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. ElipeB. Miller and M. Vallejo, Bifurcations in a non-symmetric cubic potential, Astron. Atrophys., 300 (1995), 722-725.

[9]

S. FerrerM. LaraJ.F. San JuanA. 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. FerrerH. HanffmannJ. 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. LancharesA. I. PascualJ. PalaciánP. 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. MaciejewskiW. 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. ElipeB. Miller and M. Vallejo, Bifurcations in a non-symmetric cubic potential, Astron. Atrophys., 300 (1995), 722-725.

[9]

S. FerrerM. LaraJ.F. San JuanA. 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. FerrerH. HanffmannJ. 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. LancharesA. I. PascualJ. PalaciánP. 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. MaciejewskiW. 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 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 $S_1\cap S_3\cap S_4\cap S_5\cap S_6\cap S_7\cap S_9\cap S_{11}$.
[1]

Robert Magnus, Olivier Moschetta. The non-linear Schrödinger equation with non-periodic potential: infinite-bump solutions and non-degeneracy. Communications on Pure & Applied Analysis, 2012, 11 (2) : 587-626. doi: 10.3934/cpaa.2012.11.587

[2]

Mark A. Pinsky, Alexandr A. Zevin. Stability criteria for linear Hamiltonian systems with uncertain bounded periodic coefficients. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 243-250. doi: 10.3934/dcds.2005.12.243

[3]

Shiwang Ma. Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2361-2380. doi: 10.3934/cpaa.2013.12.2361

[4]

Giuseppe Cordaro. Existence and location of periodic solutions to convex and non coercive Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 983-996. doi: 10.3934/dcds.2005.12.983

[5]

Morimichi Kawasaki, Ryuma Orita. Computation of annular capacity by Hamiltonian Floer theory of non-contractible periodic trajectories. Journal of Modern Dynamics, 2017, 11: 313-339. doi: 10.3934/jmd.2017013

[6]

Roman Šimon Hilscher. On general Sturmian theory for abnormal linear Hamiltonian systems. Conference Publications, 2011, 2011 (Special) : 684-691. doi: 10.3934/proc.2011.2011.684

[7]

V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277

[8]

César E. Torres Ledesma. Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well. Communications on Pure & Applied Analysis, 2016, 15 (2) : 535-547. doi: 10.3934/cpaa.2016.15.535

[9]

Xingyong Zhang, Xianhua Tang. Some united existence results of periodic solutions for non-quadratic second order Hamiltonian systems. Communications on Pure & Applied Analysis, 2014, 13 (1) : 75-95. doi: 10.3934/cpaa.2014.13.75

[10]

Guillaume Bal, Alexandre Jollivet. Generalized stability estimates in inverse transport theory. Inverse Problems & Imaging, 2018, 12 (1) : 59-90. doi: 10.3934/ipi.2018003

[11]

Min Zhu, Ying Wang. Blow-up of solutions to the periodic generalized modified Camassa-Holm equation with varying linear dispersion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 645-661. doi: 10.3934/dcds.2017027

[12]

Ruichao Guo, Yong Li, Jiamin Xing, Xue Yang. Existence of periodic solutions of dynamic equations on time scales by averaging. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 959-971. doi: 10.3934/dcdss.2017050

[13]

Fengming Ma, Yiju Wang, Hongge Zhao. A potential reduction method for the generalized linear complementarity problem over a polyhedral cone. Journal of Industrial & Management Optimization, 2010, 6 (1) : 259-267. doi: 10.3934/jimo.2010.6.259

[14]

Meili Li, Maoan Han, Chunhai Kou. The existence of positive periodic solutions of a generalized. Mathematical Biosciences & Engineering, 2008, 5 (4) : 803-812. doi: 10.3934/mbe.2008.5.803

[15]

Shaoyun Shi, Wenlei Li. Non-integrability of generalized Yang-Mills Hamiltonian system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1645-1655. doi: 10.3934/dcds.2013.33.1645

[16]

Yavdat Il'yasov, Nadir Sari. Solutions of minimal period for a Hamiltonian system with a changing sign potential. Communications on Pure & Applied Analysis, 2005, 4 (1) : 175-185. doi: 10.3934/cpaa.2005.4.175

[17]

Jian Zhang, Wen Zhang, Xianhua Tang. Ground state solutions for Hamiltonian elliptic system with inverse square potential. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4565-4583. doi: 10.3934/dcds.2017195

[18]

Alessandro Fonda, Andrea Sfecci. Multiple periodic solutions of Hamiltonian systems confined in a box. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1425-1436. doi: 10.3934/dcds.2017059

[19]

Tianqing An, Zhi-Qiang Wang. Periodic solutions of Hamiltonian systems with anisotropic growth. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1069-1082. doi: 10.3934/cpaa.2010.9.1069

[20]

D. Motreanu, V. V. Motreanu, Nikolaos S. Papageorgiou. Nonautonomous resonant periodic systems with indefinite linear part and a nonsmooth potential. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1401-1414. doi: 10.3934/cpaa.2011.10.1401

2017 Impact Factor: 0.972

Metrics

  • PDF downloads (11)
  • HTML views (12)
  • Cited by (0)

Other articles
by authors

[Back to Top]