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Generalized Hopf bifurcation of a non-smooth railway wheelset system

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Abstract

In this paper, we investigate the generalized Hopf bifurcation of a non-smooth railway wheelset system. It is to note that the system is a four-dimensional non-smooth differential equation. First, we show how to overcome the non-smoothness and reduce the four-dimensional system to a two-dimensional non-smooth system by the center manifold theorem. Since the two-dimensional central manifold is still non-smooth, we cannot apply the classical Hopf bifurcation theorem. Hence, we need to construct and analyze a Poincaré map so that a criterion for determining the generalized Hopf bifurcation occurring in the system is given. Finally, to demonstrate our theoretical results, we also give some numerical simulations which are presented to exhibit the corresponding bifurcation diagrams.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (11572263, 11672249, 11732014 and 11801079).

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Correspondence to Hebai Chen.

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Appendix: Some complicated formulae

Appendix: Some complicated formulae

$$\begin{aligned} k_{20}&= \frac{2C_2b_{34}b^3\omega ^2(s_2^2\lambda _3^2+2\omega ^2s_2^2+2\omega ^2s_1^2+2\omega s_1s_2\lambda _3)}{I_{wx}\lambda _3(\lambda _3^2+4\omega ^2)}, \\ k_{11}&= \frac{-4C_2b_{34}b^3\omega ^2(\omega s_2^2-\omega s_1^2-s_1s_2\lambda _3)}{I_{wx}(\lambda _3^2+4\omega ^2)}, \end{aligned}$$
$$\begin{aligned} k_{02}&= \frac{-2C_2b_{34}b^3\omega ^2(2\omega s_1s_2\lambda _3-s_1^2\lambda _3^2-2\omega ^2s_2^2-2\omega ^2s_1^2)}{I_{wx}\lambda _3(\lambda _3^2+4\omega ^2)}, \\ l_{20}&= \frac{2C_2b_{44}b^3\omega ^2(s_2^2\lambda _4^2+2\omega ^2s_2^2+2\omega ^2s_1^2+2\omega s_1s_2\lambda _4)}{I_{wx}\lambda _4(\lambda _4^2+4\omega ^2)}, \\ l_{11}&= \frac{-4C_2b_{44}b^3\omega ^2(\omega s_2^2-\omega s_1^2-s_1s_2\lambda _4)}{I_{wx}(\lambda _4^2+4\omega ^2)}, \\ l_{02}&= \frac{-2C_2b_{44}b^3\omega ^2(2\omega s_1s_2\lambda _4-s_1^2\lambda _4^2-2\omega ^2s_2^2-2\omega ^2s_1^2)}{I_{wx}\lambda _4(\lambda _4^2+4\omega ^2)}. \end{aligned}$$
$$\begin{aligned} \bar{k}_{20}&= \frac{2b}{(I_{wx}\alpha ^2+\omega ^2_2)(\alpha ^4+8\alpha ^2\omega ^2_1+16\omega ^4_1 +2\alpha ^2\omega ^2_2-8\omega ^2_1\omega ^2_2+\omega ^4_2)}(c_{34}C_2b^2\bar{s}^2_1\alpha ^5 \\&\quad +(2c_{34}C_2b^2\bar{s}_1\bar{s}_2\omega _1 +c_{44}C_2b^2\bar{s}^2_1\omega _2)\alpha ^4+(2c_{34}C_2b^2\bar{s}^2_2\omega ^2_1 +6c_{34}C_2b^2\bar{s}^2_1\omega ^2_1 \\&\quad +4c_{44}C_2b^2\bar{s}_1\bar{s}_2\omega _1\omega _2 +2c_{34}C_2b^2\bar{s}^2_1\omega ^2_2)\alpha ^3+(8c_{34}C_2b^2\bar{s}_1 \bar{s}_2\omega ^3_1 +2c_{44}C_2b^2\bar{s}^2_1\omega ^2_1\omega _2 \\&\quad +6c_{44}C_2b^2\bar{s}^2_2\omega ^2_1\omega _2+2c_{44}C_2b^2\bar{s}^2_1\omega ^3_2)\alpha ^2 +(8c_{34}C_2b^2\bar{s}^2_1\omega ^4_1+8c_{34}C_2b^2\bar{s}^2_2\omega ^4_1 \\&\quad -2c_{34}C_2b^2\bar{s}^2_1\omega ^2_1\omega ^2_2-6c_{34}C_2b^2\bar{s}^2_2\omega ^2_1\omega ^2_2 +4c_{44}C_2b^2\bar{s}_1\bar{s}_2\omega _1\omega ^3_2 +c_{34}C_2b^2\bar{s}^2_1\omega ^4_2)\alpha \\&\quad +8c_{44}C_2b^2\bar{s}^2_1\omega ^4_1\omega _2 +8c_{44}C_2b^2\bar{s}^2_2\omega ^4_1\omega _2 +8c_{34}C_2b^2\bar{s}_1\bar{s}_2\omega ^3_1\omega ^2_2 -6c_{44}C_2b^2\bar{s}^2_1\omega ^2_1\omega ^3_2 \\&\quad -2c_{44}C_2b^2\bar{s}^2_2\omega ^2_1\omega ^3_2 -2c_{34}C_2b^2\bar{s}_1\bar{s}_2\omega _1\omega ^4_2+c_{44}C_2b^2\bar{s}^2_1\omega ^5_2), \end{aligned}$$
$$\begin{aligned} \bar{k}_{11}&= \frac{2b}{I_{wx}(16\omega ^4_1-8\omega ^2_1\omega ^2_2 +\omega ^4_2+8\omega ^2_1\alpha ^2+2\omega ^2_2\alpha ^2+\alpha ^4)}(2c_{34}C_2b^2\bar{s}_1\bar{s}_2\alpha ^3 +(2c_{34}C_2b^2\bar{s}^2_2\omega _1 \\&\quad +2c_{44}C_2b^2\bar{s}_1\bar{s}_2\omega _2 -2c_{34}C_2b^2\bar{s}^2_1\omega _1)\alpha ^2+(2c_{34}C_2b^2\bar{s}_1\bar{s}_2\omega ^2_2 +8c_{34}C_2b^2\bar{s}_1\bar{s}_2\omega ^2_1 \\&\quad +4c_{44}C_2b^2\bar{s}^2_2\omega _1\omega _2-4c_{44}C_2b^2\bar{s}^2_1\omega _1\omega _2)\alpha -8c_{34}C_2b^2\bar{s}^2_1\omega ^3_1+8c_{34}C_2b^2\bar{s}^2_2\omega ^3_1 \\&\quad -8c_{44}C_2b^2\bar{s}_1\bar{s}_2\omega ^2_1\omega _2+2c_{34}C_2b^2\bar{s}^2_1\omega _1\omega ^2_2 -2c_{34}C_2b^2\bar{s}^2_2\omega _2\omega ^2_2+2c_{44}C_2b^2\bar{s}_1\bar{s}_2\omega ^3_2), \end{aligned}$$
$$\begin{aligned} \bar{k}_{02}&= \frac{2b}{I_{wx}(\alpha ^2+\omega ^2_2)(\alpha ^4 +8\alpha ^2\omega ^2_1+16\omega ^4_1+2\alpha ^2\omega ^2_2-8\omega ^2_1\omega ^2_2 +\omega ^4_2)}(c_{34}C_2b^2\bar{s}^2_2\alpha ^5 \\&\quad +(c_{44}C_2b^2\bar{s}^2_2\omega _2 -2c_{34}C_2b^2\bar{s}_1\bar{s}_2\omega _1)\alpha ^4+(2c_{34}C_2b^2\bar{s}^2_1\omega ^2_1 +6c_{34}C_2b^2\bar{s}^2_2\omega ^2_1 \\&\quad +2c_{34}C_2b^2\bar{s}^2_2\omega ^2_2 -4c_{44}C_2b^2\bar{s}_1\bar{s}_2\omega _1\omega _2)\alpha ^3+(6c_{44}C_2b^2\bar{s}^2_1\omega ^2_1\omega _2 +2c_{44}C_2b^2\bar{s}^2_2\omega ^2_1\omega _2 \\&\quad +2c_{44}C_2b^2\bar{s}^2_2\omega ^3_2-8c_{34}C_2b^2\bar{s}_1\bar{s}_2\omega ^3_1)\alpha ^2 +(8c_{34}C_2b^2\bar{s}^2_1\omega ^4_1+8c_{34}C_2b^2\bar{s}^2_2\omega ^4_1 \\&\quad -6c_{34}C_2b^2\bar{s}^2_1\omega ^2_1\omega ^2_2-2c_{34}C_2b^2\bar{s}^2_2\omega ^2_1\omega ^2_2 -4c_{44}C_2b^2\bar{s}_1\bar{s}_2\omega _1\omega ^3_2 +c_{34}C_2b^2\bar{s}^2_2\omega ^4_2)\alpha \\&\quad +8c_{44}C_2b^2\bar{s}^2_1\omega ^4_1\omega _2 +8c_{44}C_2b^2\bar{s}^2_2\omega ^4_1\omega _2 -8c_{34}C_2b^2\bar{s}_1\bar{s}_2\omega ^3_1\omega ^2_2 -2c_{44}C_2b^2\bar{s}^2_1\omega ^2_1\omega ^3_2 \\&\quad -6c_{44}C_2b^2\bar{s}^2_2\omega ^2_1\omega ^3_2 +2c_{34}C_2b^2\bar{s}_1\bar{s}_2\omega _1\omega ^4_2 +c_{44}C_2b^2\bar{s}^2_2\omega ^5_2), \end{aligned}$$
$$\begin{aligned} \bar{l}_{20}&= \frac{2b}{I_{wx}(\alpha ^2+\omega ^2_2)(\alpha ^4 +8\alpha ^2\omega ^2_1+16\omega ^4_1+2\alpha ^2\omega ^2_2 -8\omega ^2_1\omega ^2_2+\omega ^4_2)}(c_{34} C_2b^2\bar{s}^2_2\alpha ^5 \\&\quad +(2c_{44}C_2b^2\bar{s}_1\bar{s}_2\omega _1 -c_{34}C_2b^2\bar{s}^2_1\omega _2)\alpha ^4 +(6c_{44}C_2b^2\bar{s}^2_1\omega ^2_1+ 2c_{44}C_2b^2\bar{s}^2_2\omega ^2_1 \\&\quad -4c_{34}C_2b^2\bar{s}_1\bar{s}_2\omega _1\omega _2 +2c_{44}C_2b^2\bar{s}^2_1\omega ^2_2)\alpha ^3 +(8c_{44}C_2b^2\bar{s}_1\bar{s}_2\omega ^3_1 -2c_{34}C_2b^2\bar{s}^2_1\omega ^2_1\omega _2 \\&\quad -6c_{34}C_2b^2\bar{s}^2_2\omega ^2_1\omega _2 -2c_{34}C_2b^2\bar{s}^2_1\omega ^3_2)\alpha ^2 +(8c_{44}C_2b^2\bar{s}^2_1\omega ^4_1 +8c_{44}C_2b^2\bar{s}^2_2\omega ^4_1 \\&\quad -2c_{44}C_2b^2\bar{s}^2_1\omega ^2_1\omega ^2_2 -6c_{44}C_2b^2\bar{s}^2_2\omega ^2_1\omega ^2_2 -4c_{34}C_2b^2\bar{s}_1\bar{s}_2\omega _1\omega ^3_2 +c_{44}C_2b^2\bar{s}^2_1\omega ^4_2)\alpha \\&\quad -8c_{34}C_2b^2\bar{s}^2_1\omega ^4_1\omega _2-8c_{34}C_2 b^2\bar{s}^2_2\omega ^4_1\omega _2 +8c_{44}C_2b^2\bar{s}_1\bar{s}_2\omega ^3_1\omega ^2_2 +6c_{34}C_2b^2\bar{s}^2_1\omega ^2_1\omega ^3_2 \\&\quad +2c_{34}C_2b^2\bar{s}^2_2 \omega ^2_1\omega ^3_2 -2c_{44}C_2b^2\bar{s}_1\bar{s}_2\omega _1\omega ^4_2 -c_{34}C_2b^2\bar{s}^2_1\omega ^5_2), \end{aligned}$$
$$\begin{aligned} \bar{l}_{11}&= \frac{2b}{I_{wx}(\alpha ^4 +8\omega ^2_1\alpha ^2+16\omega ^4_1 +2\omega ^2_2\alpha ^2-8\omega ^2_1\omega ^2_2 +\omega ^4_2)}(2c_{44}C_2b^2\bar{s}_1\bar{s}_2\alpha ^3 +(2c_{44}C_2b^2\bar{s}^2_2\omega _1 \\&\quad -2c_{44}C_2b^2\bar{s}^2_1\omega _1 -2c_{34}C_2b^2\bar{s}_1\bar{s}_2\omega _2)\alpha ^2 +(8c_{44}C_2b^2\bar{s}_1\bar{s}_2\omega ^2_1 +4c_{34}C_2b^2\bar{s}^2_1\omega _1\omega _2 \\&\quad -4c_{34}C_2b^2\bar{s}^2_2\omega _1\omega _2 +2c_{44}C_2b^2\bar{s}_1\bar{s}_2\omega ^2_2)\alpha -8c_{44}C_2b^2 \bar{s}^2_1\omega ^3_1 +8c_{44}C_2b^2\bar{s}^2_2\omega ^3_1 \\&\quad +8c_{34}C_2b^2\bar{s}_1\bar{s}_2\omega ^2_1\omega _2 +2c_{44}C_2b^2\bar{s}^2_1\omega _1\omega ^2_2 -2c_{44}C_2b^2\bar{s}^2_2 \omega _1\omega ^2_2 -2c_{34}C_2b^2\bar{s}_1\bar{s}_2\omega ^3_2), \end{aligned}$$
$$\begin{aligned} \bar{l}_{02}&= \frac{2b}{I_{wx}(\alpha ^2 +\omega ^2_2)(\alpha ^4 +8\alpha ^2\omega ^2_1 +16\omega ^4 +2\alpha ^2\omega ^2_2 -8\omega ^2_1\omega ^2_2 +\omega ^4_2)}(c_{44} C_2b^2\bar{s}^2_2\alpha ^5 \\&\quad -(2c_{44}C_2b^2\bar{s}_1\bar{s}_2\omega _1 +c_{34}C_2b^2\bar{s}^2_2\omega _2)\alpha ^4 +(2c_{44}C_2b^2\bar{s}^2_1\omega ^2_1 +6c_{44}C_2b^2\bar{s}^2_2\omega ^2_1 \\&\quad +4c_{34}C_2b^2\bar{s}_1\bar{s}_2\omega _1\omega _2 +2c_{44}C_2b^2\bar{s}^2_2\omega ^2_2)\alpha ^3 -(8c_{44}C_2b^2\bar{s}_1\bar{s}_2\omega ^3_1 +6c_{34}C_2b^2\bar{s}^2_1\omega ^2_1\omega _2 \\&\quad +2c_{34}C_2b^2\bar{s}^2_2\omega ^2_1\omega _2 +2c_{34}C_2b^2\bar{s}^2_2\omega ^3_2)\alpha ^2 +(8c_{44}C_2b^2\bar{s}^2_1\omega ^4_1 +8c_{44}C_2b^2\bar{s}^2_2\omega ^4_1 \\&\quad -6c_{44}C_2b^2\bar{s}^2_1\omega ^2_1\omega ^2_2 +4c_{34}C_2b^2\bar{s}_1\bar{s}_2\omega _1\omega ^3_2 +c_{44}C_2b^2\bar{s}^2_2\omega ^4_2)\alpha -8c_{34}C_2b^2\bar{s}^2_1\omega ^4_1\omega _2 \\&\quad -8c_{34}C_2b^2\bar{s}^2_2\omega ^4_1\omega _2 -8c_{44}C_2b^2\bar{s}_1\bar{s}_2\omega ^3_1\omega ^2_2 +2c_{34}C_2b^2\bar{s}^2_1\omega ^2_1\omega ^3_2 +6c_{34}C_2b^2\bar{s}^2_2\omega ^2_1\omega ^3_2 \\&\quad +2c_{44}C_2b^2\bar{s}_1\bar{s}_2\omega _1\omega ^4_2 -c_{34}C_2b^2\bar{s}^2_2\omega ^5_2). \end{aligned}$$

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Miao, P., Li, D., Chen, H. et al. Generalized Hopf bifurcation of a non-smooth railway wheelset system. Nonlinear Dyn 100, 3277–3293 (2020). https://doi.org/10.1007/s11071-020-05702-7

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