Maslov index for homoclinic orbits of hamiltonian systems

Chao-Nien Chen; Xijun Hu

Annales de l'I.H.P. Analyse non linéaire (2007)

  • Volume: 24, Issue: 4, page 589-603
  • ISSN: 0294-1449

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Chen, Chao-Nien, and Hu, Xijun. "Maslov index for homoclinic orbits of hamiltonian systems." Annales de l'I.H.P. Analyse non linéaire 24.4 (2007): 589-603. <http://eudml.org/doc/78751>.

@article{Chen2007,
author = {Chen, Chao-Nien, Hu, Xijun},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {Hamiltonian system; homoclinic orbit; Maslov index; relative Morse index},
language = {eng},
number = {4},
pages = {589-603},
publisher = {Elsevier},
title = {Maslov index for homoclinic orbits of hamiltonian systems},
url = {http://eudml.org/doc/78751},
volume = {24},
year = {2007},
}

TY - JOUR
AU - Chen, Chao-Nien
AU - Hu, Xijun
TI - Maslov index for homoclinic orbits of hamiltonian systems
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 2007
PB - Elsevier
VL - 24
IS - 4
SP - 589
EP - 603
LA - eng
KW - Hamiltonian system; homoclinic orbit; Maslov index; relative Morse index
UR - http://eudml.org/doc/78751
ER -

References

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  1. [1] Abbondandolo A., A new cohomology to Morse theory of strongly indefinite functionals on Hilbert spaces, Top. Meth. Nonl. Anal.9 (1997) 325-382. Zbl0906.58007MR1491851
  2. [2] Abbondandolo A., Morse Theory for Hamiltonian Systems, Research Notes in Mathematics, vol. 425, Chapman Hall/CRC, 2001. Zbl0967.37002MR1824111
  3. [3] Abbondandolo A., Molina J., Index estimates for strongly indefinite functionals, periodic orbits and homoclinic solutions of first order Hamiltonian systems, Calc. Var.11 (2000) 395-430. Zbl0981.58009MR1808128
  4. [4] Alexander J., Gardner R., Jones C., A topological invariant arising in the stability analysis of travelling waves, J. Reine Angew. Math.410 (1990) 167-212. Zbl0705.35070MR1068805
  5. [5] Arioli G., Szulkin A., Homoclinic solutions of Hamiltonian systems with symmetry, J. Differential Equations158 (1999) 291-313. Zbl0944.37030MR1721901
  6. [6] Arnold V.I., On a characteristic class entering into the quantization condition, Funkt. Anal. Prilozh.1 (1967) 1-14, (in Russian). Zbl0175.20303MR211415
  7. [7] Atiyah M.F., Patodi V.K., Singer I.M., Spectral asymmetry and Riemannian geometry III, Math. Proc. Cambridge Philos. Soc.79 (1976) 71-99. Zbl0325.58015MR397799
  8. [8] Benci V., Rabinowitz P., Critical point theorems for indefinite functionals, Invent. Math.52 (1979) 241-273. Zbl0465.49006MR537061
  9. [9] Cappell S.E., Lee R., Miller E.Y., On the Maslov index, Comm. Pure Appl. Math.47 (1994) 121-186. Zbl0805.58022MR1263126
  10. [10] Chang K.C., Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Basel, 1993. Zbl0779.58005MR1196690
  11. [11] Chang K.C., Liu J.Q., Liu M.J., Nontrivial periodic solutions for strong resonance Hamiltonian systems, Ann. Inst. H. Poincaré Anal. Non Linéaire (1997) 103-117. Zbl0881.34061MR1437190
  12. [12] Chen C.-N., Tzeng S.-Y., Periodic solutions and their connecting orbits of Hamiltonian systems, J. Differential Equations177 (2001) 121-145. Zbl1087.37051MR1867615
  13. [13] Clement P., Felmer P., Mitidieri E., Homoclinic orbits for a class of infinite-dimensional Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci.24 (1997) 367-393. Zbl0902.35051MR1487960
  14. [14] Conley C., Zehnder E., Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math.37 (1984) 207-253. Zbl0559.58019MR733717
  15. [15] Coti Zelati V., Ekeland I., S e´r e´ E., A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann.228 (1990) 133-160. Zbl0731.34050MR1070929
  16. [16] Ding Y., Li S., Homoclinic orbits for first order Hamiltonian systems, J. Math. Anal. Appl.189 (1995) 585-601. Zbl0818.34023MR1312063
  17. [17] Ding Y., Willem M., Homoclinic orbits of a Hamiltonian system, Z. Angew. Math. Phys.50 (1999) 759-778. Zbl0997.37041MR1721793
  18. [18] Ekeland I., An index theory for periodic solutions of convex Hamiltonian systems, Proc. Symp. Pure Math.45 (1986) 395-423. Zbl0596.34023MR843575
  19. [19] Ekeland I., Convexity Methods in Hamiltonian Mechanics, Springer, Berlin, 1990. Zbl0707.70003MR1051888
  20. [20] Fei G., Relative Morse index and its application to Hamiltonian systems in the presence of symmetries, J. Differential Equations122 (1995) 302-315. Zbl0840.34032MR1355894
  21. [21] Fei G., Maslov-type index and periodic solutions of asymptotically linear systems which are resonant at infinity, J. Differential Equations121 (1995) 121-133. Zbl0831.34046MR1348538
  22. [22] Floer A., A relative Morse index for the symplectic action, Comm. Pure Appl. Math.41 (1988) 393-407. Zbl0633.58009MR933228
  23. [23] Hofer H., Wysocki K., First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems, Math. Ann.228 (1990) 483-503. Zbl0702.34039MR1079873
  24. [24] Hofer H., Zehnder E., Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser, The Analysis of Linear Partial Differential Operators, I–IV, Springer, Berlin, 1983. Zbl0837.58013
  25. [25] Lieb E.H., Loss M., Analysis, GSM, vol. 14, Amer. Math. Soc., 1997. Zbl0873.26002MR1415616
  26. [26] Long Y., Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems, Sci. China, Ser. A33 (1990) 1409-1419. Zbl0736.58022MR1090484
  27. [27] Long Y., Index Theory for Symplectic Paths with Applications, Birkhäuser, Basel, 2002. Zbl1012.37012MR1898560
  28. [28] Long Y., Zhu C., Maslov-type index theory for symplectic paths and spectral flow (II), Chin. Ann. of Math.21B (1) (2000) 89-108. Zbl0959.58017MR1762278
  29. [29] Long Y., Zhu C., Closed characteristics on compact convex hypersurfaces in R 2 n , Ann. of Math.155 (2002) 317-368. Zbl1028.53003MR1906590
  30. [30] Long Y., Zehnder E., Morse theory for forced oscillations of asymptotically linear Hamiltonian systems, in: Stoc. Proc. Phys. and Geom., World Sci., 1990, pp. 528-563. MR1124230
  31. [31] Kato T., Perturbation Theory for Linear Operators, Springer, New York, 1980. Zbl0435.47001
  32. [32] Maslov V.P., Theory of Perturbations and Asymptotic Methods, MGU, 1965, (in Russian). 
  33. [33] Rabinowitz P., Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math.31 (1978) 157-184. Zbl0358.70014MR467823
  34. [34] Rabinowitz P., Critical point theory and applications to differential equations: A survey, in: Topological Nonlinear Analysis, Progr. Nonlinear Differential Equations Appl., vol. 15, Birkhäuser, Boston, MA, 1995, pp. 464-513. Zbl0823.58009MR1322328
  35. [35] Robbin J., Salamon D., The Maslov index for paths, Topology32 (4) (1993) 827-844. Zbl0798.58018MR1241874
  36. [36] Robbin J., Salamon D., The spectral flow and the Maslov index, Bull. London Math. Soc.27 (1) (1995) 1-33. Zbl0859.58025MR1331677
  37. [37] Salamon D., Zehnder E., Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math.45 (1992) 1303-1360. Zbl0766.58023MR1181727
  38. [38] Sere E., Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z.209 (1992) 27-42. Zbl0725.58017MR1143210
  39. [39] Sere E., Looking for the Bernoulli shift, Ann. IHP-Analyse Nonlineaire10 (1993) 561-590. Zbl0803.58013MR1249107
  40. [40] Szulkin A., Zou W., Homoclinic orbits for asymptotically linear Hamiltonian systems, J. Funct. Anal.187 (2001) 25-41. Zbl0984.37072MR1867339
  41. [41] Szulkin A., Cohomology and Morse theory for strongly indefinite functionals, Math. Z.209 (1992) 375-418. Zbl0735.58012MR1152264
  42. [42] Tanaka K., Homoclinic orbits in a first order superquadratic Hamiltonian system: Convergence of subharmonic orbits, J. Differential Equations94 (1991) 315-339. Zbl0787.34041MR1137618
  43. [43] Viterbo C., A new obstruction to embedding Lagrangian tori, Invent. Math.100 (1990) 301-320. Zbl0727.58015MR1047136
  44. [44] Zhu C., Long Y., Maslov-type index theory for symplectic paths and spectral flow (I), Chin. Ann. of Math.20B (4) (1999) 413-424. Zbl0959.58016MR1752744

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