June  2016, 36(6): 3153-3225. doi: 10.3934/dcds.2016.36.3153

Homoclinic orbits with many loops near a $0^2 i\omega$ resonant fixed point of Hamiltonian systems

1. 

IUT de Lannion, Rue Edouard Branly, BP 30219, 22302 Lannion cedex, France

2. 

CEREMADE, UMR CNRS 7534 et PSL, Université Paris - Dauphine, Place du Maréchal De Lattre De Tassigny, 75775 Paris Cedex 16, France

3. 

Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse cedex 9, France

Received  September 2014 Revised  September 2015 Published  December 2015

In this paper we study the dynamics near the equilibrium point of a family of Hamiltonian systems in the neighborhood of a $0^2i\omega$ resonance. The existence of a family of periodic orbits surrounding the equilibrium is well-known and we show here the existence of homoclinic connections with several loops for every periodic orbit close to the origin, except the origin itself. The same problem was studied before for reversible non Hamiltonian vector fields, and the splitting of the homoclinic orbits lead to exponentially small terms which prevent the existence of homoclinic connections with one loop to exponentially small periodic orbits. The same phenomenon occurs here but we get round this difficulty thanks to geometric arguments specific to Hamiltonian systems and by studying homoclinic orbits with many loops.
Citation: Tiphaine Jézéquel, Patrick Bernard, Eric Lombardi. Homoclinic orbits with many loops near a $0^2 i\omega$ resonant fixed point of Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3153-3225. doi: 10.3934/dcds.2016.36.3153
References:
[1]

P. Bernard, Homoclinic orbit to a center manifold, Calc. Var., 17 (2003), 121-157. doi: 10.1007/s00526-002-0162-0.

[2]

P. Bernard, Homoclinic orbits in families of hypersurfaces with hyperbolic periodic orbits, J. Differential Equations, 180 (2002), 427-452. doi: 10.1006/jdeq.2001.4062.

[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. doi: 10.1016/0022-0396(69)90108-9.

[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. doi: 10.1016/0167-2789(87)90049-2.

[8]

C. Grotta Ragazzo, Irregular dynamics and homoclinic orbits to Hamiltonian Saddle-Centers, Comm. Pure App. Math. L, 50 (1997), 105-147. doi: 10.1002/(SICI)1097-0312(199702)50:2<105::AID-CPA1>3.0.CO;2-G.

[9]

C. Grotta Ragazzo, On the stability of double homoclinic loops, Comm. Math. Phys., 184 (1997), 251-272. doi: 10.1007/s002200050060.

[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. doi: 10.1007/978-0-85729-112-7.

[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. doi: 10.1017/S0308210500021119.

[12]

G. Iooss and K. Kirchgässner, Travelling waves in a chain of coupled nonlinear oscillators, Comm. Math. Phys., 211 (2000), 439-464. doi: 10.1007/s002200050821.

[13]

G. Iooss and E. Lombardi, Normal forms with exponentially small remainder: Application to homoclinic connections for the reversible $0^{2+}\i\omega$ resonance, C. R. Math. Acad. Sci. Paris, 339 (2004), 831-838. doi: 10.1016/j.crma.2004.10.002.

[14]

G. Iooss and M.-C. Pérouème, Perturbed homoclinic solutions in 1:1 resonance vector fields, J. Differential Equations, 102 (1993), 62-88. doi: 10.1006/jdeq.1993.1022.

[15]

T. Jézéquel, Formes normales de champs de vecteurs: restes exponentiellement petits dans le cas non autonome périodique et orbites homoclines à plusieurs boucles au voisinage de la résonance $0^2\i\omega$ 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. doi: 10.1016/0022-0396(82)90058-4.

[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. doi: 10.1007/s002050050029.

[18]

E. Lombardi, Oscillatory Integrals and Phenomena Beyond all Algebraic Orders, Lecture Notes in Mathematics, 1741, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104102.

[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. doi: 10.1007/BF01048157.

[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. doi: 10.1002/cpa.3160110208.

[22]

H. Rüssmann, Über das Verhalten analytischer Hamiltonscher Differentialgleichungen in der Nähe einer Gleichgewichtslösung, Math. Annalen, 154 (1964), 285-300. doi: 10.1007/BF01362565.

[23]

S. M. Sun, Non-existence of truly solitary waves in water with small surface tension, Proc. Roy. London A, 455 (1999), 2191-2228. doi: 10.1098/rspa.1999.0399.

show all references

References:
[1]

P. Bernard, Homoclinic orbit to a center manifold, Calc. Var., 17 (2003), 121-157. doi: 10.1007/s00526-002-0162-0.

[2]

P. Bernard, Homoclinic orbits in families of hypersurfaces with hyperbolic periodic orbits, J. Differential Equations, 180 (2002), 427-452. doi: 10.1006/jdeq.2001.4062.

[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. doi: 10.1016/0022-0396(69)90108-9.

[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. doi: 10.1016/0167-2789(87)90049-2.

[8]

C. Grotta Ragazzo, Irregular dynamics and homoclinic orbits to Hamiltonian Saddle-Centers, Comm. Pure App. Math. L, 50 (1997), 105-147. doi: 10.1002/(SICI)1097-0312(199702)50:2<105::AID-CPA1>3.0.CO;2-G.

[9]

C. Grotta Ragazzo, On the stability of double homoclinic loops, Comm. Math. Phys., 184 (1997), 251-272. doi: 10.1007/s002200050060.

[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. doi: 10.1007/978-0-85729-112-7.

[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. doi: 10.1017/S0308210500021119.

[12]

G. Iooss and K. Kirchgässner, Travelling waves in a chain of coupled nonlinear oscillators, Comm. Math. Phys., 211 (2000), 439-464. doi: 10.1007/s002200050821.

[13]

G. Iooss and E. Lombardi, Normal forms with exponentially small remainder: Application to homoclinic connections for the reversible $0^{2+}\i\omega$ resonance, C. R. Math. Acad. Sci. Paris, 339 (2004), 831-838. doi: 10.1016/j.crma.2004.10.002.

[14]

G. Iooss and M.-C. Pérouème, Perturbed homoclinic solutions in 1:1 resonance vector fields, J. Differential Equations, 102 (1993), 62-88. doi: 10.1006/jdeq.1993.1022.

[15]

T. Jézéquel, Formes normales de champs de vecteurs: restes exponentiellement petits dans le cas non autonome périodique et orbites homoclines à plusieurs boucles au voisinage de la résonance $0^2\i\omega$ 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. doi: 10.1016/0022-0396(82)90058-4.

[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. doi: 10.1007/s002050050029.

[18]

E. Lombardi, Oscillatory Integrals and Phenomena Beyond all Algebraic Orders, Lecture Notes in Mathematics, 1741, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104102.

[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. doi: 10.1007/BF01048157.

[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. doi: 10.1002/cpa.3160110208.

[22]

H. Rüssmann, Über das Verhalten analytischer Hamiltonscher Differentialgleichungen in der Nähe einer Gleichgewichtslösung, Math. Annalen, 154 (1964), 285-300. doi: 10.1007/BF01362565.

[23]

S. M. Sun, Non-existence of truly solitary waves in water with small surface tension, Proc. Roy. London A, 455 (1999), 2191-2228. doi: 10.1098/rspa.1999.0399.

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