Invariant curves near Hamiltonian-Hopf bifurcations of four-dimensional symplectic maps

Nonlinearity

Volumn 17, Issue 2, 2004, Pages 691-710

  Jorba, Àngel ^a   Ollé, Mercè ^b  

^a UNIVERSITY OF BARCELONA   (Spain)

^b UNIVERSITAT POLITÈCNICA DE CATALUNYA   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 1642328412     PISSN: 09517715     EISSN: None     Source Type: Journal    
DOI: 10.1088/0951-7715/17/2/019     Document Type: Article

References (31)

1. [BF93] Bridges T J and Furter J E 1993 Singularity theory and equivariant symplectic maps Lecture Notes in Mathematics (Berlin: Springer) p 1558
2. [BCM95] Bridges T J, Cushman R H and Mackay R S 1995 Dynamics near an irrational collision of eigenvalues for symplectic mappings Fields Inst. Commun. 4 61-79
3. [BHJ+02] Broer H W, Hanßmann H, Jorba À, Villanueva J and Wagener F O O 2002 Normal-internal resonances in quasiperiodically forced oscillators: a conservative approach Preprint
4. [CB94] Contopoulos G and Barbanis B 1994 Periodic orbits and their bifurcations in a 3-D system Celestial Mech. 59 279-300
5. [CG88] Contopoulos G and Giorgilli A 1988 Bifurcations and complex instability in a 4-dimensional symplectic mapping Meccanica 23 19-28
6. [CJ00] Castellà E and Jorba À 2000 On the vertical families of two-dimensional tori near the triangular points of the bicircular problem Celestial Mech. 76 35-54
7. [Fro72] Froeschlé C 1972 Numerical study of a four-dimensional mapping Astron. Astrophys. 16 172-89
8. 1 Celestial Mech. 56 541-62
9. [Had85] Hadjidemetriou J D 1985 The stability of resonant orbits in planetary systems Resonances in ihe Motion of Planets, Satellites and Asteroids vol 3 Math. Dyn. Astronom. Ser. University of São Paulo pp 1-18
10. [Heg85] Heggie D G 1985 Bifurcation at complex instability Celestial Mech. 35 357-82
11. [Jor00] Jorba À 2000 On practical stability regions for the motion of a small particle close to the equilateral points of the real Earth-Moon system Hamiltonian Systems and Celestial Mechanics (HAMSYS-98) vol 6, ed J Delgado et al, World Scientific Monograph Series in Mathematics (Singapore: World Scientific) pp 197-213
12. [Jor01] Jorba À 2001 Numerical computation of the normal behaviour of invariant curves of n-dimensional maps Nonlinearity 14 943-76
13. [JV97a] Jorba À and Villanueva J 1997 On the normal behaviour of partially elliptic lower dimensional tori of Hamiltonian systems Nonlinearity 10 783-822
14. [JV97b] Jorba À and Villanueva J 1997 On the persistence of lower dimensional invariant tori under quasi-periodic perturbations J. Nonlinear Sci. 7 427-73
15. [JV98] Jorba À and Villanueva J 1998 Numerical computation of normal forms around some periodic orbits of the restricted three body problem Physica D 114 197-229
16. [JV01] Jorba À and Villanueva J 2001 The fine geometry of the Cantor families of invariant tori in Hamiltonian systems European Congress of Mathematics (Barcelona, 2000) vol II Prog. Math. vol 202 (Basel: Birkhäuser) pp 557-64
17. [KO98] Krauskopf B and Osinga H 1998 Globalizing two-dimensional unstable manifolds of maps Int. J. Bifur. Chaos Appl. Sci. Eng. 8 483-503
18. [Mee85] van der Meer J 1985 The Hamiltonian Hopf Bifurcation (Lecture Notes in Mathematics vol 1160) (New York: Springer)
19. [Mey98] Meyer KR 1998 The evolution of the stable and unstable manifold of an equilibrium point Celestial Mech. 70 159-65
20. [MH92] Meyer K R and Hall G R 1992 Introduction to Hamiltonian Dynamical Systems and the N-Body Problem (New York: Springer)
21. [Oll00] Ollé M 2000 Numerical exploration of bifurcation phenomena associated with complex instability Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems (Minneapolis, MN, 1997) vol 119 IMA Vol. Math. Appl. (New York: Springer) pp 319-26
22. [OP98a] Ollé M and Pacha J R 1998 Dynamics and bifurcation near the transition from stability to complex instability: some examples Actas del XV CEDYA/V CMA Universidad de Vigo, pp 339-44
23. [OP98b] Ollé M and Pfenniger D 1998 Vertical orbital structure and the Lagrangian points in barred galaxies Astron. Astrophys. 334 829-39
24. [OP99a] Ollé M and Pacha J R 1999 The 3d elliptic RTBP: periodic orbits which bifurcate from limiting restricted problems: complex instability Astron. Astrophys. 351 1149-64
25. [OP99b] Ollé M and Pfenniger D 1999 Bifurcation at complex instability Hamiltonian Systems with Three or More Degrees of Freedom ed C Simó NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. (S'Agaró, Spain, 19-30 June 1995) (Dordrecht: Kluwer) pp 518-22
26. [OPV] Ollé M, Pacha J R and Villanueva J 2001 Dynamics and bifurcation near the transition from stability to complex instability Proc. 4th International Symp. HAMSYS-2001 (Guanajuato, México, 19-24 March) at press
27. [Pac02] Pacha J R 2002 On the quasi-periodic Hamiltonian Andronov-Hopf bifurcation PhD thesis Universitat Politècnica de Catalunya
28. [Pfe85a] Pfenniger D 1985 Numerical study of complex instability. I mappings Astron. Astrophys. 150 97-111
29. [Pfe85b] Pfenniger D 1985 Numerical study of complex instability. II barred galaxy bulges Astron. Astrophys. 150 112-28
30. [Pfe90] Pfenniger D 1990 Stability of the Lagrangian points in stellar bars Astron. Astrophys. 230 55-66
31. [Sim94] Simó C 1994 Averaging under fast quasiperiodic forcing Hamiltonian Mechanics: Integrability and Chaotic Behaviour vol 331, ed J Seimenis NATO Adv. Sci. Inst. Ser. B Phys. (Toruń, Polland, 28 June-2 July 1993) (New York: Plenum) pp 13-34