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

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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 A, B, C.

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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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  1. Departamento de Matemática y Física Aplicadas, Facultad de Ingeniería, Universidad Católica de la Ssma, Concepción, Alonso de Ribera 2850, Concepción, VIII-Región, Chile

    Marco Uribe & Margarita Quispe

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  1. Marco Uribe
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  2. Margarita Quispe
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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 31, 743–765 (2023). https://doi.org/10.1007/s12591-020-00526-8

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