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Existence of periodic solutions and their stability for a sextic galactic potential function

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Abstract

The periodic solutions for the Hamiltonian function governing the sextic galactic potential function in accordance with two different methods are investigated. The first method is applied using the averaging theory of first order. The sufficient conditions on the parameters for the stability are given and analyzed. The numerical examples of families of periodic orbits are introduced. Meanwhile, the second method is presented using Lyapunov’s theorem for the holomorphic integral, where the periodic solutions depend on the type of the equilibrium points.

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Acknowledgement

The authors would like to thank the anonymous referee for their comments which substantially improved the presentation of their paper.

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Authors and Affiliations

  1. Mathematics Department, Faculty of Education, Ain Shams University, Cairo, Egypt

    F. M. El-Sabaa & H. M. Gad

  2. Mathematics Department, Faculty of Science, Tanta University, Tanta, 31527, Egypt

    T. S. Amer

  3. Physics and Engineering Mathematics Department, Faculty of Engineering, Tanta University, Tanta, 31734, Egypt

    M. A. Bek

  4. Mathematics Department, Faculty of Advanced Basic Science, Galala University, New Galala City, 43511, Egypt

    M. A. Bek

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  1. F. M. El-Sabaa
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  2. T. S. Amer
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  3. H. M. Gad
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  4. M. A. Bek
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Correspondence to H. M. Gad.

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Appendices

Appendix 1

$$\begin{aligned} &M_{0} = 5BC^{2} + D\bigl[3C^{2} - (5A + 3C)D\bigr],\\ &M_{1} = BC^{2} + D\bigl[C^{2} - (A + C)D\bigr], \\ &M_{2} = 5BC^{2} + D\bigl[C^{2} - (5A + C)D \bigr],\\ & M_{3} = B + C - 2D + l_{1}, \\ &M_{4} = A - B - 3C + 3D,\qquad M_{5} = A - 2C + D + l_{1}, \\ &M_{6} = BC + C^{2} - AD - D^{2} + (C - D)l_{1},\\ & M_{7} = - (B + C - 2D) + l_{1}, \\ &M_{8} = A - 2C + D - l_{1},\\ & M_{9} = - BC - C^{2} + D(A + D) + (C - D)l_{1}, \\ &M_{10} = 5B + C - 2D + l_{2},\\ & M_{11} = 5(A - B) + 3(D - C), \\ &M_{12} = 5A - 2C + D + l_{2},\\ & M_{13} = 5BC + C^{2} - D(5A + D) + (C - D)l_{2}, \\ &M_{14} = - (5B + C - 2D) + l_{2},\\ & M_{15} = 5A - 2C + D - l_{2}, \\ &M_{16} = - 5BC - C^{2} + D(5A + D) + (C - D)l_{2}, \end{aligned}$$

Appendix 2

$$\begin{aligned} &N_{1} = M_{4}\bigl\{ \bigl[AB - BC + C^{2} - (A + C)D + D^{2}\bigr]\\ &\phantom{N_{1} =}{}\times \bigl[( - C\bigl(3AB + B^{2} + 2AC - 3BC - 2C^{2}\bigr) \\ &\phantom{N_{1} =}{}+ (A - C) (A + 3B + 4C)D + ( - 3A + 2B + 4C)D^{2} \\ &\phantom{N_{1} =}{}- 2D^{3}\bigr] + \bigl[A^{2}\bigl( - BC + 3BD + 2D(C - 2D)\bigr) \\ &\phantom{N_{1} =}{}+ (2C - D) \bigl(2B^{2}C+ B(C - 2D) (C + D) \\ &\phantom{N_{1} =}{} + (C - 2D) \bigl(C^{2} - D^{2}\bigr) \bigr) + A\bigl( - 3B^{2}C + DB^{2} \\ &\phantom{N_{1} =}{}- (C - 2D) (2C - D) (C + D) + B\bigl(D^{2} - C^{2}\bigr) \bigr)\bigr]l_{1}\bigr\} , \\ &N_{2} = M_{4}\bigl\{ \bigl[AB - BC + C^{2} - (A + C)D + D^{2}\bigr]\\ &\phantom{N_{1} =}{}\times\bigl[( - C\bigl(3AB + B^{2} + 2AC- 3BC - 2C^{2}\bigr) \\ &\phantom{N_{1} =}{} + (A - C) (A + 3B + 4C)D + ( - 3A + 2B + 4C)D^{2} \\ &\phantom{N_{1} =}{}- 2D^{3}\bigr] + \bigl[A^{2}\bigl(BC - 3BD - 2D(C - 2D) \bigr) \\ &\phantom{N_{1} =}{}- (2C - D) \bigl(2B^{2}C + B(C- 2D) (C + D) \\ &\phantom{N_{1} =}{}+ (C - 2D) \bigl(C^{2} - D^{2}\bigr)\bigr) + A\bigl(3B^{2}C - DB^{2} \\ &\phantom{N_{1} =}{}+ (C - 2D) (2C - D) (C + D) - B\bigl(D^{2} - C^{2}\bigr) \bigr)\bigr]l_{1}\bigr\} ,\\ &N_{3} = M_{11}\bigl\{ \bigl[25AB - 5BC + C^{2} - (5A + C)D + D^{2}\bigr] \\ &\phantom{N_{1} =}{}\times\bigl[C\bigl( - 25B(3A + B)+ 5( - 2A + 3B)C + 2C^{2}\bigr) \\ &\phantom{N_{1} =}{} + (5A - C) (5A + 15B + 4C)D \\ &\phantom{N_{1} =}{}+ ( - 15A + 10B + 4C)D^{2} - 2D^{3}\bigr]\\ &\phantom{N_{1} =}{} + \bigl[ - 25A^{2}\bigl(5B(C - 3D) - 2D(C - 2D)\bigr) \\ &\phantom{N_{1} =}{}+ (2C - D) \bigl(50B^{2}C + 5B(C - 2D) (C + D) \\ &\phantom{N_{1} =}{}+ (C - 2D) \bigl(C^{2} - D^{2}\bigr)\bigr) \\ &\phantom{N_{1} =}{}- 5A\bigl(25B^{2}(3C - D) + (C - 2D) (2C - D) (C + D) \\ &\phantom{N_{1} =}{}- 5B \bigl(D^{2} - C^{2}\bigr)\bigr)\bigr]l_{2} \bigr\} , \\ &N_{4} = M_{11}\bigl\{ \bigl[25AB - 5BC + C^{2} - (5A + C)D + D^{2}\bigr]\\ &\phantom{N_{1} =}{}\times \bigl[C\bigl( - 25B(3A + B)+ 5( - 2A + 3B)C + 2C^{2}\bigr) \\ &\phantom{N_{1} =}{} + (5A - C) (5A + 15B + 4C)D + ( - 15A \\ &\phantom{N_{1} =}{}+ 10B + 4C)D^{2} - 2D^{3}\bigr]\\ &\phantom{N_{1} =}{} + \bigl[25A^{2} \bigl(5B(C - 3D) - 2D(C - 2D)\bigr) \\ &\phantom{N_{1} =}{}- (2C - D) \bigl(50B^{2}C + 5B(C - 2D) (C + D)\\ &\phantom{N_{1} =}{} + (C - 2D) \bigl(C^{2} - D^{2}\bigr)\bigr) \\ &\phantom{N_{1} =}{}+ 5A\bigl(25B^{2}(3C - D) + (C - 2D) (2C - D) (C + D)\\ &\phantom{N_{1} =}{} - 5B \bigl(D^{2} - C^{2}\bigr)\bigr)\bigr]l_{2} \bigr\} . \end{aligned}$$

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El-Sabaa, F.M., Amer, T.S., Gad, H.M. et al. Existence of periodic solutions and their stability for a sextic galactic potential function. Astrophys Space Sci 366, 74 (2021). https://doi.org/10.1007/s10509-021-03981-z

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