_{2I}-K_{4I})\sigma ^2 (3+2\sigma )\nonumber \\&\quad -4(K_{2R}+K_{4R})\sigma (1+\sigma )\xi _1-(K_{2I}-K_{4I})(1+2\sigma )\xi _1^2-(K_{2R}\nonumber \\&\quad +K_{4R})\xi _1^3)+K_{2I}(-4(2K_{2R}+K_{4R})\sigma ^2 (3+2\sigma )\nonumber \\&\quad +4K_{4I}\sigma (1+\sigma )\xi _1-(2K_{2R}+K_{4R})(1+2\sigma )\xi _1^2+K_{4I}\xi _1^3\nonumber \\&\quad +4(2K_{2R}+K_{4R})\sigma \xi _1\xi _2+K_{4I}(12\sigma ^2+\xi _1^2)\xi _2)\nonumber \\&\quad -K_{2I}^2 (4\sigma \xi _1+\xi _1^2 (\xi _1+\xi _2)+4\sigma ^2 (\xi _1+3\xi _2))\nonumber \\&\quad +K_{2R}(K_{4I}(\xi _1^2+2\sigma (6\sigma +4\sigma ^2+\xi _1(\xi _1-2\xi _2)))\nonumber \\&\quad +(K_{2R}+K_{4R})(4\sigma \xi _1+\xi _1^2 (\xi _1+\xi _2)\nonumber \\&\quad +4\sigma ^2 (\xi _1+3\xi _2)))))\sin (3\delta _2), \end{aligned}$$
(112)
$$\begin{aligned} p_4&=-128(2(4\sigma ^2+\xi _1^2)^2 (1+\xi _2^2)^2-8\gamma _0(4\sigma ^2+\xi _1^2)(1+\xi _2^2)(\xi _1^2\nonumber \\&\quad +2\sigma (1+2\sigma +\xi _2^2))+\gamma _0^4 (3f_0^2 (K_0+K_{2R}\nonumber \\&\quad +2(K_0+K_{2R})\sigma +K_{2I}(\xi _1+\xi _2))+2(1+4\sigma (1+\sigma )\nonumber \\&\quad +(\xi _1+\xi _2)^2)^2)+\gamma _0^2 (192\sigma ^3 (1+\xi _2^2)+64\sigma ^4 (3+\xi _2^2)\nonumber \\&\quad +12\sigma \xi _1(f_0^2 (K_{2I}-K_{2R}\xi _2)+4\xi _1(1+\xi _2^2))+4\sigma ^2 (3f_0^2 (K_0+K_{2R}+K_{2I}\xi _2)\nonumber \\&\quad +12(1+\xi _2^2)^2+8\xi _1^2 (3+\xi _2^2)+8\xi _1(\xi _2+\xi _2^3))\nonumber \\&\quad +\xi _1^2 (4+12\xi _1^2+3f_0^2 (K_0-K_{2R}-K_{2I}\xi _2)+4\xi _2(\xi _1+\xi _2)(2+\xi _2(\xi _1+\xi _2))))\nonumber \\&\quad -\gamma _0^3 (128\sigma ^4+64\sigma ^3 (3+\xi _2^2)\nonumber \\&\quad +4\sigma ^2 (3f_0^2 (K_0+K_{2R})+8(3+2\xi _1^2+2\xi _1\xi _2+3\xi _2^2))+\xi _1(8\xi _1(1+(\xi _1+\xi _2)^2)\nonumber \\&\quad +3f_0^2 (2K_{2I}+K_0\xi _1-K_{2R}(\xi _1+2\xi _2)))\nonumber \\&\quad +4\sigma (3f_0^2 (K_0+K_{2R}+K_{2I}(\xi _1+\xi _2))+4((1+\xi _2^2)^2+\xi _1^2 (3+\xi _2^2)+2\xi _1(\xi _2+\xi _2^3))))), \end{aligned}$$
(113)
$$\begin{aligned} p_3&= 128f_0\gamma _0((4\sigma ^2+\xi _1^2)(1+\xi _2^2)+\gamma _0^2 (1+4\sigma (1+\sigma )+(\xi _1+\xi _2)^2)\nonumber \\&\quad -2\gamma _0(\xi _1^2+2\sigma (1+2\sigma +\xi _2^2))) (((K_{2R}-K_{4R})(\gamma _0-2\sigma )+(K_{2I}+K_{4I})\xi _1)\cos (3\delta _2)\nonumber \\&\quad +((K_{2I}-K_{4I})(\gamma _0-2\sigma )-(K_{2R}+K_{4R})\xi _1)\sin (3\delta _2)), \end{aligned}$$
(114)
$$\begin{aligned} p_2&=-128(2(4\sigma ^2+\xi _1^2)^2 (1+\xi _2^2)^2-8\gamma _0(4\sigma ^2+\xi _1^2)(\xi _2^2)(\xi _1^2+2\sigma (1+2\sigma +\xi _2^2))\nonumber \\&\quad +\gamma _0^4 (3f_0^2 (K_0+K_{2R}+2(K_0+K_{2R})\sigma +K_{2I}(\xi _1+\xi _2))\nonumber \\&\quad +2(1+4\sigma (1+\sigma )+(\xi _1+\xi _2)^2)^2) +\gamma _0^2 (192\sigma ^3 (1+\xi _2^2)+64\sigma ^4 (3+\xi _2^2)+12\sigma \xi _1(f_0^2 (K_{2I}-K_{2R}\xi _2)\nonumber \\&\quad +4\xi _1(1+\xi _2^2))+4\sigma ^2 (3f_0^2 (K_0+K_{2R}+K_{2I}\xi _2)\nonumber \\&\quad +12(1+\xi _2^2)^2+8\xi _1^2 (3+\xi _2^2)+8\xi _1(\xi _2+\xi _2^3))+\xi _1^2 (4+12\xi _1^2+3f_0^2 (K_0-K_{2R}-K_{2I}\xi _2)\nonumber \\&\quad +4\xi _2(\xi _1+\xi _2)(2+\xi _2(\xi _1+\xi _2)))) -\gamma _0^3 (128\sigma ^4+64\sigma ^3 (3+\xi _2^2)+4\sigma ^2 (3f_0^2 (K_0+K_{2R})\nonumber \\&\quad +8(3+2\xi _1^2+2\xi _1\xi _2+3\xi _2^2))+\xi _1(8\xi _1(1+(\xi _1+\xi _2)^2)+3f_0^2 (2K_{2I}+K_0\xi _1-K_{2R}(\xi _1+2\xi _2)))\nonumber \\&\quad +4\sigma (3f_0^2 (K_0+K_{2R}+K_{2I}(\xi _1+\xi _2)) +4((1+\xi _2^2)^2+\xi _1^2 (3+\xi _2^2)+2\xi _1(\xi _2+\xi _2^3))))), \end{aligned}$$
(115)
$$\begin{aligned} p_0&= 256f_0^2 \gamma _0^2 ((4\sigma ^2+\xi _1^2)(1+\xi _2^2)+\gamma _0^2(1+4\sigma (1+\sigma )\nonumber \\&\quad +(\xi _1+\xi _2)^2)-2\gamma _0(\xi _1^2+2\sigma (1+2\sigma +\xi _2^2))). \end{aligned}$$
(116)

Development backbone curve with variable rigidity

$$\begin{aligned} p_{bc,0}&=-256(\gamma _0+2((-1)+\gamma _0)\sigma )^4, \end{aligned}$$
(117)
$$\begin{aligned} p_{bc,1}&=768K_0(\gamma _0-2\sigma )(\gamma _0+2((-1)+\gamma _0)\sigma )^3, \end{aligned}$$
(118)
$$\begin{aligned} p_{bc,2}&=-16(\gamma _0-2\sigma )^2(\gamma _0+2((-1)+\gamma _0)\sigma )^2(54K_0^2-19K_{2I}^2-19K_{2R}^2-K_{4I}^2-K_{4R}^2\nonumber \\&\quad +(-2K_{2I}K_{4I}+2K_{2R}K_{4R})\cos (6\delta _2) \nonumber \\&\quad +2(K_{2R}K_{4I}+K_{2I}K_{4R})\sin (6\delta _2)), \end{aligned}$$
(119)
$$\begin{aligned} p_{bc,3}&=24(\gamma _0-2\sigma )^3(\gamma _0+2((-1)+\gamma _0)\sigma )(18K_0^3+4K_{2I}K_{2R}K_{4I}\nonumber \\&\quad -2K_{2I}^2K_{4R}+2K_{2R}^2K_{4R}-K_0(19 K_{2I}^2+19K_{2R}^2+K_{4I}^2+K_{4R}^2) \nonumber \\&\quad +(3K_{2I}^2K_{2R}-2K_{2I}K_{4I}(K_0+ K_{4R})\nonumber \\&\quad -K_{2R}(K_{2R}^2-K_{4I}^2-2K_0K_{4R}+K_{4R}^2))\cos (6\delta _2) \nonumber \\&\quad +(K_{2I}^3+2K_{2R}K_{4I}(K_0-K_{4R})\nonumber \\&\quad +K_{2I}(-3K_{2R}^2-K_{4I}^2+2K_0K_{4R}+K_{4R}^2))\sin (6\delta _2)), \end{aligned}$$
(120)
$$\begin{aligned} p_{cb,4}&=-9(\gamma _0-2\sigma )^4(9K_0^4+K_0(8K_{2I}K_{2R}K_{4I}-4K_{2I}^2K_{4R}+4K_{2R}^2K_{4R})\nonumber \\&\quad +(K_{2I}^2+K_{2R}^2)(8K_{2I}^2+8K_{2R}^2-K_{4I}^2-K_{4R}^2) \nonumber \\&\quad -K_0^2(19K_{2I}^2+19K_{2R}^2+K_{4I}^2+K_{4R}^2)-2(K_0^2(K_{2I}K_{4I}-K_{2R}K_{4R})\nonumber \\&\quad +(K_{2I}^2+K_{2R}^2)(K_{2I}K_{4I}-K_{2R}K_{4R}) \nonumber \\&\quad +K_0(-3K_{2I}^2K_{2R}+K_{2R}^3-K_{2R}K_{4I}^2+2K_{2I}K_{4I}K_{4R}+K_{2R}K_{4R}^2))\cos (6\delta _2)\nonumber \\&\quad +2(K_0^2K_{2R}K_{4I}+K_{2I}K_{4R}) \nonumber \\&\quad +(K_{2I}^2+K_{2R}^2)(K_{2R}K_{4I}+K_{2I}K_{4R})+K_0(K_{2I}^3-2K_{2R}K_{4I}K_{4R}\nonumber \\&\quad + K_{2I}(-3K_{2R}^2-K_{4I}^2+K_{4R}^2)))\sin (6\delta _2). \end{aligned}$$
(121)

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Labetoulle, A., Ture Savadkoohi, A. & Gourdon, E. Detection of different dynamics of two coupled oscillators including a time-dependent cubic nonlinearity. Acta Mech 233, 259–290 (2022). https://doi.org/10.1007/s00707-021-03119-w

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  • DOI: https://doi.org/10.1007/s00707-021-03119-w

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