The Hénon and Heiles problem in three dimensions. I. Periodic orbits near the origin

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 8, Issue 6, 1998, Pages 1199-1213

  Ferrer, S ^a   Lara, M ^b   Palacian J ^c   San Juan, J F ^d   Viartola, A ^a   Yanguas, P ^c  

^a UNIVERSITY OF ZARAGOZA   (Spain)

^b Real Instituto y Observatorio de la Armada   (Spain)

^c PUBLIC UNIVERSITY OF NAVARRE   (Spain)

^d UNIVERSITY OF LA RIOJA   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0032093326     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127498000942     Document Type: Article

References (25)

1. Churchill, R. C., Pecelli, G. & Rod, D. L. [1979] "A survey of the Hénon-Heiles Hamiltonian with applications to related examples," in Stochastic Behavior in Classical and Quantum Hamiltonian Systems, eds. Casati, G. & Ford, J., Lecture Notes in Physics 93 (Springer Verlag, Berlin), pp. 76-136.
2. Churchill, R. C., Kummer, M. & Rod, D. L. [1983] "On averaging, reduction, and symmetry in Hamiltonian systems," J. Diff. Eq. 49, 359-414.
3. Cushman, R. H., Ferrer, S. & Hanßmann, H. [1998] "Singular reduction of axially-symmetric perturbations of the isotropic harmonic oscillator," submitted to Nonlinearity.
4. de Zeeuw, T. [1985] "Motion in the core of a triaxial potential," Mon. Not. R. Astr. Soc. 215, 731-760.
5. Deprit, A. [1981] "The elimination of the parallax in satellite theory," Celes. Mech. 24, 111-153.
6. Deprit, A. [1991] "The Lissajous transformation. I: Basics," Celes. Mech. Dyn. Astron. 51, 201-225.
7. Deprit, A. & Henrard, J. [1967] "Natural families of periodic orbits," Astron. J. 72, 158-172.
8. Ferrer, S. & Gárate, J. [1996] ́On perturbed 3D elliptic oscillators: A case of critical inclination in Galactic Dynamics,́ in New Trends for Hamiltonian Systems and Celestial Mechanics, eds. Lacomba, E. A. & Llibre, J., Advanced Series in Nonlinear Dynamics 8 (World Scientific, Singapore), pp. 179-197.
9. Ferrer, S., Lara, M., Palacián, J. & Yanguas, P. [1997a] ́On perturbed oscillators in 1-1-1 resonance: Critical inclination in the 3D Hénon-Heiles potential,́ in Proc. NATO-ASI Hamiltonian Systems with Three or More Degrees of Freedom, ed. Simó, C. (Kluwer Academic Publishers), pp. 378-382.
10. Ferrer, S., Palacián, J., San Juan, J. F., Viartola, A. & Yanguas, P. [1997b] ́On the Hénon and Heiles system in three dimensions: The role of the axial symmetry,́ Phys. Lett. A228, 255-260.
11. Ferrer, S., Lara, M., Palacián, J., San Juan, J. F., Viartola, A. & Yanguas, P. [1998] ́The Hénon and Heiles problem in three dimensions. II. Relative equilibria and bifurcations in the reduced system,́ Int. J. Bifurcations and Chaos 8(6), 1215-1229.
12. 3," Am. J. Phys. A33, 207-211.
13. Gutzwiller, M. C. [1990] Chaos in Classical and Quantum Mechanics, Interdisciplinary Applied Mathematics 1, (Springer-Verlag, New York).
14. Hénon, M. & Heiles, C. [1964] "The applicability of the third integral of motion: Some numerical experiments," Astron. J. 69, 73-79.
15. Hoveijn, I. [1993] Aspects of Resonance in Dynamical Systems, PhD thesis, Universiteit Utrecht, Utrecht.
16. Lara, M. [1996] "On numerical continuation of families of periodic orbits in a parametric potential," Mech. Res. Commun. 23, 291-298.
17. Lara, M., Deprit, A. & Elipe, A. [1995] "Numerical continuation of families of frozen orbits in the zonal problem of artificial satellite theory," Celes. Mech. Dyn. Astron. 62, 167-181.
18. Lieberman, M. A. [1992] Regular and Chaotic Dynamics, 2nd ed., Applied Mathematical Sciences 38, (Springer-Verlag, New York).
19. Moser, J. [1970] "Regularization of Kepler's problem and the averaging method on a manifold," Commun. Pure Appl. Math. 23, 609-636.
20. Robe, H. [1985] "Periodic orbits in a triaxial galaxy," Astron. Astrophys. 142, 351-354.
21. Sanders, J. A. & Verhulst, F. [1985] Averaging Methods in Nonlinear Dynamical Systems, Applied Mathematical Sciences 59 (Springer-Verlag, New York).
22. Verhulst, F. [1979] "Discrete symmetric dynamical systems at the main resonances with applications to axisymmetric galaxies," Phil. Trans. Roy. Soc. London A290, 435-465.
23. Verhulst, F. [1984] "Invariant manifolds and periodic solutions of three degrees of freedom Hamiltonian systems," in Trends and Applications of Pure Mathematics to Mechanics, eds. Ciarlet, P. G. & Roseau, M., Lecture Notes in Physics 195 (Springer-Verlag, Berlin), pp. 413-422.
24. ™ Book (Addison-Wesley Publishing Company, New York).
25. Yanguas, P. [1998] Integrability, Normalization and Symmetries of Hamiltonian Systems in 1-1-1 Resonance, PhD thesis, Universidad Pública de Navarra, Pamplona.