# 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
##### References:
 [1] B. Barbanis, Escape regions of a quartic potential, Celest. Mech. Dyn. Astron., 48 (1990), 57-77. doi: 10.1007/BF00050676. [2] A. Buică and J. Llibre, Averaging methods for finding periodic orbits via Brouwer degree, Bull. Sci. Math., 128 (2004), 7-22. doi: 10.1016/j.bulsci.2003.09.002. [3] N. D. Caranicolas, A map for a group of resonant cases in quartic galactic hamiltonian, J. Astrophys. Astron., 22 (2001), 309-319. doi: 10.1007/BF02702274. [4] N. D. Caranicolas, Orbits in global and local galactic potentials, Astron. Astrophys. Trans., 23 (2004), 241-252. doi: 10.1080/10556790410001704668. [5] N. D. Caranicolas and G. I. Karanis, Motion in a potential creating a weak bar structure, Astron. Astrophys., 342 (1999), 389-394. [6] N. D. Caranicolas and N. D. Zotos, Investigating the nature of motion in 3D perturbed elliptic oscillators displaying exact periodic orbits, Nonlinear Dyn., 69 (2012), 1795-1805. doi: 10.1007/s11071-012-0386-2. [7] G. Contopoulos, Orbits in highly perturbed dynamical systems. Ⅱ. Stability of periodic orbits, Astron. J., 75 (1970), 108-130. doi: 10.1086/110949. [8] A. Elipe, B. Miller and M. Vallejo, Bifurcations in a non-symmetric cubic potential, Astron. Atrophys., 300 (1995), 722-725. [9] S. Ferrer, M. Lara, J.F. San Juan, A. Viatola and P. Yanguas, The Hénon and Heiles problem in three dimensions. Ⅰ. Periodic orbits near the origin, Int. J. Bifurcat. Chaos Appl. Sci. Engrg., 8 (1998), 1199-1213. doi: 10.1142/S0218127498000942. [10] S. Ferrer, H. Hanffmann, J. Palacián and P. Yanguas, On perturbed oscillators in 1-1-1: Resonance: the case of axially symmetric cubic potentials, J. Geom. Phys., 40 (2002), 320-369. doi: 10.1016/S0393-0440(01)00041-9. [11] A. Giorgilli and L. Galgani, Formal integrals for an autonomous Hamiltonian system near an equilibrium point, Celest. Mech., 17 (1978), 267-280. doi: 10.1007/BF01232832. [12] H. Hanffmann and B. Sommer, A degenerate bifurcation in the Hénon-Heiles family, Celest. Mech. Dyn. Astron., 81 (2001), 249-261. doi: 10.1023/A:1013252302027. [13] H. Hanffmann and J. C. van der Meer, On the Hamiltonian Hopf bifurcation in the 3D Hénon-Heiles family, J. Dyn. Differ. Equ., 14 (2002), 675-695. doi: 10.1023/A:1016343317119. [14] M. Hénon and C. Heiles, The applicability of the third integral of motion: some numerical experiments, Astron. J., 69 (1964), 73-79. doi: 10.1086/109234. [15] G. I. Karanis and L. Ch. Vozikis, Fast detection of chaotic behavior in galactic potentials, Astron. Nachr., 329 (2008), 403-412. doi: 10.1002/asna.200710835. [16] V. Lanchares, A. I. Pascual, J. Palacián, P. Yanguas and J. P. Salas, Perturbed ion traps: A generalization of the three-dimensional Heénon-Heiles problem, Chaos, 12 (2002), 87-99. doi: 10.1063/1.1449957. [17] J. Llibre and L. Jiménez-Lara, Periodic orbits and non-integrability of Hénon-Heiles systems, J. Phys. A: Math. Theor., 44 (2011), 205103, 14 pp. [18] N. G. Lloyd, Degree Theory, Cambridge Tracts in Mathematics, Cambridge Univesity Press, Cambridge, New York-Melbourne, 1978. [19] A. Maciejewski, W. Radzki and S. Rybicki, Periodic trajectories near degenerate equilibria in the Hénon-Heiles and Yang-Mills Hamiltonian systems, J. Dyn. Diff. Equ., 17 (2005), 475-488. doi: 10.1007/s10884-005-4577-0. [20] F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag, Berlin, 1996. [21] E. E. Zotos, Application of new dynamical spectra of orbits in Hamiltonian systems, Nonlinear Dyn., 69 (2012), 2041-2063. doi: 10.1007/s11071-012-0406-2. [22] E. E. Zotos, The fast norm vector indicator (FNVI) method: A new dynamical parameter for detecting order and chaos in Hamiltonian systems, Nonlinear Dyn., 70 (2012), 951-978. doi: 10.1007/s11071-012-0504-1.

show all references

##### References:
 [1] B. Barbanis, Escape regions of a quartic potential, Celest. Mech. Dyn. Astron., 48 (1990), 57-77. doi: 10.1007/BF00050676. [2] A. Buică and J. Llibre, Averaging methods for finding periodic orbits via Brouwer degree, Bull. Sci. Math., 128 (2004), 7-22. doi: 10.1016/j.bulsci.2003.09.002. [3] N. D. Caranicolas, A map for a group of resonant cases in quartic galactic hamiltonian, J. Astrophys. Astron., 22 (2001), 309-319. doi: 10.1007/BF02702274. [4] N. D. Caranicolas, Orbits in global and local galactic potentials, Astron. Astrophys. Trans., 23 (2004), 241-252. doi: 10.1080/10556790410001704668. [5] N. D. Caranicolas and G. I. Karanis, Motion in a potential creating a weak bar structure, Astron. Astrophys., 342 (1999), 389-394. [6] N. D. Caranicolas and N. D. Zotos, Investigating the nature of motion in 3D perturbed elliptic oscillators displaying exact periodic orbits, Nonlinear Dyn., 69 (2012), 1795-1805. doi: 10.1007/s11071-012-0386-2. [7] G. Contopoulos, Orbits in highly perturbed dynamical systems. Ⅱ. Stability of periodic orbits, Astron. J., 75 (1970), 108-130. doi: 10.1086/110949. [8] A. Elipe, B. Miller and M. Vallejo, Bifurcations in a non-symmetric cubic potential, Astron. Atrophys., 300 (1995), 722-725. [9] S. Ferrer, M. Lara, J.F. San Juan, A. Viatola and P. Yanguas, The Hénon and Heiles problem in three dimensions. Ⅰ. Periodic orbits near the origin, Int. J. Bifurcat. Chaos Appl. Sci. Engrg., 8 (1998), 1199-1213. doi: 10.1142/S0218127498000942. [10] S. Ferrer, H. Hanffmann, J. Palacián and P. Yanguas, On perturbed oscillators in 1-1-1: Resonance: the case of axially symmetric cubic potentials, J. Geom. Phys., 40 (2002), 320-369. doi: 10.1016/S0393-0440(01)00041-9. [11] A. Giorgilli and L. Galgani, Formal integrals for an autonomous Hamiltonian system near an equilibrium point, Celest. Mech., 17 (1978), 267-280. doi: 10.1007/BF01232832. [12] H. Hanffmann and B. Sommer, A degenerate bifurcation in the Hénon-Heiles family, Celest. Mech. Dyn. Astron., 81 (2001), 249-261. doi: 10.1023/A:1013252302027. [13] H. Hanffmann and J. C. van der Meer, On the Hamiltonian Hopf bifurcation in the 3D Hénon-Heiles family, J. Dyn. Differ. Equ., 14 (2002), 675-695. doi: 10.1023/A:1016343317119. [14] M. Hénon and C. Heiles, The applicability of the third integral of motion: some numerical experiments, Astron. J., 69 (1964), 73-79. doi: 10.1086/109234. [15] G. I. Karanis and L. Ch. Vozikis, Fast detection of chaotic behavior in galactic potentials, Astron. Nachr., 329 (2008), 403-412. doi: 10.1002/asna.200710835. [16] V. Lanchares, A. I. Pascual, J. Palacián, P. Yanguas and J. P. Salas, Perturbed ion traps: A generalization of the three-dimensional Heénon-Heiles problem, Chaos, 12 (2002), 87-99. doi: 10.1063/1.1449957. [17] J. Llibre and L. Jiménez-Lara, Periodic orbits and non-integrability of Hénon-Heiles systems, J. Phys. A: Math. Theor., 44 (2011), 205103, 14 pp. [18] N. G. Lloyd, Degree Theory, Cambridge Tracts in Mathematics, Cambridge Univesity Press, Cambridge, New York-Melbourne, 1978. [19] A. Maciejewski, W. Radzki and S. Rybicki, Periodic trajectories near degenerate equilibria in the Hénon-Heiles and Yang-Mills Hamiltonian systems, J. Dyn. Diff. Equ., 17 (2005), 475-488. doi: 10.1007/s10884-005-4577-0. [20] F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag, Berlin, 1996. [21] E. E. Zotos, Application of new dynamical spectra of orbits in Hamiltonian systems, Nonlinear Dyn., 69 (2012), 2041-2063. doi: 10.1007/s11071-012-0406-2. [22] E. E. Zotos, The fast norm vector indicator (FNVI) method: A new dynamical parameter for detecting order and chaos in Hamiltonian systems, Nonlinear Dyn., 70 (2012), 951-978. doi: 10.1007/s11071-012-0504-1.
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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