Hyperbolic characteristics on star-shaped hypersurfaces

Chun-Gen Liu; Yiming Long

Annales de l'I.H.P. Analyse non linéaire (1999)

  • Volume: 16, Issue: 6, page 725-746
  • ISSN: 0294-1449

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Liu, Chun-Gen, and Long, Yiming. "Hyperbolic characteristics on star-shaped hypersurfaces." Annales de l'I.H.P. Analyse non linéaire 16.6 (1999): 725-746. <http://eudml.org/doc/78481>.

@article{Liu1999,
author = {Liu, Chun-Gen, Long, Yiming},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {hypersurface; Maslov-type index; stability of closed characteristics},
language = {eng},
number = {6},
pages = {725-746},
publisher = {Gauthier-Villars},
title = {Hyperbolic characteristics on star-shaped hypersurfaces},
url = {http://eudml.org/doc/78481},
volume = {16},
year = {1999},
}

TY - JOUR
AU - Liu, Chun-Gen
AU - Long, Yiming
TI - Hyperbolic characteristics on star-shaped hypersurfaces
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 1999
PB - Gauthier-Villars
VL - 16
IS - 6
SP - 725
EP - 746
LA - eng
KW - hypersurface; Maslov-type index; stability of closed characteristics
UR - http://eudml.org/doc/78481
ER -

References

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  1. [1] A. Bahri and H. Berestycki, Forced vibrations of superquadratic Hamiltonian systems, Acta Mathematica, Vol. 152, 1984, pp. 143-197. Zbl0592.70027MR741053
  2. [2] H. Berestycki, J.M. Lasry, G. Mancini and B. Rof, Existence of multiple periodic orbits on starshaped Hamiltonian systems, Comm. pure Appl. Math., Vol. 38, 1985, pp. 253-289. Zbl0569.58027MR784474
  3. [3] V. Brousseau, Espaces de Krein et index des systémes hamiltoniens, Ann. Inst. H. Poincaré, Anal. non linéaire, Vol. 7, 1990, pp. 525-560. Zbl0719.58030MR1079571
  4. [4] K.C. Chang, Infinite dimensional Morse theory and multiple solution problems, Birkhäuser, Boston, 1993. Zbl0779.58005MR1196690
  5. [5] C. Conley and E. Zehnder, Maslov-type index theory for flows and periodic solutions for Hamiltonian equations, Commun. Pure Appl. Math., Vol. 37, 1984, pp. 207-253. Zbl0559.58019MR733717
  6. [6] G. DELL'ANTONIO, Variational calculus and stability of periodic solutions of a class of Hamiltonian systems, SISSA Ref. (185/92/FM (Oct. 1992)). MR1301373
  7. [7] G. DELL'ANTONIO, B. D'Onofrio and I. Ekeland, Les systém hamiltoniens convexes et pairs ne sont pas ergodiques en general, C. R. Acad. Sci. Paris, t. 315, Series I, 1992, pp. 1413-1415. Zbl0768.70014MR1199013
  8. [8] D. Dong and Y. Long, The Iteration Formula of the Maslov-type Index Theory with Applications to Nonlinear Hamiltonian Systems, Trans. Amer. Math. Soc., Vol. 349, 1997, pp. 2619-2661. Zbl0870.58024MR1373632
  9. [9] I. Ekeland and H. Hofer, Convex Hamiltonian energy surfaces and their closed trajectories, Comm. Math. Physics, Vol. 113, 1987, pp. 419-467. Zbl0641.58038MR925924
  10. [10] I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer, Berlin, 1990. Zbl0707.70003MR1051888
  11. [11] I. Ekeland, An index theory for periodic solutions of convex Hamiltonian systems, In Nonlinear Funct. Anal. and its Appl. Proc. Symposia in pure Math., Vol. 45-1, 1986, pp. 395-423. Zbl0596.34023MR843575
  12. [12] I. Ekeland, Une thórie de Morse pour les systèms hamiltoniens convexes, Ann. Inst. Henri poincaté. Anal. Non Linéair, Vol. 1, 1984, pp. 19-78. Zbl0537.58018MR738494
  13. [13] G. Fei and Q. Qiu, Periodic solutions of asymptotically linear Hamiltonian systems, Preprint, 1996, Chinese Ann. of Math. (To appear). Zbl0884.58081MR1480013
  14. [14] N. Ghoussoub, Location, multiplicity and Morse indices of minimax critical points, J. Reine Angew Math., Vol. 417, 1991, pp. 27-76. Zbl0736.58011MR1103905
  15. [15] C. Liu, Monotonicity of the Maslov-type index and the ω-index theory, Acta of Nankai University (Chinese). to appear. 
  16. [16] C. Liu and Y. Long, An optimal increasing estimate of the Maslov-type indices for iterations, Chinese Sci. Bull, Vol. 42, 1997, pp. 2275-2277 (Chinese edition), Vol. 43, 1998, pp. 1063-1066 (English edition). Zbl1002.53055MR1663290
  17. [17] Y. Long, Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems, Science in China (Scientia Sinica), Vol. Series A.33, 1990, pp. 1409-1419. Zbl0736.58022MR1090484
  18. [18] Y. Long, A Maslov-type theory and asymptotically linear Hamiltonian systems, In Dyna. Syst. and Rel. Topics. K. Shiraaiwa ed. World Sci., 1991, pp. 333-341. MR1164899
  19. [19] Y. Long, The Index Theory of Hamiltonian Systems with Applications, (In Chinese)Science Press, Beijing, 1993. 
  20. [20] Y. Long, Bott formula of the Maslov-type index theory, Nankai Inst. of Math. Nankai Univ. Preprint, 1995, Revised 1996, 1997, Pacific J. Math., Vol. 187, 1999, pp. 113-149. Zbl0924.58024MR1674313
  21. [21] Y. Long, Hyperbolic closed characteristics on compact convex smooth hypersurfaces in R2n, Nankai Inst. of Math. Preprint, 1996, Revised 1997, J. Diff. Equa., Vol. 150, 1998, pp. 227-249. Zbl0915.58032MR1658613
  22. [22] Y. Long, A Maslov-type index theory for symplectic paths, Top. Meth. Nonl. Anal., Vol. 10, 1997, pp. 47-78. Zbl0977.53075MR1646611
  23. [23] Y. Long and E. Zehnder, Morse theory for forced oscillations of asymptotically linear Hamiltonian systems, Stoc. Proc. Phys. and Geom., S. Albeverio et al. ed. World Sci., 1990, pp. 528-563. MR1124230
  24. [24] P.H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., Vol. 31, 1978, pp. 157-184. Zbl0358.70014MR467823
  25. [25] P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conf. Ser. in Math. Amer. Math. Soc., Vol. 65, 1986. Zbl0609.58002MR845785
  26. [26] C. Viterbo, A Proof of the Weinstein conjecture in R2n, Ann. IHP. Analyse nonlinéaire, Vol. 4, 1987, pp. 337-357. Zbl0631.58013MR917741
  27. [27] C. Viterbo, Equivariant Morse theory for starshaped Hamiltonian systems, Trans. Amer. Math. Soc., Vol. 311, 1989, pp. 621-655. Zbl0676.58030MR978370
  28. [28] T. Wang and G. Fei, Subharmonicsfor superquadratic Hamiltonian systems via the iteration method of the Maslov-type index theory, Preprint, 1996. 
  29. [29] A. Weinstein, On the Hypotheses of Rabinowitz' Periodic Orbit Theorems, J. Diff. Equa., Vol. 33, 1979, pp. 353-358. Zbl0388.58020MR543704
  30. [30] J.A. Yorke, periods of periodic solutions and the Lipschitz contant, Proc. Amer. Math. Soc., Vol. 22, 1969, pp. 509-512. Zbl0184.12103MR245916

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