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Dynamics of a 3D autonomous quadratic system with an invariant algebraic surface

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Abstract

An invariant algebraic surface is calculated for a 3D autonomous quadratic system. Also, the dynamics near finite singularities and near infinite singularities on the invariant algebraic surface is analyzed. Furthermore, pitchfork bifurcation is analyzed using center manifold theorem and a first integral of this quadratic system for some special parameters is provided. Finally, the dynamics of this system at infinity using the Poincare compactification in \(R^3\) is investigated and the singularly degenerate heteroclinic cycles are presented by a first integral and verified by numerical simulations.

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Acknowledgments

The author acknowledges the referees and the editor for carefully reading this paper and suggesting many helpful comments. This work was supported by the Natural Science Foundation of Shaanxi Province (Grant No. 2011EJ001 ), the Scientific Research Program Funded by Shaanxi Provincial Education Department (Grant No.12JK1077, 12JK1073), and the Scientific Research Foundation of Xijing University (Grant No. XJ130114, XJ130245, XJ130244).

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Authors and Affiliations

  1. Department of Mathematics, Xijing University, Xi’an , 710123, People’s Republic of China

    Zhen Wang, Xiaojian Xi & Yongxin Li

  2. School of Mathematics and Physics, China University of Geosciences, Wuhan , 430074, People’s Republic of China

    Zhouchao Wei

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  1. Zhen Wang
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  2. Zhouchao Wei
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  3. Xiaojian Xi
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  4. Yongxin Li
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Correspondence to Zhen Wang.

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Wang, Z., Wei, Z., Xi, X. et al. Dynamics of a 3D autonomous quadratic system with an invariant algebraic surface. Nonlinear Dyn 77, 1503–1518 (2014). https://doi.org/10.1007/s11071-014-1395-0

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