On the influence of resonances on the asymptotic behavior of trajectories of nonlinear systems in critical cases

Abstract

This paper is devoted to the study of the asymptotic behavior of an essentially nonlinear system with resonant frequencies. Namely, it is assumed that the matrix of linear approximation of the system has several subsets of multiple purely imaginary eigenvalues. For such systems, the paper presents sufficient conditions for the asymptotic stability of the equilibrium regardless of forms higher than the third order. The main result is a power estimate for the norm of solutions of a system. A method for computing the coefficient of such an estimate is also proposed with use of the center manifold reduction and the normal form theory. It is shown that the order of the decay estimate varies for cases of a diagonalizable matrix of linear approximation and for a matrix containing a \(2 \times 2\) Jordan block. As an example, the decay estimate and the Lyapunov function are constructed explicitly for a spring-pendulum system with partial dissipation.

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Acknowledgments

The author is grateful to Prof. Frank Allgöwer for valuable suggestions and to Prof. Alexander Zuyev for fruitful discussions and constant attention to this work.

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Correspondence to Victoria Grushkovskaya.

Additional information

This work is supported by the Alexander von Humboldt Foundation.

Appendices

Appendix 1: Coefficients of the model system

In this appendix we present explicit expression for functions \(F_{1j}\), \(F_{2j}\) in Eq. (16).

$$\begin{aligned} \begin{aligned} F_{1\alpha }&=r_\alpha ^2{\sum _{j\in \mathcal {M}_s\setminus \{\alpha \}}}r_j\Big ((a_{\alpha j}+\tilde{a}_{\alpha j})\cos \theta _{\alpha _j}\\&-(b_{\alpha j}-\tilde{b}_{\alpha j})\sin {\theta _{\alpha j}}\Big )\\&+\,r_\alpha {\sum _{j_1,j_2\in \mathcal {M}_s\setminus \{\alpha \}}}r_{j_1}r_{j_2}\Big (a_{\alpha j_1j_2}\cos {(\theta _{\alpha j_1}+\theta _{\alpha j_2})}\\&+b_{\alpha j_1j_2}\sin {(\theta _{\alpha j_1}+\theta _{\alpha j_2})}\Big )\\&+\,r_\alpha {\sum _{l=1}^L}{\sum _{\underset{j_1\ne j_2}{j_1,j_2\in \mathcal {M}_l\setminus \{\alpha \}}}}r_{j_1}r_{j_2}\Big (a_{\alpha \alpha j_1j_2}\cos {\theta _{j_1 j_2}} \\&-b_{\alpha \alpha j_1j_2}\sin {\theta _{j_1j_2}} \Big )\\&+{\sum _{j_1\in \mathcal {M}_s\setminus \{\alpha \}}}{\sum _{j_2\ne \alpha }}r_{j_1}r_{j_2}^2 \Big (a_{\alpha j_1j_2j_2}\cos {\theta _{\alpha j_1}}\\&+b_{\alpha j_1j_2j_2}\sin {\theta _{\alpha j_1}}\Big )\\&+{\sum _{j_1\in \mathcal {M}_s\setminus \{\alpha \}}}{\sum _{l=1}^L}{\sum _{\underset{j_2\ne j_3}{j_2,j_3\in \mathcal {M}_l\setminus \{\alpha \}}}}r_{j_1}r_{j_2}r_{j_3}\\&\times \Big (a_{\alpha j_1j_2j_3}\cos {(\theta _{\alpha j_1}+\theta _{j_3j_2})} +b_{\alpha j_1j_2j_3}\sin {(\theta _{\alpha j_1}+\theta _{j_3j_2})}\Big ), \end{aligned} \end{aligned}$$
$$\begin{aligned} F_{2\alpha }= & {} r_\alpha ^2{\sum _{j\in \mathcal {M}_s\setminus \{\alpha \}}}r_j\Big ((b_{\alpha j}+\tilde{b}_{\alpha j})\cos {\theta _{\alpha j}}\\&+\,\,(a_{\alpha j}-\tilde{a}_{\alpha j})\sin {\theta _{\alpha j}}\Big )\\&+r_\alpha {\sum _{j_1,j_2\in \mathcal {M}_s\setminus \{\alpha \}}}r_{j_1}r_{j_2}\Big (b_{\alpha j_1j_2}\cos {(\theta _{\alpha j_1}+\theta _{\alpha j_2})}\\&-a_{\alpha j_1j_2}\sin {(\theta _{\alpha j_1}+\theta _{\alpha j_2})}\Big )\\&+\,\,r_\alpha {\sum _{l=1}^L}{\sum _{\underset{j_1\ne j_2}{j_1,j_2\in \mathcal {M}_l\setminus \{\alpha \}}}}r_{j_1}r_{j_2}\Big (b_{\alpha \alpha j_1j_2}\cos {\theta _{j_1 j_2}}\\&+a_{\alpha \alpha j_1j_2}\sin {\theta _{j_1 j_2}} \Big )\\&+{\sum _{j_1\in \mathcal {M}_s\setminus \{\alpha \}}}{\sum _{j_2\ne \alpha }}r_{j_1}r_{j_2}^2 \Big (b_{\alpha j_1j_2j_2}\cos {\theta _{\alpha j_1}}\\&{-}\,a_{\alpha j_1j_2j_2}\sin {\theta _{\alpha j_1}}\Big ){+}{\sum _{j_1\in \mathcal {M}_s\setminus \{\alpha \}}}{\sum _{l=1}^L}{\sum _{\underset{j_2\ne j_3}{j_2,j_3\in \mathcal {M}_l\setminus \{\alpha \}}}}r_{j_1}r_{j_2}r_{j_3}\\&\times \Big (b_{\alpha j_1j_2j_3}\cos {(\theta _{\alpha j_1}{+}\theta _{j_3j_2})}{-}a_{\alpha j_1j_2j_3}\sin {(\theta _{\alpha j_1}{+}\theta _{j_3j_2})}\Big ),\\&\alpha \in \mathcal {M}_s,\;s=\overline{1,L},\\ F_{1\beta }= & {} r_\beta \sum _{l=1}^L\sum \limits _{\underset{j_1{\ne }j_2}{j_1,j_2{\in } \mathcal {M}_l,}}r_{j_1}r_{j_2}\big (a_{\beta \beta j_1j_2} \cos {\theta _{j_1 j_2}}\\&- b_{\beta \beta j_1j_2}\sin {\theta _{j_1 j_2}}\big ),\\ F_{2\beta }= & {} r_\beta \sum _{l=1}^L\sum \limits _{\underset{j_1{\ne } j_2}{j_1,j_2{\in } \mathcal {M}_l,}}r_{j_1}r_{j_2}\big (b_{\beta \beta j_1j_2} \cos {\theta _{j_1 j_2}}\\&+a_{\beta \beta j_1j_2}\sin {\theta _{j_1 j_2}}\big ), \end{aligned}$$

\(\beta {\in }\mathcal {M}_0\), where \(\theta _{jk}=\theta _j-\theta _k\), and coefficients of functions \(F_s\) are defined from coefficients of Eq. (14) as follows:

$$\begin{aligned} A_{jk}= & {} \left\{ \begin{aligned} \mathrm{ReA}_{jjkk}+\mathrm{ReA}_{jkjk},&\quad \text {if }j,k\in \mathcal {M}_s,\\ j\ne k,\;s=\overline{1,L},&\\ \mathrm{ReA}_{jjkk},&\quad \text {otherwise}; \end{aligned} \right. \\ B_{jk}= & {} \left\{ \begin{aligned} \mathrm{ImA}_{jjkk}+\mathrm{ImA}_{jkjk},&\quad \text {if }j,k\in \mathcal {M}_s,\\ j\ne k,\;s=\overline{1,L},&\\ \mathrm{ImA}_{jjkk},&\quad \text {otherwise}; \end{aligned} \right. \\ a_{jj j_1 j_2}= & {} \left\{ \begin{aligned} \mathrm{Re}&\mathrm{A}_{jjj_1j_2}+\mathrm{Rea}_{j j_1j j_2},\text { if }j{\in } \mathcal {M}_s,\\&j_1,j_2{\in } \mathcal {M}_l{\setminus }\{j\},\;j_1\ne j_2,\;l,s=\overline{1,L},\\ \mathrm{Re}&\mathrm{A}_{jjj_1j_2},\qquad \qquad \ \ \text { if }j{\notin } \mathcal {M}_s,\\&\,j_1,j_2\in \mathcal {M}_l,\;j_1\ne j_2,\;s=\overline{1,L}; \end{aligned} \right. \\ b_{jj j_1 j_2}= & {} \left\{ \begin{aligned} \mathrm{Im}&\mathrm{A}_{jjj_1j_2}+\mathrm{Ima}_{j j_1j j_2},\text { if }j{\in } \mathcal {M}_s,\\&j_1,j_2{\in } \mathcal {M}_l{\setminus }\{j\},\;j_1\ne j_2,\;l,s=\overline{1,L},\\ \mathrm{Im}&\mathrm{A}_{jjj_1j_2},\qquad \qquad \ \ \text { if }j{\notin } \mathcal {M}_s,\\&j_1,j_2\in \mathcal {M}_l,\;j_1\ne j_2,\;s=\overline{1,L}; \end{aligned} \right. \end{aligned}$$
$$\begin{aligned}&a_{\alpha j}=\mathrm{ReA}_{\alpha \alpha \alpha j},\;b_{\alpha j}=\mathrm{ImA}_{\alpha \alpha \alpha j};\\&\tilde{a}_{\alpha j}=\mathrm{ReA}_{\alpha j\alpha \alpha }, \; \tilde{b}_{\alpha j}=\mathrm{ImA}_{\alpha j\alpha \alpha };\\&a_{\alpha j_1j_2}=\mathrm{ReA}_{\alpha j_1j_2\alpha },\;b_{\alpha j_1j_2}=\mathrm{ImA}_{\alpha j_1j_2\alpha };\\&a_{\alpha j_1j_2j_2}=\mathrm{ReA}_{\alpha j_1j_2j_2},\;b_{\alpha j_1j_2j_2}=\mathrm{Im}_{\alpha j_1j_2j_2};\\&a_{\alpha j_1j_2j_3}=\mathrm{ReA}_{\alpha j_1j_2j_3},\;b_{\alpha j_1j_2j_3}=\mathrm{Im}_{\alpha j_1j_2j_3}. \end{aligned}$$

Equation (16) with terms up to the third order is called to be the model system:

$$\begin{aligned} \begin{aligned}&\dot{r}_{s}=r_s\sum _{j=1}^qA_{sj}r_j^2+F_{1s}(r,\theta ),\\&r_{s}\dot{\theta }_s=r_s\sum _{j=1}^qB_{s j}r_j^2 +F_{2s}(r,\theta )+\omega _sr_s,\quad s=\overline{1,q}.\\ \end{aligned} \end{aligned}$$

Thus, the study of the asymptotic stability regardless of forms higher than the third order for the trivial solution of system (1) is reduced to studying the asymptotic stability of the invariant set \(\{(r,\theta ):r=0\}\) of the model system.

Appendix 2: Asymptotic stability conditions for system (17)

The following statement provides asymptotic stability conditions for a class of systems (17) with nonnegative diagonal coefficients.

Theorem 6

It is sufficient for asymptotic stability of the invariant set \(\big \{(r,\theta ):r=0\big \}\) of system (17) that there exists constants \(c_j>0\), \(c_{1j_1j_2}\), and \(c_{2j_1\alpha _2}\), (\(c_{1j_1j_2}=c_{1j_2j_1}\), \(c_{2j_1j_2}=-c_{2j_2j_1}\), \(j=\overline{1,q}\), \(j_1,j_2\in \mathcal {M}_l\), \(l=\overline{1,L}\)), satisfying the following conditions in the cone \(\mathcal {K}\):

  1. 1.

    \(\displaystyle {\sum _{j=1}^q}c_j\rho _s^2-{\sum _{s=1}^L}{\sum _{\underset{j_1\ne j_2}{j_1,j_2\in \mathcal {M}_s}}}\rho _{j_1}\rho _{j_2}\big (c_{1j_1j_2}^2 +c_{2j_1j_2}^2\big )^{1/2}\ge 0\);

  2. 2.

    \(\displaystyle W_1(\rho )\le 0\),

where \(W_1(\rho )\) is a quadratic form defined as follows:

$$\begin{aligned} W_1(\rho )= & {} 2{\sum _{j\in \mathcal {M}_0}}c_jA_{jj}\rho _j^2+2{\sum _{\underset{j_1\ne j_2}{j_1,j_2=1}}^q}c_jA_{j_1j_2}\rho _{j_1}\rho _{j_2}\nonumber \\&+\,\,{\sum _{l=1}^L}{\sum _{j_1\in \mathcal {M}_l}}\rho _{j_1}^2\Big ( 2c_{j_1}A_{j_1j_1} +\nonumber \\&{\sum _{j_2\in \mathcal {M}_l\setminus \{j_1\}}} \big ( w_{1j_1j_2}+ 2(c_{1j_1j_2}a_{j_2j_1j_1j_1}\nonumber \\&{-}c_{2j_1j_2}b_{j_2j_1j_1j_1})\big )\Big )\nonumber \\&+{\sum _{l=1}^L}{\sum _{\underset{j_1\ne j_2}{j_1,j_2\in \mathcal {M}_l}}}\rho _{j_1}\rho _{j_2}\big ( 2(c_{1j_1j_2}\tilde{a}_{j_1j_2}\nonumber \\&+c_{2j_1j_2} \tilde{b}_{j_1j_2})+w_{2j_1j_2}\big )\nonumber \\&+{\sum _{j_1=1}^q}{\sum _{l=1}^L} {\sum _{\underset{j_2\ne j_3}{j_2,j_3\in \mathcal {M}_l\setminus \{j_1\}}}}\rho _{j_1}\rho _{j_2} \Big (2(c_{1j_2j_3}a_{j_3j_2j_1j_1}\nonumber \\&-\,c_{2j_2j_3}b_{j_3j_2j_1j_1})+ w_{1j_1j_2j_3}\Big )\nonumber \\&+\,\frac{1}{2}{\sum _{l_1,l_2=1}^L}{\sum _{\underset{j_1\ne j_2}{j_1,j_2\in \mathcal {M}_{l_1}}}}\sum _{j_3\in \mathcal {M}_{l_2}\setminus \{j_1\}}\rho _{j_1}\rho _{j_3}w_{2j_1j_2j_3},\nonumber \\ \end{aligned}$$
(42)

with

$$\begin{aligned}&w_{1j_1j_2}=c_{j_1}(d_{j_1j_2}+\tilde{d}_{j_1j_2}) + c_{j_2}d_{j_2j_1j_1j_1}+\big (c_{1j_1j_2}^2\\&\qquad +\,c_{2j_1j_2}^2\big )^{1/2} \Big ( \big (A_{j_1j_1}^2+B_{j_1j_1}^2\big )^{1/2}+ \big (A_{j_2j_1}^2+B_{j_2j_1}^2\big )^{1/2}\\&\qquad +\,\frac{1}{2}{\sum _{j_3\in \mathcal {M}_l\setminus \{j_2\}}}\big (d_{j_2j_1j_3} +d_{j_2j_3j_1}\big ) \Big )\\&\qquad +{\sum _{j_3\in \mathcal {M}_l\setminus \{j_1,j_2\}}} \Big ( \frac{c_{j_2}}{2}\big (d_{j_2j_1j_1j_3} +d_{j_2j_1j_3j_1}\big )+\big (c_{1j_1j_2}^2\\&\qquad +\,c_{2j_1j_2}^2\big )^{1/2} d_{j_1j_3j_2j_2} +\big (c_{1j_2j_3}^2+c_{2j_2j_3}^2\big )^{1/2}d_{j_2j_1j_1j_1}\\&\qquad +{\sum _{j_4\in \mathcal {M}_l\setminus \{j_1,j_2\}}}\big ((c_{1j_1j_2}^2+c_{2j_1j_2}^2)^{1/2} \big (d_{j_2j_3j_1j_4}\\&\qquad +\,d_{j_2j_3j_4j_1}\big )+(c_{1j_2j_3}^2+c_{2j_2j_3}^2)^{1/2} \big (d_{j_2j_1j_1j_4} +d_{j_2j_1j_4j_1} \big ) \big ) \Big ),\\&w_{2j_1j_2}=c_{j_1}( d_{j_1j_2}+ \tilde{d}_{j_1j_2}+d_{j_1j_2j_2j_2})+2\big (c_{1j_1j_2}^2+ c_{2j_1j_2}^2\big )^{1/2}d_{j_1j_2}\\&\qquad +\,{\sum _{j_3\in \mathcal {M}_l\setminus \{j_1\}}}\big (c_{j_1}+\frac{1}{2}( c_{1j_1j_2}^2+c_{2j_1j_2}^2)^{1/2}\big )(d_{j_1j_2j_3} +d_{j_1j_3j_2}) \\&\qquad +{\sum _{j_3\in \mathcal {M}_l\setminus \{j_1,j_2\}}}\Big ( \frac{c_{j_1}}{2}\big (d_{j_1j_2j_2j_3} +d_{j_1j_2j_3j_2}\\&\qquad +\,d_{j_1j_3j_2j_3} +d_{j_1j_3j_3j_2}\big ) +\big (c_{1j_1j_2}^2+c_{2j_1j_2}^2\big )^{1/2}(d_{j_1j_3}+\tilde{d}_{j_1j_3})\\&\qquad +\,\big (c_{1j_1j_3}^2+c_{2j_1j_3}^2\big )^{1/2}(d_{j_1j_2} +\tilde{d}_{j_1j_2})+\frac{1}{2} {\sum _{j_4\in \mathcal {M}_l\setminus \{j_1,j_3\}}}\big (c_{1j_1j_3}^2\\&\qquad +\,c_{2j_1j_3}^2\big )^{1/2}(d_{j_1j_2j_4} +d_{j_1j_4j_2}+d_{j_3j_2j_4}+d_{j_3j_4j_2})\Big ),\\&w_{1j_1j_2j_3}=c_{j_1}\Big (d_{j_1j_1j_2j_3}+d_{j_1j_1j_3j_2}\big )+\big (c_{j_2}^2+c_{j_3}^2\big )^{1/2} \\&\qquad \times \,(d_{j_2j_3j_1j_1}+d_{j_3j_2j_1j_1})+ 2\big (c_{1j_2j_3}^2 +c_{2j_2j_3}^2\big )^{1/2}\big (A_{j_3j_1}^2\\&\qquad +\,B_{j_3j_1}^2\big )^{1/2}+{\sum _{j_4\in \mathcal {M}_l\setminus \{j_2,j_3\}}}\Big (\big (c_{1j_2j_3}^2 +c_{2j_2j_3}^2\big )^{1/2} d_{j_3j_4j_1j_1}\\&\qquad +\,\big (c_{1j_3j_4}^2+c_{2j_3j_4}^2\big )^{1/2}d_{j_3j_2j_1j_1}^2\Big ),\\&w_{2j_1j_2j_3}=\sum _{j_4\in \mathcal {M}_s\setminus {j_1}}\Big ((1-\delta _{j_2j_3})(1-\delta _{j_2j_4})\big ( c_{j_1}(d_{j_1j_2j_3j_4}\\&\qquad +\,d_{j_1j_2j_4j_3} +d_{j_2j_1j_3j_4}+ d_{j_2j_1j_4j_3})\big )+ \big ( (c_{1j_1j_2}^2+c_{2j_1j_2}^2)^{1/2}\\&\qquad (d_{j_1j_1j_3j_4}+d_{j_1j_1j_4j_3}+ d_{j_2j_2j_3j_4}+ d_{j_2j_2j_4j_2})\\&\qquad +{\sum _{j_5\in \mathcal {M}_l\setminus \{j_1,j_2\}}}\big ((c_{1j_1j_2}^2+c_{2j_1j_2}^2)^{1/2} (d_{j_2j_5j_3j_4} +d_{j_2j_5j_4j_3})\\&\qquad +\,(c_{1j_2j_5}^2+c_{2j_2j_5}^2)^{1/2} (d_{j_2j_1j_3j_4} +d_{j_2j_1j_4j_3})\big ) \big )\Big ), \end{aligned}$$

\(\delta _{j_1j_2}\) is the Kronecker delta, and \(d_{\alpha }=(a_\alpha ^2+b_{\alpha }^2)^{1/2}\), \(\tilde{d}_{\alpha }=(\tilde{a}_\alpha ^2+\tilde{b}_{\alpha }^2)^{1/2}\), for all indices \(\alpha \).

Proof

Under assumptions of the theorem, a function

$$\begin{aligned} V_1(r)= & {} {\sum _{s=1}^q}c_sr_s^2+{\sum _{s=1}^L}{\sum _{\underset{\alpha _1\ne \alpha _2}{\alpha _1, \alpha _2\in \mathcal {M}_s}}}r_{\alpha _1}r_{\alpha _2} \big (c_{1\alpha _1\alpha _2}\cos {\theta _{\alpha _1\alpha _2}}\nonumber \\&+c_{2\alpha _1\alpha _2}\sin {\theta _{\alpha _1\alpha _2}}\big ) \end{aligned}$$
(43)

is the Lyapunov function for system (17) and its time derivative along the trajectories of system (17) can be estimated from the above by function (42) with \(\rho _s=r_s^2\). \(\square \)

Appendix 3: Construction of the model system (37)

In this appendix we briefly describe the center manifold reduction for system (35). With linear transformation

$$\begin{aligned} \xi _s= & {} -\omega _1x_1,\,\xi _2=-\omega _2x_2,\,w_1=x_3,\,w_2=x_4,\nonumber \\ w_3= & {} x_5,\,\eta _1=x_6,\,\eta _2=x_7,\,w_4=x_8,\nonumber \\ w_5= & {} x_9,\,w_6=x_{10}, \end{aligned}$$
(44)

system (35) takes the form

$$\begin{aligned} \begin{aligned} \dot{\xi }_s&=-\omega _s\eta _s,\,\dot{\eta }_s=\omega _s\xi _s+X_s(\xi ,\eta ,w),\quad s=1,2,\\ \dot{w}_j&=\sum _{k=1}^5b_{jk}w_k+W_j(\xi ,\eta ,w),\quad j=\overline{1,6},\\ \end{aligned} \end{aligned}$$
(45)

where

$$\begin{aligned}&X_1(\xi ,\eta ,w)={-}\frac{\kappa _1^2(M+m)}{\omega _1gMm^2}\xi _1w_1{-}\frac{\kappa _1\kappa _2}{\omega _1Mmg}\xi _1w_2\\&\quad +\frac{\kappa \kappa _1}{\omega _1Mmg}\xi _1w_3{-}\frac{\nu _1\kappa _1}{\omega _1Mmg}\xi _1w_4\\&\quad {-}\frac{\nu _2\kappa _1}{\omega _1Mmg}\xi _1w_5 {-}\frac{2\kappa _1}{mg}\eta _1w_4\\&\quad {-}\frac{\kappa _1(M{-}3m)}{Mm\omega _1^3}\xi _1^3+ \frac{\kappa _1}{2M\omega _1\omega _2^2}\xi _1\xi _2^2\\&\quad +\frac{\kappa _1^3(M+m)}{g^2Mm^3\omega _1}\xi _1w_1^2+ \frac{\kappa _1^2\kappa _2}{\omega _1Mm^2g^2}\xi _1w_1w_2 \\&\quad {-}\,\frac{\kappa \kappa _1^2}{\omega _1Mm^2g^2}\xi _1w_1w_3+\frac{\nu _1\kappa _1^2}{\omega _1Mm^2g^2}\xi _1w_1w_4\\&\quad +\,\frac{\nu _2\kappa _1^2}{\omega _1Mm^2g^2}\xi _1w_1w_5+\frac{2\kappa _1^2}{m^2g^2}w_1\eta _1w_4,\\&X_2(\xi ,\eta ,w)= {-}\frac{\kappa _1\kappa _2}{\omega _2Mmg}\xi _2w_1{-}\frac{\kappa _2^2(M+m)}{\omega _2gMm^2}\xi _2w_2\\&\quad +\frac{\kappa \kappa _2}{\omega _2Mmg}\xi _2w_3{-}\frac{\nu _1\kappa _2}{\omega _2Mmg}\xi _2w_4\\&\quad {-}\frac{\nu _2\kappa _2}{\omega _2Mmg}\xi _2w_5{-}\frac{2\kappa _2}{mg}\eta _2w_5\\&\quad +\frac{\kappa _2}{\omega _1^2\omega _2 2M}\xi _1^2\xi _2{-}\frac{\kappa _1(M{-}3m)}{\omega _2^3Mm}\xi _2^3\\&\quad + \frac{\kappa _1\kappa _2^2}{\omega _2Mm^2g^2}\xi _2w_1w_2+\frac{\kappa _2^3(M+m)}{\omega _2g^2Mm^3}\xi _2w_2^2\\&\quad {-}\,\frac{\kappa \kappa _2^2}{\omega _2Mm^2g^2}\xi _2w_2w_3 +\omega _2\frac{\nu _1\kappa _2^2}{\omega _2Mm^2g^2}\xi _2w_2w_4\\&\quad +\,\frac{\nu _2\kappa _2^2}{\omega _2Mm^2g^2}\xi _2w_2w_5+\frac{2\kappa _2^2}{m^2g^2}w_2\eta _2w_5, \end{aligned}$$
$$\begin{aligned}&b_{jk}=1,\text { for }\,k=j+3, \text { and }0,\text { otherwise },\quad j=1,2,3,\\&b_{41}={-}\frac{\kappa _1(M+m)}{Mm},\,b_{42}={-}b_{62}={-}\frac{\kappa _2}{M},\\&b_{43}=b_{53}={-}b_{63}=\frac{\kappa }{M},\\&b_{44}={-}\frac{\nu _1(M+m)}{M},\,b_{45}={-}b_{65}={-}\frac{\nu _2}{M},\\&b_{51}={-}b_{61}={-}\frac{\kappa _1}{M},\\&b_{52}={-}\frac{\kappa _2(M+m)}{Mm},\,b_{54}={-}b_{64}={-}\frac{\nu _1}{M},\\&b_{55}={-}\frac{\nu _2(M+m)}{M},\\&W_j(\xi ,\eta ,w)\equiv 0,\text { for }j=1,2,3,\\&W_4(\xi ,\eta ,w)={-}\frac{(M{-}m)g}{2\omega _1^2M}\xi _1^2{+}\frac{mg}{2\omega _2^2M}\xi _2^2 {+}\frac{mg}{\kappa _1}\eta _1^2{+}\frac{\kappa _1}{\omega _1^2M}\xi _1^2w_1\\&\quad +\,\frac{\omega _2^2\kappa _2}{2M}\xi _1^2w_2{-}\frac{\kappa }{2\omega _1^2M}\xi _1^2w_3{+}\frac{\nu _1}{\omega _1^2M}\xi _1^2w_4{+}\frac{\nu _2}{2\omega _1^2M}\xi _1^2w_5\\&\quad +\,\frac{\kappa _2}{2\omega _2^2M}\xi _2^2w_2 {+}\frac{\nu _2}{2\omega _2^2M}\xi _2^2w_5{+}w_1\eta _1^2,\\ \end{aligned}$$
$$\begin{aligned} \begin{aligned} W_5(\xi ,\eta ,w)=&\frac{mg}{2\omega _1^2M}\xi _1^2{-}\frac{(M{-}m)g}{2\omega _2^2M}\xi _2^2 +\frac{mg}{\kappa _2}\eta _2^2+\frac{\kappa _1}{\omega _1^2M}\xi _1^2w_1\\&+\,\frac{\nu _1}{2\omega _1^2M}\xi _1^2w_4+\frac{\kappa _1}{2\omega _2^2M}\xi _2^2w_1+\frac{\kappa _2}{\omega _2^2M}\xi _2^2w_2 \\&{-}\frac{\kappa }{2\omega _2^2M}\xi _2^2w_3+\frac{\nu _1}{2\omega _2^2M}\xi _2^2w_4\\&+\,\frac{\nu _2}{\omega _2^2M}\xi _2^2w_5+w_2\eta _2^2,\\ W_6(\xi ,\eta ,w)=&{-}\frac{mg}{2\omega _1^2M}\xi _1^2{-}\frac{mg}{2\omega _2^2M}\xi _2^2{-}\frac{\kappa _1}{2\omega _1^2M}\xi _1^2w_1\\&{-} \frac{\nu _1}{\omega _1^2M}\xi _1^2w_4{-}\,\frac{\kappa _2}{2\omega _2^2M}\xi _2^2w_2\\&{-}\,\frac{\nu _2}{2\omega _2^2M}\xi _2^2w_5,\\ \end{aligned} \end{aligned}$$

Accordingly with the center manifold theory [3, 20], we introduce variables \(\zeta \) such that the second-order terms in the equations for \(\dot{w}_4\), \(\dot{w}_5\), \(\dot{w}_6\) vanish: \( \zeta _j=w_j-v_j(\xi ,\eta ),\quad j=\overline{1,6},\) where \(v_j(\xi ,\eta )\) are the second-order polynomial with constant coefficients. Then system (45) can be rewritten as

$$\begin{aligned} \dot{\xi }_s= & {} -\omega _s\eta _s,\quad s=1,2,\nonumber \\ \dot{\eta }_s= & {} \omega _s\xi _s+\widetilde{X}_s(\xi ,\eta )+\sum _{k=1}^2(\xi _kP_{s1}(\zeta )+\eta _kP_{s2}(\zeta ))+\cdots ,\nonumber \\ \dot{\zeta }_j= & {} \sum _{k=1}^5b_{jk}\zeta _k+\sum _{k=1}^6\zeta _kZ_k(\xi ,\eta )+{\cdots },\quad j=\overline{1,6},\nonumber \\ \end{aligned}$$
(46)

where \(P_{s1}(\zeta )\), \(P_{s2}(\zeta )\) contain linear and quadratic terms with respect to \(\zeta \), \(\widetilde{X}_s(\xi ,\eta )\) are the second-order forms, \(Z_k(\xi ,\eta )\) are the third-order forms, and by dots we mean terms of higher than the third orders. The next step is to eliminate linear with respect to \(\zeta \) terms from the equations for \(\dot{\eta }\). For this purpose, we put

$$\begin{aligned} y_{1s}= & {} \xi _s,\, y_{2s}= \eta _s+\sum _{j=1}^6\zeta _j(\psi _{j1}^{(s)}\xi _1+\psi _{j2}^{(s)}\xi _2\\&+\,\psi _{j3}^{(s)}\eta _1+\psi _{j4}^{(s)}\eta _2), \end{aligned}$$

where \(\psi _{jk}^{(s)}\) are constant coefficients, \(s=1,2\). Then we introduce complex conjugate variables by the substitution \( z_s=y_{1s}+iy_{2s}\), \(\bar{z}_s=y_{1s}-iy_{2s}\), \( s=1,2\), and come to the system of type (6). We decompose it into a critical subsystem

$$\begin{aligned} \dot{z}_s=i\omega _s z_s+Y_s(z,\bar{z}), \dot{\bar{z}}_s=-i\omega _s\bar{z}_s+\overline{Y_s}(z,\bar{z}),\quad s=1,2, \end{aligned}$$
(47)

and a stable subsystem

$$\begin{aligned} \dot{\zeta }_j=\sum _{k=1}^5b_{jk}\zeta _k,\quad j=\overline{1,6}. \end{aligned}$$
(48)

By applying transformations (12) and (15), we get system of type (16) and then come to the model system.

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Grushkovskaya, V. On the influence of resonances on the asymptotic behavior of trajectories of nonlinear systems in critical cases. Nonlinear Dyn 86, 587–603 (2016). https://doi.org/10.1007/s11071-016-2909-8

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Keywords

  • Asymptotic behavior
  • Decay estimate
  • Essentially nonlinear system
  • Resonance
  • Asymptotic stability
  • Lyapunov function

Mathematics Subject Classification

  • 34D05
  • 34D20
  • 70E55