Periodic orbits of perturbed non-axially symmetric potentials in 1:1:1 and 1:1:2 resonances

Motserrat Corbera; Jaume Llibre; Claudia Valls

doi:10.3934/dcdsb.2018101

Article Contents

2018, Volume 23, Issue 6: 2299-2337. Doi: 10.3934/dcdsb.2018101

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Periodic orbits of perturbed non-axially symmetric potentials in 1:1:1 and 1:1:2 resonances

1.
Departament d'enginyeries, Universitat de Vic - Universitat Central de Catalunya (UVic-UCC), C. de la Laura, 13, 08500 Vic, Barcelona, Spain
2.
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Barcelona, Spain
3.
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1049-001, Lisboa, Portugal

Received: January 2017

Revised: November 2017

Early access: July 2018

Published: June 2018

The first two authors are partially supported by MINECO grants MTM2013-40998-P and MTM2016-77278-P. The second author is also supported by an AGAUR grant 2014 SGR568. The third author is partially supported by FCT/Portugal through UID/MAT/04459/2013.

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Abstract

We analytically study the Hamiltonian system in $ \mathbb{R}^6$ with Hamiltonian

$H = 1/2 (p_x^2+p_y^2+p_z^2)+\frac{1}{2} (ω_1^2 x^2+ω_2^2 y^2+ ω_3^2 z^2)+ \varepsilon(a z^3 + z (b x^2 +c y^2)),$

being $ a,b,c∈\mathbb{R}$ with $ c\ne 0$, $ \varepsilon$ a small parameter, and $ ω_1$, $ ω_2$ and $ ω_3$the unperturbed frequencies of the oscillations along the $ x$, $ y$ and $ z$ axis, respectively. For $ |\varepsilon|>0$ small, using averaging theory of first and second order we find periodic orbits in every positive energy level of $ H$ whose frequencies are $ ω_1 = ω_2 = ω_3/2$ and $ ω_1 = ω_2 = ω_3$, respectively (the number of such periodic orbits depends on the values of the parameters $ a,b,c$). We also provide the shape of the periodic orbits and their linear stability.

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Mathematics Subject Classification: Primary: 34C25; Secondary: 37C10, 34C29.

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Figure 1. The plot of the regions $S_i$.

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Figure 2. Examples of the intersection of the regions $S_i$. a) the case $\cap_{i = 1}^{11} S_i = \emptyset$. b) the case where only one condition $S_i$ is satisfied. The top of the upper region corresponds to $S_2$, the bottom of the upper region to $S_8$, the left hand side region to $S_6$ and the right hand side region to $S_7$. c) the case where 8 different conditions $S_i$ are satisfied simultaneously. The upper region corresponds to $S_1\cap S_3\cap S_5\cap S_6\cap S_7\cap S_8\cap S_9\cap S_{11}$ and the lower one to $S_1\cap S_3\cap S_4\cap S_5\cap S_6\cap S_7\cap S_9\cap S_{11}$.

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