Figure 1.
The plot of the regions $S_i$ .
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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
| 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 |
We analytically study the Hamiltonian system in
| $ \mathbb{R}^6$ |
| $H = 1/2 (p_x^2+p_y^2+p_z^2)+\frac{1}{2} (ω_1^2 x^2+ω_2^2 y^2+ ω_3^2 z^2)+ \varepsilon(a z^3 + z (b x^2 +c y^2)),$ |
being
| $ a,b,c∈\mathbb{R}$ |
| $ c\ne 0$ |
| $ \varepsilon$ |
| $ ω_1$ |
| $ ω_2$ |
| $ ω_3$ |
| $ x$ |
| $ y$ |
| $ z$ |
| $ |\varepsilon|>0$ |
| $ H$ |
| $ ω_1 = ω_2 = ω_3/2$ |
| $ ω_1 = ω_2 = ω_3$ |
| $ a,b,c$ |
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 and 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. |
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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