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

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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  • 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.

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


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

    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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