Continuation of Relative Periodic Orbits in a Class of Triatomic Hamiltonian Systems

James, Guillaume; Noble, Pascal; Sire, Yannick

doi:10.1016/j.anihpc.2008.10.002

James, Guillaume ; Noble, Pascal ; Sire, Yannick

Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 4, pp. 1237-1264

MR

DOI : 10.1016/j.anihpc.2008.10.002

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     author = {James, Guillaume and Noble, Pascal and Sire, Yannick},
     title = {Continuation of {Relative} {Periodic} {Orbits} in a {Class} of {Triatomic} {Hamiltonian} {Systems}},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1237--1264},
     year = {2009},
     publisher = {Elsevier},
     volume = {26},
     number = {4},
     doi = {10.1016/j.anihpc.2008.10.002},
     mrnumber = {2542723},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2008.10.002/}
}

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James, Guillaume; Noble, Pascal; Sire, Yannick. Continuation of Relative Periodic Orbits in a Class of Triatomic Hamiltonian Systems. Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 4, pp. 1237-1264. doi: 10.1016/j.anihpc.2008.10.002

Bibliographie
Cité par

[1] Abraham R., Marsden J. E., Foundations of Mechanics, second ed., Addison-Wesley Publishing Company, 1987. | Zbl | MR

[2] Arnol'D V. I., Ordinary Differential Equations, Springer-Verlag, 1992. | Zbl | MR

[3] Aubry S., Discrete Breathers in Anharmonic Models With Acoustic Phonons, Ann. Inst. H. Poincaré Phys. Théor. 68 (1998) 381-420. | Zbl | MR | Numdam

[4] Buffoni B., Toland J., Analytic Theory of Global Bifurcation, Princeton Ser. Appl. Math., 2003. | Zbl | MR

[5] Cretegny T., Livi R., Spicci M., Breather Dynamics in Diatomic FPU Chains, Physica D 119 (1998) 88-98.

[6] Dhooge A., Govaerts W., Kuznetsov Yu. A., MATCONT: a Matlab Package for Numerical Bifurcation Analysis of ODEs, ACM Trans. Math. Software 29 (2) (2003) 141-164, Software cl_matcont available at:, http://www.matcont.ugent.be/. | Zbl | MR

[7] Doedel E. J. et al. , Computation of Periodic Solutions of Conservative Systems With Application to the 3-Body Problem, Int. J. Bifurcation and Chaos 13 (6) (2003) 1353-1381. | Zbl | MR

[8] Fura J., Rybicki S., Periodic Solutions of Second Order Hamiltonian Systems Bifurcating From Infinity, Ann. Institut H. Poincaré Anal. Non Linéaire 24 (3) (2007) 471-490. | Zbl | MR | Numdam

[9] Iooss G., Joseph D. D., Elementary Stability and Bifurcation Theory, Springer, 1980. | Zbl | MR

[10] James G., Noble P., Weak Coupling Limit and Localized Oscillations in Euclidean Invariant Hamiltonian Systems, J. Nonlinear Sci. 18 (2008) 433-461. | MR

[11] Kato T., Perturbation Theory for Linear Operators, Springer-Verlag, 1966. | Zbl | MR

[12] Kielhöfer H., Bifurcation Theory. an Introduction With Applications to PDEs, Applied Mathematical Sciences, vol. 156, Springer-Verlag, 2004. | Zbl | MR

[13] Kozin I. N., Pavlichenkov I. M., Bifurcation in Rotational Spectra of Nonlinear ${AB}_{2}$ Molecules, J. Chem. Phys. 104 (11) (1996) 4105-4113.

[14] Kozin I. N., Roberts R. M., Tennyson J., Symmetry and Structure of Rotating $H_{3}^{+}$ , J. Chem. Phys. 111 (1) (1999) 140-150.

[15] Lerman E., Tokieda T., On Relative Normal Modes, C. R. Acad. Sci. Paris, Sér. I 328 (1999) 413-418. | Zbl | MR

[16] Littlejohn R. G., Mitchell K. A., Aquilanti V., Cavalli S., Body Frames and Frame Singularities for Three-Atom Systems, Phys. Rev. A 58 (5) (1998) 3705-3717.

[17] Livi R., Spicci M., Mackay R. S., Breathers on a Diatomic FPU Chain, Nonlinearity 10 (1997) 1421-1434. | Zbl | MR

[18] Mackay R. S., Aubry S., Proof of Existence of Breathers for Time-Reversible Or Hamiltonian Networks of Weakly Coupled Oscillators, Nonlinearity 7 (1994) 1623-1643. | Zbl | MR

[19] Mackay R. S., Optic Discrete Breathers in Euclidean Invariant Systems, I, J. Nonlin. Sci. Num. Sim. 1 (2000) 99-103. | Zbl | MR

[20] Marin J. L., Aubry S., Breathers in Nonlinear Lattices: Numerical Calculation From the Anticontinuous Limit, Nonlinearity 9 (1996) 1501-1528. | Zbl | MR

[21] Marsden J. E., Hughes T., Mathematical Foundations of Elasticity, Dover Publications, 1994. | MR

[22] Mawhin J., Willem M., Critical Point Theory and Hamiltonian Systems, Springer, New York, 1989. | Zbl | MR

[23] Meyer K. R., Periodic Solutions of the N-Body Problem, Lecture Notes in Mathematics, vol. 1719, Springer-Verlag, 1999. | Zbl | MR

[24] Montaldi J., Persistance D'orbites Périodiques Relatives Dans Les Systèmes Hamiltoniens Symétriques, C. R. Acad. Sci. Paris, Sér. I 324 (1997) 553-558. | Zbl | MR

[25] Montaldi J. A., Roberts R. M., Relative Equilibria of Molecules, J. Nonlinear Sci. 9 (1999) 53-88. | Zbl | MR

[26] Moser J., Periodic Orbits Near an Equilibrium and a Theorem by Alan Weinstein, Comm. Pure Appl. Math. 29 (1976) 727-747. | Zbl | MR

[27] Muñoz-Almaraz F. J. et al. , Continuation of Periodic Orbits in Conservative and Hamiltonian Systems, Physica D 181 (2003) 1-38. | Zbl | MR

[28] Ortega J.-P., Relative Normal Modes for Nonlinear Hamiltonian Systems, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003) 665-704. | Zbl | MR

[29] Ortega J.-P., Ratiu T. S., Persistance Et Différentiabilité De L'ensemble Des Éléments Critiques Relatifs Dans Les Systèmes Hamiltoniens Symétriques, C. R. Acad. Sci. Paris, Sér. I 325 (1997) 1107-1111. | Zbl | MR

[30] Prosmiti R., Farantos S. C., Periodic Orbits, Bifurcation Diagrams and the Spectroscopy of $C_{2} H_{2}$ System, J. Chem. Phys. 103 (9) (1995) 3299-3314.

[31] Prosmiti R., Farantos S. C., Guo H., Assigning the Transition From Normal to Local Vibrational Mode in ${SO}_{2}$ by Periodic Orbits, Chem. Phys. Lett. 311 (1999) 241-247.

[32] Prosmiti R. et al. , A Combined Classical/Quantum Study of the Photodissociation Dynamics of ${NeBr}_{2} (B)$ Near the ${Br}_{2} (B)$ Dissociation Limit, Chem. Phys. Lett. 359 (2002) 229-236.

[33] Roberts R. M., Sousa Dias M. E.R., Bifurcation of Relative Equilibria, Nonlinearity 10 (1997) 1719-1738. | Zbl | MR

[34] Sbano L., Symmetric Solutions in Molecular Potentials, in: Gaeta G., Prinari B., Rauch S., Terracini S. (Eds.), Symmetry and Perturbation Theory SPT2004, Proceedings of Cala Gonone Workshop, Italy, 30 May-6 June 2004, World Scientific, 2005. | Zbl | MR

[35] L. Sbano, J. Southall, Periodic solutions of the N-body problem with Lennard-Jones potential, Mathematics Institute preprint, Univ. Warwick, 2007.

[36] Schmidt D. S., Hopf's Bifurcation Theorem and the Center Theorem of Liapunov, in: Marsden J. E., Mccracken M. (Eds.), The Hopf Bifurcation Theorem and Its Applications, Appl. Math. Sci., vol. 19, Springer, New York, 1976, pp. 95-104.

[37] Sepulchre J.-A., Mackay R. S., Localized Oscillations in Conservative Or Dissipative Networks of Weakly Coupled Autonomous Oscillators, Nonlinearity 10 (1997) 679-713. | Zbl | MR

[38] Weinstein A., Normal Modes for Nonlinear Hamiltonian Systems, Invent. Math. 20 (1973) 47-57. | Zbl | MR

[39] Wulff C., Persistence of Relative Equilibria in Hamiltonian Systems With Noncompact Symmetry, Nonlinearity 16 (2003) 67-91. | Zbl | MR

[40] Wulff C., Persistence of Hamiltonian Relative Periodic Orbits, J. Geom. Phys. 48 (2003) 309-338. | Zbl | MR

[41] Yanao T., Takatsuka K., Kinematic Effects Associated With Molecular Frames in Structural Isomerization Dynamics of Clusters, J. Chem. Phys. 120 (2004) 8924-8936.

[42] Yanao T., Koon W. S., Marsden J. E., Kevrekidis I. G., Gyration-Radius Dynamics in Structural Transitions of Atomic Clusters, J. Chem. Phys. 126 (2007) 124102.

[43] Zevin A. A., Global Continuation of Lyapunov Centre Orbits in Hamiltonian Systems, Nonlinearity 12 (1999) 1339-1349. | Zbl | MR

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