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Linearizability and critical period bifurcations of a generalized Riccati system

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Abstract

In this paper, we investigate the isochronicity and linearizability problem for a cubic polynomial differential system which can be considered as a generalization of the Riccati system. Conditions for isochronicity and linearizability are found. The global structure of systems of the family with an isochronous center is determined. Furthermore, we find the order of weak center and study the problem of local bifurcation of critical periods in a neighborhood of the center.

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Notes

  1. One can download linearizability quantities \(i_1\), \(j_1\), \(\ldots \), \(i_8\), \(j_8\) and the Singular code to perform the decomposition of the variety from http://teacher.shnu.edu.cn/_upload/article/files/79/14/f36e87e342b8b0d6977e6debdeb3/3b818cf4-a6f7-4f07-8669-f4b78e48f733.txt.

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Acknowledgements

The first author acknowledges the financial support from the Slovenian Research Agency (research core funding No. P1-0306). The second author is partially supported by a CAPES grant. The third author has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Grant Agreement No. 655212, and is partially supported by the National Natural Science Foundation of China (No. 11431008) and the RFDP of Higher Education of China grant (No. 20130073110074). The first, second and third authors are also supported by Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme, FP7-PEOPLE-2012-IRSES-316338. The forth author is partially supported by the National Natural Science Foundation of China (No. 11501370). The first author thanks Professor Maoan Han for fruitful discussions on the work.

Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, People’s Republic of China

    Valery G. Romanovski & Yun Tian

  2. Faculty of Electrical Engineering and Computer Science, University of Maribor, Smetanova 17, 2000, Maribor, Slovenia

    Valery G. Romanovski

  3. Faculty of Natural Science and Mathematics, University of Maribor, Koroška c. 160, 2000, Maribor, Slovenia

    Valery G. Romanovski

  4. Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, 2000, Maribor, Slovenia

    Valery G. Romanovski & Yilei Tang

  5. Instituto de Ciências Matemáticas e de Computação - USP, Avenida Trabalhador São-carlense, 400, São Carlos, 13566-590, Brazil

    Wilker Fernandes

  6. School of Mathematical Science, Shanghai Jiao Tong University, Dongchuan Road 800, Shanghai, 200240, People’s Republic of China

    Yilei Tang

Authors
  1. Valery G. Romanovski
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  2. Wilker Fernandes
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  3. Yilei Tang
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  4. Yun Tian
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Corresponding author

Correspondence to Yilei Tang.

Appendix

Appendix

Here are listed the first two pairs of the linearizability quantities of system (1.3).

$$\begin{aligned} i_1= & {} 10 a_{02}^2 + 9 a_{03} + 4 b_{02}^2 - a_{02} b_{11} + b_{11}^2 - 3 b_{12} \\&+ 10 b_{02} b_{20} + 10 b_{20}^2 - 9 b_{30},\\ j_1= & {} 2 a_{02} b_{02} - b_{02} b_{11} - b_{11} b_{20} + b_{21},\\ i_2= & {} 168 a_{02}^2 a_{03} - 272 a_{02}^2 b_{02}^2 - 72 a_{03} b_{02}^2 - 32 b_{02}^4 \\&-\, 112 a_{02}^3 b_{11} - 42 a_{02} a_{03} b_{11} + 40 a_{02} b_{02}^2 b_{11} \\&-\, 21 a_{03} b_{11}^2- 18 b_{02}^2 b_{11}^2 - 21 a_{02} b_{11}^3 - b_{11}^4 \\&+\, 12 a_{02}^2 b_{12} - 48 b_{02}^2 b_{12} + 72 a_{02} b_{11} b_{12} - 3 b_{11}^2 b_{12} \\&+\, 18 b_{12}^2 - 48 a_{02}^2 b_{02} b_{20} - 132 a_{03} b_{02} b_{20} \\&-\, 80 b_{02}^3 b_{20} - 286 a_{02} b_{02} b_{11} b_{20} + 47 b_{02} b_{11}^2 b_{20} \\&-\, 144 b_{02} b_{12} b_{20} - 160 a_{02}^2 b_{20}^2 - 102 a_{03} b_{20}^2\\&+\, 12 b_{02}^2 b_{20}^2 - 306 a_{02} b_{11} b_{20}^2 + 61 b_{11}^2 b_{20}^2 \\&-\, 114 b_{12} b_{20}^2 + 260 b_{02} b_{20}^3 + 200 b_{20}^4\\&-\, 30 a_{02} b_{02} b_{21} - 39 b_{02} b_{11} b_{21}+ 84 a_{02} b_{20} b_{21} \\&-\, 96 b_{11} b_{20} b_{21} + 27 b_{21}^2 + 132 a_{02}^2 b_{30}+ 81 a_{03} b_{30} \\&-\, 66 b_{02}^2 b_{30} + 207 a_{02} b_{11} b_{30} - 6 b_{11}^2 b_{30}- 6 a_{02}^2 b_{11}^2 \\&+\, 81 b_{12} b_{30} - 498 b_{02} b_{20} b_{30} - 498 b_{20}^2 b_{30} + 135 b_{30}^2,\\ j_2= & {} 224 a_{02}^3 b_{02} + 240 a_{02} a_{03} b_{02} - 16 a_{02} b_{02}^3 \\&-\, 184 a_{02}^2 b_{02} b_{11} + 6 a_{03} b_{02} b_{11} + 40 b_{02}^3 b_{11} \\&+\, 124 a_{02} b_{02} b_{11}^2 - 11 b_{02} b_{11}^3 - 156 a_{02} b_{02} b_{12} \\&+ \,54 b_{02} b_{11} b_{12} - 120 a_{02}^3 b_{20} - 108 a_{02} a_{03} b_{20} \\&+ \,104 a_{02} b_{02}^2 b_{20}+ 40 a_{02}^2 b_{11} b_{20} + 27 a_{03} b_{11} b_{20}\\&+ \,38 b_{02}^2 b_{11} b_{20} + 77 a_{02} b_{11}^2 b_{20} - 8 b_{11}^3 b_{20}\\&+ \,24 a_{02} b_{12} b_{20} + 39 b_{11} b_{12} b_{20} + 140 a_{02} b_{02} b_{20}^2 \\&- \,64 b_{02} b_{11} b_{20}^2 - 120 a_{02} b_{20}^3 - 50 b_{11} b_{20}^3 \\&- \,48 a_{02}^2 b_{21} - 45 a_{03} b_{21} - 42 b_{02}^2 b_{21} - 87 a_{02} b_{11} b_{21} \\&- \,27 b_{12} b_{21} - 6 b_{02} b_{20} b_{21} + 6 b_{11}^2 b_{21}+ 30 b_{20}^2 b_{21}\\&-\, 270 a_{02} b_{02} b_{30} + 105 b_{02} b_{11} b_{30} + 108 a_{02} b_{20} b_{30}\\&+ \,84 b_{11} b_{20} b_{30} - 36 b_{21} b_{30}. \end{aligned}$$

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Romanovski, V.G., Fernandes, W., Tang, Y. et al. Linearizability and critical period bifurcations of a generalized Riccati system. Nonlinear Dyn 90, 257–269 (2017). https://doi.org/10.1007/s11071-017-3659-y

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