Existence and Stability of Periodic Orbits for a Hamiltonian System with Homogeneous Potential of Degree Five

Abstract

In this paper we consider the autonomous Hamiltonian system with two degrees of freedom associated to the function \(H=\dfrac{1}{2} (x^2+y^2)+ \frac{1}{2}(p_x^2+ p_y^2)+ V_5(x, y)\), where \(V_5(x,y)=\Big (\dfrac{A}{5}x^5+Bx^3y^2+\dfrac{C}{5}xy^4\Big )\) which is related to a homogeneous potential of degree five. We prove the existence of different families of periodic orbits and the type of stability is analyzed through the averaging theory which guarantee the existence of such orbits on adequate sets defined by the parameters ABC.

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Acknowledgements

The first author was partially supported by Proyecto DINREG 01/2017, Dirección de Investigación de la Universidad Católica de la Ssma. Concepción Chile. The second author was partially supported by Facultad de Ingeniería, Universidad Católica de la Ssma. Concepción. The authors would like to thank the referees for their valuable comments and suggestions to improve the quality of this paper.

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Correspondence to Marco Uribe.

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Appendix

Appendix

In this section we give the algebraic expressions that are used in the proof of Theorems 1 and 2.

$$\begin{aligned} L_{1 }&= (-20 A C+225 B^2-80 B C+16 C^2) (40 C^4 (701 A^2-3854 A B+5070 B^2 )\\&\quad +50 C^3 (632 A^3-3462 A^2 B+4825 A B^2+2070 B^3 )+125 C^2 (8 A^4-840 A^3 B\\&\quad -892 A^2 B^2+11343 A B^3-14875 B^4 )+1875 B C (-4 A^4+14 A^3 B+774 A^2 B^2\\&\quad -1804 A B^3+545 B^4 )+84375 B^3 (2 B-A) (A+3 B) (10 B-A)+256 C^5 (56 A\\&\quad -185 B)+3072 C^6 ).\\ L_{2}&= \sqrt{ (-20 A C+225 B^2-80 B C+16 C^2)} (2240 C^5 (33 A^2-295 A B+595 B^2 )\\&\quad +400 C^4 (109 A^3-1645 A^2 B+6254 A B^2-5585 B^3 )+500 C^3 (200 A^4-16 A^3 B\\&\quad +754 A^2 B^2-92 A B^3-12645 B^4 )-28125 B^2 C (8 A^4-64 A^3 B-578 A^2 B^2+796 A B^3\\&\quad +1185 B^4 )+625 C^2 (16 A^5-112 A^4 B-3984 A^3 B^2+3616 A^2 B^3-3979 A B^4\\&\quad +34665 B^5 )+1265625 B^4 (2 B-A) (A+3 B) (10 B-A)+512 C^6 (97 A-430 B)\\&\quad +12288 C^7 ).\\ L_{3}&= (5 C (4 A^2+28 A B-77 B^2 )-225 B^2 (A+3 B)+12 C^2 (7 A-4 B) ) (4 C^2 (4 A^2\\&\quad +A B+4 B^2 )+5 B C (4 A^2-20 A B+7 B^2 )-225 A B^3 ).\\ L_4= & {} 945 A^2-2730 A B+698 A C+1725 B^2-850 B C+105 C^2. \\ L_5&=\left( 6825 A B -3490 A C -7575 B^2 +5615 B C -882 C^2 \right) .\\ L_6&=\left( 3490 A C+6525 B^2-9710 B C+2142 C^2\right) .\\ L_7&= 13608 (-125 C^4 (38316 A^2-18388 A B+34103 B^2 )+46875 B^3 (101 B\\&\quad -91 A) (100 A^2-91 A B+7 B^2 )+625 C^3 (-87948 A^3+110376 A^2 B-24133 A B^2\\&\quad +27791 B^3 )-1875 C^2 (70204 A^4-192364 A^3 B+75471 A^2 B^2+37310 A B^3\\&\quad +9330 B^4 )-3125 B C (-65520 A^4+39271 A^3 B+178533 A^2 B^2-156765 A B^3\\&\quad +11683 B^4 )+140 C^5 (349 A+3124 B)-18816 C^6).\\ L_8&=L_7+\sqrt{4 \left( 30 L_4 L_6-9 L_5^2\right) {}^3+L_7^2}.\\ L_9&= \left( \dfrac{ 18 \root 3 \of {2} L_5^2-6 \root 3 \of {L_8} L_5-60 \root 3 \of {2} L_4 L_6+\root 3 \of {4L_8^2} }{900 L_4^2 \root 3 \of {L_8^2}}\right) ^2. \end{aligned}$$

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Uribe, M., Quispe, M. Existence and Stability of Periodic Orbits for a Hamiltonian System with Homogeneous Potential of Degree Five. Differ Equ Dyn Syst (2020). https://doi.org/10.1007/s12591-020-00526-8

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Keywords

  • Hamiltonian systems
  • Periodic orbits
  • Stability
  • Averaging theory

Mathematics Subject Classification

  • Primary 34C29
  • 37J25
  • 34C25
  • Secondary 85A05