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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

Motserrat Corbera 1, , Jaume Llibre 2, and  Claudia Valls 3,

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.
Keywords: Galactic potential, generalized Hénon-Heiles hamiltonian, periodic solutions, averaging theory, linear stability.
Mathematics Subject Classification: Primary: 34C25; Secondary: 37C10, 34C29.
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

N. D. Caranicolas, Orbits in global and local galactic potentials, Astron. Astrophys. Trans., 23 (2004), 241-252.  doi: 10.1080/10556790410001704668.  Google Scholar

[5]

N. D. Caranicolas and G. I. Karanis, Motion in a potential creating a weak bar structure, Astron. Astrophys., 342 (1999), 389-394.   Google Scholar

[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.  Google Scholar

[7]

G. Contopoulos, Orbits in highly perturbed dynamical systems. Ⅱ. Stability of periodic orbits, Astron. J., 75 (1970), 108-130.  doi: 10.1086/110949.  Google Scholar

[8]

A. Elipe, B. Miller and M. Vallejo, Bifurcations in a non-symmetric cubic potential, Astron. Atrophys., 300 (1995), 722-725.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

N. G. Lloyd, Degree Theory, Cambridge Tracts in Mathematics, Cambridge Univesity Press, Cambridge, New York-Melbourne, 1978.  Google Scholar

[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.  Google Scholar

[20]

F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag, Berlin, 1996.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

N. D. Caranicolas, Orbits in global and local galactic potentials, Astron. Astrophys. Trans., 23 (2004), 241-252.  doi: 10.1080/10556790410001704668.  Google Scholar

[5]

N. D. Caranicolas and G. I. Karanis, Motion in a potential creating a weak bar structure, Astron. Astrophys., 342 (1999), 389-394.   Google Scholar

[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.  Google Scholar

[7]

G. Contopoulos, Orbits in highly perturbed dynamical systems. Ⅱ. Stability of periodic orbits, Astron. J., 75 (1970), 108-130.  doi: 10.1086/110949.  Google Scholar

[8]

A. Elipe, B. Miller and M. Vallejo, Bifurcations in a non-symmetric cubic potential, Astron. Atrophys., 300 (1995), 722-725.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

N. G. Lloyd, Degree Theory, Cambridge Tracts in Mathematics, Cambridge Univesity Press, Cambridge, New York-Melbourne, 1978.  Google Scholar

[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.  Google Scholar

[20]

F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag, Berlin, 1996.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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