Homoclinic orbits with many loops near a 02iω resonant fixed point of Hamiltonian systems

Discrete and Continuous Dynamical Systems- Series A

Volumn 36, Issue 6, 2016, Pages 3153-3225

  Jézéquel, Tiphaine ^a   Bernard, Patrick ^b   Lombardi, Eric ^c  

^a IUT de Lannion   (France)

^b UNIVERSITÉ PARIS DAUPHINE   (France)

^c INSTITUT DE MATHÉMATIQUES DE TOULOUSE   (France)

Author keywords

02i ; Exponentially small phenomena; Generalized solitary waves; Gevrey; Hamiltonian systems; Homoclinic orbits with several loops; Invariant manifolds; KAM; Liapunoff theorem; Normal forms

Indexed keywords

EID: 84954304532     PISSN: 10780947     EISSN: 15535231     Source Type: Journal    
DOI: 10.3934/dcds.2016.36.3153     Document Type: Article

References (23)

1. P. Bernard, Homoclinic orbit to a center manifold, Calc. Var., 17 (2003), 121-157.
2. P. Bernard, Homoclinic orbits in families of hypersurfaces with hyperbolic periodic orbits, J. Differential Equations, 180 (2002), 427-452.
3. P. Bernard, C. Grotta Ragazzo and P. A. S. Salomao, Homoclinic orbits near saddle-center fixed points of Hamiltonian systems with two degrees of freedom, Geometric Methods in Dynamics (I) - Volume in honor of Jacob Palis, Astérisque, 286 (2003), 151-165.
4. G. D. Birkhoff, Dynamical Systems, A.M.S. Coll. Publications, vol. 9, (1927), reprinted 1966.
5. C. Conley, Twist mappings, linking, analyticity, and periodic solutions which pass close to an unstable periodic solution, in Topological Dynamics (Sympos, Colorado State Univ., Ft. Collins, Colo., 1967), Benjamin, New York, 1968, 129-153.
6. C. Conley, On the ultimate behavior of orbits with respect to an unstable critical point. I Oscillating, asymptotic and capture orbits, Journ. Diff. Eqns., 5 (1969), 136-158.
7. C. Elphick, E. Tirapegui, M. E. Brachet, P. Coullet and G. Iooss, A simple global characterization for normal forms of singular vector fields, Physica D, 29 (1987), 95-127.
8. C. Grotta Ragazzo, Irregular dynamics and homoclinic orbits to Hamiltonian Saddle-Centers, Comm. Pure App. Math. L, 50 (1997), 105-147.
9. C. Grotta Ragazzo, On the stability of double homoclinic loops, Comm. Math. Phys., 184 (1997), 251-272.
10. M. Haragus and G. Iooss, Local Bifurcations, Center Manifolds, and Normal Forms in Infinite Dimensional Dynamical Systems, Springer Verlag London, Ltd., London; EDP Sciences, Les Ulis, 2011.
11. G. Iooss and K. Kirchgässner, Water waves for small surface tension: An approach via normal form, Proceedings of the Royal Society of Edinburgh, 122 (1992), 267-299.
12. G. Iooss and K. Kirchgässner, Travelling waves in a chain of coupled nonlinear oscillators, Comm. Math. Phys., 211 (2000), 439-464.
13. 2+iω resonance, C. R. Math. Acad. Sci. Paris, 339 (2004), 831-838.
14. G. Iooss and M.-C. Pérouème, Perturbed homoclinic solutions in 1:1 resonance vector fields, J. Differential Equations, 102 (1993), 62-88.
15. 2iω Hamiltonienne, Phd thesis, Université Paul Sabatier, Toulouse, 2011.
16. K. Kirchgässner, Wave solutions of reversible systems and applications, J. Diff. Equ., 45 (1982), 113-127.
17. E. Lombardi, Orbits homoclinic to exponentially small periodic orbits for a class of reversible systems. Application to water waves, Arch. Rationnal Mech. Anal., 137 (1997), 227-304.
18. E. Lombardi, Oscillatory Integrals and Phenomena Beyond all Algebraic Orders, Lecture Notes in Mathematics, 1741, Springer-Verlag, Berlin, 2000.
19. A. Mielke, P. Holmes and O. O'Reilly, Cascades of homoclinic orbits to, and chaos near, a Hamiltonian saddle center, J. Dyn. Diff. Eqns., 4 (1992), 95-126.
20. J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Gottingen Math.-Phys., 1962 (1962), 1-20.
21. J. Moser, On the generalization of a theorem of A. Liapounoff, Comm. Pure Appl. Math., 11 (1958), 257-271.
22. H. Rüssmann, Über das Verhalten analytischer Hamiltonscher Differentialgleichungen in der Nähe einer Gleichgewichtslösung, Math. Annalen, 154 (1964), 285-300.
23. S. M. Sun, Non-existence of truly solitary waves in water with small surface tension, Proc. Roy. London A, 455 (1999), 2191-2228.