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

Motserrat Corbera; Jaume Llibre; Claudia Valls

doi:10.3934/dcdsb.2018101

Article Contents

2018, Volume 23, Issue 6: 2299-2337. Doi: 10.3934/dcdsb.2018101

This issue Previous Article Quantized vortex dynamics and interaction patterns in superconductivity based on the reduced dynamical law Next Article On a coupled SDE-PDE system modeling acid-mediated tumor invasion

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

Early access: July 2018

Published: June 2018

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.

Abstract / Introduction Full Text(HTML) Figure(2) Related Papers Cited by

Abstract

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:

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

Citation:

Full Text(HTML)

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

Download: Full-size image PowerPoint slide

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 $S_1\cap S_3\cap S_4\cap S_5\cap S_6\cap S_7\cap S_9\cap S_{11}$.

Download: Full-size image PowerPoint slide

Related Papers

Cited by

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.