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Integrability of 3-dim polynomial systems with three invariant planes

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Abstract

We investigate the problem of integrability for a family of three-dimensional autonomous polynomial systems of ODEs. Necessary and sufficient conditions for the existence of two independent analytic first integrals for systems of the family are given. The linearizability of the systems is studied as well.

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Notes

  1. The proposition states that if an n-dim autonomous analytic system with a nonzero diagonal matrix of the linear approximation admits n−1 independent formal integrals, then there exists a convergent normalizing transformation to a Poincaré–Dulac normal form.

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Acknowledgements

Valery Romanovski acknowledges the support of the study by the Slovenian Research Agency and the Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme, FP7-PEOPLE-2012-IRSES-316338. Zhaoping Hu acknowledges the support by a grant of “The First-class Discipline of Universities in Shanghai” and Slovenian Research Agency, Slovene Human Resources Development and Scholarship Fund. The work on the paper started while V.R. was visiting the Faculty of Mechanics and Mathematics of al-Farabi Kazakh National University. He thanks the university for the invitation and the members of the Faculty for warm hospitality during his stay there. We also thank the referees for careful reading and helpful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, 200444, P.R. China

    Zhaoping Hu

  2. Al-Farabi Kazakh National University, Al-Farabi Av. 71, Almaty, Kazakhstan

    Maira Aldazharova & Tamasha M. Aldibekov

  3. Center for Applied Mathematics and Theoretical Physics and Faculty of Natural Science and Mathematics, University of Maribor, 2000, Maribor, Slovenia

    Valery G. Romanovski

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  1. Zhaoping Hu
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  2. Maira Aldazharova
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  3. Tamasha M. Aldibekov
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  4. Valery G. Romanovski
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Correspondence to Valery G. Romanovski.

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Hu, Z., Aldazharova, M., Aldibekov, T.M. et al. Integrability of 3-dim polynomial systems with three invariant planes. Nonlinear Dyn 74, 1077–1092 (2013). https://doi.org/10.1007/s11071-013-1025-2

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  • DOI: https://doi.org/10.1007/s11071-013-1025-2

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