# American Institute of Mathematical Sciences

August  2018, 23(6): 2299-2337. doi: 10.3934/dcdsb.2018101

## 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 Published  July 2018

Fund Project: 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

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.
Citation: Motserrat Corbera, Jaume Llibre, Claudia Valls. Periodic orbits of perturbed non-axially symmetric potentials in 1:1:1 and 1:1:2 resonances. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2299-2337. doi: 10.3934/dcdsb.2018101
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##### References:
The plot of the regions $S_i$.
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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