Periodic solutions for a dumbbell satellite equation

Abstract

In this paper, we study the existence of at least two geometrically distinct periodic solutions for a differential equation which models the planar oscillations of a dumbbell satellite under the influence of the gravity field generated by an oblate body, considering the effect of the zonal harmonic parameter \(J_{2}\). And at least one of such two periodic solutions is unstable. The proof is based on the version of the Poincaré–Birkhoff theorem due to Franks. Moreover, we also study the existence and multiplicity of periodic solutions and subharmonic solutions with winding number.

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Acknowledgements

We would like to express our great thanks to the referees for their valuable suggestions. We also would like to show our thanks to Professor Jifeng Chu (Shanghai Normal University) for his constant supervision and support. Zaitao Liang was jointly supported by the Key Program of Scientific Research Fund for Young Teachers of Anhui University of Science and Technology (QN2018109). Fangfang Liao was supported by the National Natural Science Foundation of China (Grant No. 11701375) and QingLan project of Jiangsu Province.

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Correspondence to Zaitao Liang.

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Liang, Z., Liao, F. Periodic solutions for a dumbbell satellite equation. Nonlinear Dyn 95, 2469–2476 (2019). https://doi.org/10.1007/s11071-018-4709-9

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Keywords

  • Dumbbell satellite
  • Geometrically distinct periodic solutions
  • Unstable
  • Poincaré–Birkhoff theorem

Mathematics Subject Classification

  • 34C25
  • 37C25