## Appendix A  
List of symbols  
\(m_{1}\) is the mass of the pipe. \(m_{ncs}\) is the mass of an NES. \(c_{1}\) is the damping of the pipe. \(c_{ncs}\) is the damping of an NES. \(k_{1}\) is the structural stiffness of the pipe. \(k_{ncs}\) is the cubic stiffness of an NES. \(x_{1}\), \(\dot{x}_{1}\), \(\dot{x}_{1}\) are the displacement, velocity and acceleration of the pipe. \(x_{2}\), \(\dot{x}_{2}\), \(\ddot{x}_{2}\) are the displacement, velocity and acceleration of an NES. \(\Omega\) is the vortex shedding frequency. \(z\) is the relative displacement of the pipe and NES. \(\varepsilon\) is the small value parameter. \(\sigma\) is the coordination factor. \(D_{i}(i=1,2)\) is the differential operator. cc is the conjugate complex. A is the undetermined complex function of the variable \(t_{1}\). B\({}_{2}\) is the undetermined complex function of \(z_{12}\). \(a(t_{1})\), \(b(t_{1})\), \(a(t_{1})\), \(\beta(t_{1})\) is the function related to \(t_{1}\). \(\lambda\) is the eigen value of the characteristic equation. \(\delta_{i}\) is the coefficient of the characteristic equation. The equations in the cartesian coordinate system are as follows:  
\[\left\{ \begin{array}{l}\dot{p}_{1}=\frac{1}{8}\frac{3 xp_{2}{}^{3}2_{2}+3 xq_{2} ^{3}-4\xi_{1}\omega p_{1}+4\xi_{2}\omega p_{2}+8\omega\omega q_{1}}{\omega}\\ \dot{q}_{1}=-\frac{1}{8}\frac{3 xp_{3}{}^{3}+3 xp_{2}q_{2}+8\omega p_{1}+4\xi_{1 }\omega q_{1}-4\xi_{2}\omega q_{2}+4F}{16\
u_{2}\omega^{2}+4\xi_{2}^{2}}(6\kappa p _{2}{}^{2}\rho_{2}{}^{2}q_{2}\omega+6\kappa p_{2}{}^{2}\delta\omega^{3}_{2} \omega+3 xp_{2}{}^{3}\rho_{2}\xi_{2}+6\kappa p_{2}{}^{2}\rho_{2}q_{2}\omega\\ +3 xp_{2}{}p_{2}q_{2}{}^{2}\xi_{2}+6\kappa p_{2}q_{2}{}^{2}\omega-8p_{1}\rho_{2} \xi_{1}\omega^{2}+8p_{1}{}^{2}\xi_{2}\omega^{2}-16\sigma p_{2}{}^{2}q_{2}\omega \\ -8\rho_{2}{}^{2}q_{2}\omega^{3}+3 xp_{2}{}^{3}\xi_{2}+3 xp_{2}q_{2}{}^{2}\xi_{2} +4p_{2}\rho_{2}\xi_{2}\omega^{2}+4\rho_{2}q_{1}\xi_{1}\xi_{2}\omega-4\sigma q_{ 2}\xi_{2}^{2}\omega\\ -4\sigma q_{2}\xi_{2}{}^{2}-4q_{2}\xi_{2}{}^{2}\omega+4F\rho_{2}\xi_{2})\\ \dot{q}_{2}=\frac{1}{16\omega^{2}\omega^{2}+4\xi_{2}^{2}}(6\kappa p_{2}{}^{2} \rho_{2}{}^{2}\omega+6\kappa p_{2}{}^{2}q_{2}{}^{2}\omega+6\kappa p_{2}{}^{3} \rho_{2}\omega-3 xp_{2}{}^{2}\rho_{2}q_{2}\xi_{2}\\ +6\kappa p_{2}\rho_{2}q_{2}{}^{2}\omega-3\kappa p_{2}\rho_{2}{}^{3}\xi_{2}-16p_ {2}\rho_{2}\omega^{2}-8p_{2}\rho_{2}{}^{2}\omega^{3}\\ +8\rho_{2}{}^{2}q_{1}\xi_{1}\omega^{2}-8\rho_{2}{}^{2}q_{2}\xi_{2}\omega^{2}-3 \kappa p_{2}{}^{2}q_{2}\xi_{2}-3\kappa q_{2}{}^{3}\xi_{2}+4p_{1}p_{2}\xi_{1}\xi _{2}\omega-\\ 4p_{2}\rho_{2}\xi_{2}{}^{2}\omega-4\rho_{2}q_{2}\xi_{2}\omega^{2}+8F\rho_{2}{} ^{2}\omega-4p_{2}\sigma_{5}{}^{2}-4p_{2}\xi_{2}{}^{2}\omega)\\ \end{array} \right. \tag{1}\]  
The Jacobian matrix is as follows:  
\[\left.\begin{array}{l}J(p_{1},q_{1},p_{2},q_{2})=\\ \begin{vmatrix}-\frac{\xi_{1}}{2}&\frac{3p_{2}q_{2}\kappa+2\xi_{2}\omega}{4 \omega}&\sigma&\frac{3p_{2}{}^{2}\kappa+9q_{2}{}^{2}\kappa}{8\omega}\\ \frac{2\xi_{1}\omega^{2}{}^{2}}{4\omega^{2}\rho_{2}{}^{2}+5^{2}}&J_{22}&\frac{ -\xi_{1}\xi_{2}\omega\rho_{2}}{4\omega^{2}\rho_{2}^{2}+5^{2}}&J_{24}\\ -\sigma&-\frac{-9p_{2}{}^{2}\kappa+3q_{2}{}^{2}\kappa}{8\omega}&\frac{-\xi_{1 }}{4\omega}  
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