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Fractional modified Duffing–Rayleigh system and its synchronization

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Abstract

Chaotic vibrations, stability and synchronization are important topics in nonlinear dynamics, and thus are studied in this paper for a new chaotic system with quadratic and cubic nonlinearly. The modified Duffing–Rayleigh system with piecewise quadratic function is presented firstly. Then, by taking the Melnikov function method, necessary conditions for chaotic motion of the modified Duffing–Rayleigh system are given. Fractional modified Duffing–Rayleigh oscillator is also discussed, and results of computer simulation demonstrate the chaotic dynamic behaviors of the fractional-order modified Duffing–Rayleigh system with order less than 2. Furthermore, generalized projective synchronization of two fractional modified Duffing–Rayleigh oscillators is explored by the active control technology. Adaptive synchronization and parameter identification of a fractional modified Duffing–Rayleigh oscillator with unknown parameters are also investigated. Numerical results validate the effectiveness and applicability of the proposed synchronization schemes.

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Acknowledgements

This work was supported by Grants from the National Natural Science Foundation of China (Nos. 11526109, 61379021), Natural Science Foundation of Fujian (Nos. 2015J05011, 2016J01671, JK2014028, JA14200), and the outstanding youth foundation of the Education Department of Fujiang Province.

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Authors and Affiliations

  1. College of Computer, Minnan Normal University, Zhangzhou, 363000, Fujian, China

    Yan-Lan Zhang

  2. School of Mathematics and Statistics, Minnan Normal University, Zhangzhou, 363000, Fujian, China

    Chang-Qing Li

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Correspondence to Yan-Lan Zhang.

Appendix

Appendix

$$\begin{aligned}&\int _{-\infty }^{+\infty }\frac{1}{\cosh ^{2}\left( \sqrt{a}t\right) }\times \tanh ^{2}\left( \sqrt{a}t\right) \hbox {d}t\\&\quad =\frac{1}{\sqrt{a}}\mu \int _{-\infty }^{+\infty } \frac{1}{\cosh ^{2}\left( \sqrt{a}t\right) }\times \tanh ^{2}\left( \sqrt{a}t\right) d\left( \sqrt{a}t\right) \\&\quad =\frac{1}{\sqrt{a}}\mu \int _{-\infty }^{+\infty }\tanh ^{2}\left( \sqrt{a}t\right) d(\tanh \sqrt{a}t)\\&\quad = \frac{1}{3\sqrt{a}}\mu \times \tanh ^{3}\left( \sqrt{a}t\right) |_{-\infty }^{+\infty }\\&\quad =\frac{2}{3\sqrt{a}}\mu ,\\&\int _{-\infty }^{+\infty }\left| \frac{1}{\cosh \left( \sqrt{a}t\right) }\times \tanh \left( \sqrt{a}t\right) \right| ^{3}\hbox {d}t\\&\quad =2 \int _{0}^{+\infty } \frac{1}{\cosh ^{3}\left( \sqrt{a}t\right) }\times \tanh ^{3}\left( \sqrt{a}t\right) \hbox {d}t\\&\quad =2 \sqrt{\frac{1}{a}}\times \int _{0}^{+\infty }\hbox {sech}^{3}\left( \sqrt{a}t\right) \times \tanh ^{3}\left( \sqrt{a}t\right) d\left( \sqrt{a}t\right) \\&\quad =-2 \sqrt{\frac{1}{a}}\times \int _{0}^{+\infty }\hbox {sech}^{2}\left( \sqrt{a}t\right) \\&\qquad \times \tanh ^{2}\left( \sqrt{a}t\right) d\hbox {sech}\left( \sqrt{a}t\right) \\&\quad =-\frac{2}{3}\sqrt{\frac{1}{a}}\times \int _{0}^{+\infty }\tanh ^{2}\left( \sqrt{a}t\right) d\hbox {sech}^{3}\left( \sqrt{a}t\right) \\&\quad =-\frac{2}{3}\sqrt{\frac{1}{a}}\times (\tanh ^{2}\left( \sqrt{a}t\right) \hbox {sech}^{3}\left( \sqrt{a}t\right) |_{0}^{+\infty }\\&\qquad -\,\int _{0}^{+\infty }\hbox {sech}^{3}\left( \sqrt{a}t\right) d\tanh ^{2}\left( \sqrt{a}t\right) )\\&\quad =\frac{4}{3}\sqrt{\frac{1}{a}}\int _{0}^{+\infty }\hbox {sech}^{5}\left( \sqrt{a}t\right) \tanh \left( \sqrt{a}t\right) d \left( \sqrt{a}t\right) \\&\quad =-\frac{4}{3}\sqrt{\frac{1}{a}}\int _{0}^{+\infty }\hbox {sech}^{4}\left( \sqrt{a}t\right) d\hbox {sech }\left( \sqrt{a}t\right) \\&\quad =-\frac{4}{15}\sqrt{\frac{1}{a}}\hbox {sech }\left( \sqrt{a}t\right) |_{0}^{+\infty }\\&\quad =\frac{4}{15}\sqrt{\frac{1}{a}}.\\&\qquad -\,\sqrt{\frac{2 a^{2}}{b}}\int _{-\infty }^{+\infty }\frac{\tanh \left( \sqrt{a}t\right) }{\cosh \left( \sqrt{a}t\right) } \cos \omega ( t-t_{0})\hbox {d}t\\&\quad =-\sqrt{\frac{2 a^{2}}{b}}\sin (\omega t_{0})F\times \int _{-\infty }^{+\infty }\frac{\tanh \left( \sqrt{a}t\right) }{\cosh \left( \sqrt{a}t\right) }\sin (\omega t)\hbox {d}t\\&\quad =-\frac{\pi \omega \sqrt{2b}}{b}F\times \frac{\sin (\omega t_{0})}{\cosh \frac{\pi \omega }{2\sqrt{a}}}. \end{aligned}$$

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Zhang, YL., Li, CQ. Fractional modified Duffing–Rayleigh system and its synchronization. Nonlinear Dyn 88, 3023–3041 (2017). https://doi.org/10.1007/s11071-017-3430-4

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