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Periodic solutions of hamiltonian equations

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Dynamics and Processes

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References

  1. J.C. Alexander / J.A. Yorke: "Global Bifurcation of periodic orbits", Amer. J. Math. 100 (1978), 263–292

    Article  MathSciNet  MATH  Google Scholar 

  2. H. Amann / E. Zehnder: "Nontrivial Solutions for a Class of Nonresonance Problems and Applications to Nonlinear Differential Equations", Annali Sc. Norm. Sup. Pisa, Serie IV, Vol. VII (1980), 539–603

    MathSciNet  MATH  Google Scholar 

  3. H. Amann / E. Zehnder: "Periodic solutions of Asymptotically linear Hamiltonian systems", Manus. Math. 32, (1980), 149–189

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Ambrosetti: "Recent Advances in the study of the existence of periodic orbits of Hamiltonian systems", Preprint, SISSA, Trieste (1982), 1–19

    Google Scholar 

  5. A. Ambrosetti / G. Mancini: "Solutions of Minimal period for a Class of Convex Hamiltonian systems", Math. Ann. 255 (1981), 405–421

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Ambrosetti / G. Mancini: "On a theorem by Ekeland und Lasry concerning the number of periodic Hamiltonian trajectories", J. Diff. Equ. 43 (1982), 249–256

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. A. Ambrosetti / P.H. Rabinowitz: "Dual variational methods in critical point theory and applications", Journal Functional Analysis 14, (1973), 349–381

    Article  MathSciNet  MATH  Google Scholar 

  8. V.I. Arnold: Proceedings of Symposia in Pure Mathematics, Vol. XXVIII A.M.S. (1976), p.66

    ADS  Google Scholar 

  9. V.I. Arnold: "Mathematical Methods of Classical Mechanics", (Appendix 9), Springer 1978

    Google Scholar 

  10. A. Bahri / H. Berestycki: "Points critiques de perturbations de fonctionnelles paires et applications", C.R. Acad. Sc. Paris, t. 291 série A (1980), 189–192

    MathSciNet  MATH  Google Scholar 

  11. A. Bahri / H. Berestycki: "Existence d'une infinité de solutions périodiques de certains systèmes hamiltoniens en présence d'un terme de contrainte", C.R. Acad. Sc. Paris, t. 292, série A (1981), 315–318

    MathSciNet  MATH  Google Scholar 

  12. W. Ballmann / G. Thorbergsson / W. Ziller: "Closed geodesics on positively curved manifolds", Annals of Math. 116, (1982), 231–247

    Article  MathSciNet  MATH  Google Scholar 

  13. W. Ballmann / W. Ziller: "On the number of closed geodesics on a compact Riemannian manifold", Duke Math. J. 49 (1982), 629–632

    Article  MathSciNet  MATH  Google Scholar 

  14. V. Bangert: "Closed Geodesics on Complete Surfaces", Math. Ann. 251 (1980), 83–96

    Article  MathSciNet  MATH  Google Scholar 

  15. V. Bangert / W. Klingenberg: "Homology generated by iterated closed geodesics", Preprint Freiburg, Bonn, 1981

    MATH  Google Scholar 

  16. A. Banyaga: "Sur la structure du groupe des difféomorphismes qui presêrvent une forme symplectique", Comment. Math. Helvetici 53 (1978), 174–227

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Banyaga: "On fixed points of symplectic maps", Preprint

    Google Scholar 

  18. V. Benci: "Some critical point theorems and applications", Comm. Pure Appl. Math. 33 (1980)

    Google Scholar 

  19. V. Benci / P.H. Rabinowitz: "Critical point theorems for indefinite functionals", Inv. math. 52 (1979), 336–352

    Article  MathSciNet  MATH  Google Scholar 

  20. M. Berger: "On periodic solutions of second order Hamiltonian systems", J. Math. Anal. Appl. 29 (1970), 512–522

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. M. Berger: "Periodic solutions of second order dynamical systems and isoperimetric variational problems", Amer. J. Math. 93 (1971) 1–10

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. M. Berger: "On a family of periodic solutions of Hamiltonian systems", J. Diff. Eq. 10 (1971), 17–26

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. G.D. Birkhoff: "An extension of Poincaré's last geometric theorem", Acta Math. 47 (1925), 297–311

    Article  MathSciNet  MATH  Google Scholar 

  24. G.D. Birkhoff: "Une generalization à n-dimensions du dernier théorème de géometrie de Poincaré", Comp. Rend. Acad. Sci. 192, (1931), 196–198

    MATH  Google Scholar 

  25. G.D. Birkhoff / D.C. Lewis: "On the periodic motions near a given periodic motion of a dynamical system", Ann. Mat. Pura Appl. 12 (1933), 117–133

    Article  MathSciNet  MATH  Google Scholar 

  26. R. Bott: "Marston Morse and his mathematical works", Bull (New Series) A.M.S. 3 (1980), 907–950

    Article  MathSciNet  MATH  Google Scholar 

  27. R. Bott: "Lectures on Morse theory, old and new", Bulletin (New Series) A.M.S. 7 (2) (1982), 331–358

    Article  MathSciNet  MATH  Google Scholar 

  28. M. Bottkol: "Bifurcation of periodic orbits on manifolds and Hamiltonian systems, Thesis, New York University 1977

    Google Scholar 

  29. H. Brezis / J.M. Coron / L. Nirenberg: "Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz", Comm. Pure Appl. Math 33 (1980), 667–689

    Article  MathSciNet  MATH  Google Scholar 

  30. M. Brown / W.D. Neumann: "Proof of the Poincaré-Birkhoff fixed point theorem", Michigan Math. Journ. 24 (1977), 21–31

    Article  MathSciNet  MATH  Google Scholar 

  31. K.C. Chang: "Solutions of asymptotically linear operator equations via Morse theory", Comm. Pure and Appl. Math 34 (1981), 693–712

    Article  MathSciNet  MATH  Google Scholar 

  32. A. Chenciner: "Points périodiques de longues periodes au voisinage d'une bifurcations de Hopf dégenerée de difféomorphismes de IR2", Preprint 1982

    Google Scholar 

  33. S.N. Chow / J. Mallet-Paret: "The Fuller Index and Global Hopf Bifurcation", J. of Diff. Equ. 29 (1978), 66–85

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. S.N. Chow / J. Mallet-Paret: "Periodic solutions near an equilibrium of a nonpositive definite Hamiltonian system", Michigan State Univ. Preprint

    Google Scholar 

  35. S.N. Chow / J. Mallet-Paret / J.A. Yorke: "Global Hopf Bifurcation from a multiple eigenvalue", Nonlinear Analysis, Theory, Meth. and Appl. 2 (1978), 753–763

    Article  MathSciNet  MATH  Google Scholar 

  36. F. Clarke: "A classical variational principle for periodic Hamiltonian trajectories", Proc. Amer. Math. Soc. 76 (1979), 186–188

    MathSciNet  MATH  Google Scholar 

  37. F. Clarke: "Periodic solutions to Hamiltonian inclusions", J. Diff. Equ. 40 (1981), 1–6

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. F. Clarke / I. Ekeland: "Nonlinear Oscillations and Boundary-Value Problems for Hamiltonian systems", Archive Rat. Mech. and Analysis, in press

    Google Scholar 

  39. F. Clarke / I. Ekeland: "Hamiltonian Trajectories Having Prescribed Minimal Periods", Comm. on Pure and Appl. Math. 33 (1980), 103–116

    Article  MathSciNet  MATH  Google Scholar 

  40. C.C. Conley: "Isolated invariant sets and the Morse index", CBMS Regional Conf. Series in Math 38 (1978) A.M.S. Providence R.I.

    Google Scholar 

  41. C.C. Conley / E. Zehnder: "Morse type index theory for flows and periodic solutions for Hamiltonian equations", to appear in Comm. Pure and Appl. Math.

    Google Scholar 

  42. C.C. Conley / E. Zehnder: "An index theory for periodic solutions of a Hamiltonian system", to appear in the Proceedings of the Rio Conference on Dynamical systems

    Google Scholar 

  43. C.C. Conley / E. Zehnder: "The Birkhoff-Lewis fixed point theorem and a conjecture of V. Arnold", Preprint FIM, ETH Zürich (1982), 1–26

    Google Scholar 

  44. H. Duistermaat: "On periodic solutions near equilibrium points of conservative systems", Arch. Rat. Mech. Anal. 45 (1972), 143–160

    Article  MathSciNet  MATH  Google Scholar 

  45. H. Duistermaat: "On the Morse Index in variational calculus", Adv. in Math. 21 (1976), 173–195

    Article  MathSciNet  MATH  Google Scholar 

  46. H. Duistermaat: "Periodic solutions near equilibrium points of Hamiltonian systems", Utrecht, Dept. of Math., Preprint Nr. 156 (1980)

    Google Scholar 

  47. I. Ekeland: "Periodic solutions of Hamiltonian equations and a theorem of P. Rabinowitz", J. Differential Equations 34 (1979), 523–534

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. I. Ekeland: "La théorie des perturbations au voisinage des systèmes Hamiltonian convexes", École Polytechnique, Centre de Mathématiques, Exposé no VII (1981)

    Google Scholar 

  49. I. Ekeland: "Oscillations de systèmes Hamiltoniens non linéaires III", Bull. Soc. Math. France 109 (1981), 297–330

    MathSciNet  MATH  Google Scholar 

  50. I. Ekeland: "Forced oscillations for Nonlinear Hamiltonian Systems II" in Advances in Mathematics, volume en l'honneur de Laurent Schwartz, Nachbin, éd. 1981, Academic Press

    Google Scholar 

  51. I. Ekeland: "Dualité et stabilité des systèmes hamiltoniens", C.R. Acad. Sc. Paris 294 Série I (1982) 673–676.

    MathSciNet  MATH  Google Scholar 

  52. I. Ekeland / J.M. Lasry: "On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface", Ann. of Math. 112 (1980), 283–319

    Article  MathSciNet  MATH  Google Scholar 

  53. E.R. Fadell / P.H. Rabinowitz: "Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems", Inv. Math. 45 (1978), 139–174

    Article  ADS  MathSciNet  MATH  Google Scholar 

  54. H. Gluck / W. Ziller: "Existence of periodic motions of conservative systems", University of Pennsylvania (1980), Preliminary draft

    Google Scholar 

  55. W.B. Gordon: "A theorem on the existence of periodic solutions to Hamiltonian systems with convex potentials", J. Diff. Eq. 10 (1971), 324–335

    Article  ADS  MATH  Google Scholar 

  56. T.C. Harris: "Periodic solutions of arbitrary long period in Hamiltonian systems", J. Diff. Eq. 4 (1968), 131–141

    Article  ADS  MATH  Google Scholar 

  57. J. Horn: "Beiträge zur Theorie der kleinen Schwingungen", Zeit. Math. Phys. 48 (1903), 400–434

    MATH  Google Scholar 

  58. H. Jacobowitz: "Periodic solutions of x″+f(x,t)=o via the Poincaré-Birkhoff theorem", J. Diff. Eq. 20 (1976), 37–52

    Article  ADS  MathSciNet  MATH  Google Scholar 

  59. W. Klingenberg: "Lectures on closed geodesics", Grundlehren Vol. 230, Springer 1978

    Google Scholar 

  60. A. Lyapunov: "Problème générale de la stabilite du mouvement", Ann. Fac. Sci. Toulouse (2) (1907), 203–474

    Google Scholar 

  61. J. Moser: "Periodic orbits near an equilibrium and a theorem by Alan Weinstein", Comm. Pure Appl. Math. 29 (1976), 727–747

    Article  ADS  MathSciNet  MATH  Google Scholar 

  62. J. Moser: "Proof of a generalized form of a fixed point theorem due to G.D. Birkhoff", Springer Lecture Notes in Mathematics, Vol. 597: Geometry and Topology (1977), 464–494

    Article  MathSciNet  Google Scholar 

  63. J. Moser: "A fixed point theorem in symplectic geometry", Acta Math. 141 (1978), 17–34

    Article  ADS  MathSciNet  MATH  Google Scholar 

  64. H. Poincaré: "Méthodes nouvelles de la mécanique célèste", Vol. 3, chap. 28, Gauthier Villars, Paris (1899)

    Google Scholar 

  65. J. Pöschel: "Integrability of Hamiltonian Systems on Cantor Sets", Comm. Pure Appl. Math. 35 (1982) 653–696

    Article  MathSciNet  MATH  Google Scholar 

  66. P.H. Rabinowitz: "A variational method for finding periodic solutions of differential equations", Nonlinear Evolution Equations (M.G. Crandall, editor), Academic Press (1978), 225–251

    Google Scholar 

  67. P.H. Rabinowitz: "Some minimax theorems and applications to nonlinear partial differential equations", Nonlinear Analysis, A Collection of Papers in Honor of Erich H. Rothe, 161–177, Academic Press 1978

    Google Scholar 

  68. P.H. Rabinowitz: "Periodic solutions of Hamiltonian systems", Comm. Pure Appl. Math. 31 (1978), 157–184

    Article  MathSciNet  Google Scholar 

  69. P.H. Rabinowitz: "Periodic solutions of a Hamiltonian system on a prescribed energy surface", J. Diff. Eq. 33 (1979), 336–352

    Article  ADS  MathSciNet  MATH  Google Scholar 

  70. P.H. Rabinowitz: "Periodic solutions of Hamiltonian systems: a survey", SIAM J. Math. Anal. 13 (1982), 343–352

    Article  MathSciNet  MATH  Google Scholar 

  71. P.H. Rabinowitz: "On periodic solutions of large norm of some ordinary and partial differential equations", Ergodic Theory and Dynamical Systems, Proc. Sp. Yr.-Maryland 79–80, A. Katok, ed., to appear

    Google Scholar 

  72. P.H. Rabinowitz: "Subharmonic solutions of Hamiltonian systems", Comm. Pure Appl. Math. XXXIII (1980), 609–633

    Article  MathSciNet  MATH  Google Scholar 

  73. J.A. Sanders: "Are higher order resonances really interesting?", Celestial Mech. 16 (1978), 421–440

    Article  ADS  MathSciNet  MATH  Google Scholar 

  74. D.S. Schmidt: "Periodic solutions near a resonant equilibrium of a Hamiltonian system", Celestial Mech. 9 (1974), 81–103

    Article  ADS  MathSciNet  MATH  Google Scholar 

  75. H. Seifert: "Periodische Bewegungen mechanischer Systeme", Math. Z. 51 (1948), 197–216

    Article  MathSciNet  MATH  Google Scholar 

  76. A. Weinstein: "Normal modes for nonlinear Hamiltonian systems, Inv. Math 20 (1973), 47–57

    Article  ADS  MathSciNet  MATH  Google Scholar 

  77. A. Weinstein: "Lectures on symplectic manifolds", CBMS, Regional conf. series in Math. 29 (1977)

    Google Scholar 

  78. A. Weinstein: "Bifurcations and Hamilton's Principle", Math. Z. 159 (1978), 235–248

    Article  MathSciNet  MATH  Google Scholar 

  79. A. Weinstein: "Periodic orbits for convex Hamiltonian systems", Ann. Math. 108 (1978), 507–518

    Article  MathSciNet  MATH  Google Scholar 

  80. A. Weinstein: "On the hypotheses of Rabinowitz' periodic orbit theorems", J. Diff. Eq. 33 (1979), 353–358

    Article  ADS  MathSciNet  MATH  Google Scholar 

  81. M. Willem: "On the number of Periodic Hamiltonian orbits on a convex surface", Preprint (1983)

    Google Scholar 

  82. H. Amann: "Gewöhnliche Differentialgleichungen" Teubner 1983

    Google Scholar 

  83. P.A. Schweitzer: "Counterexamples to the Seifert conjecture and opening closed leaves of foliations", Annals of Math. 100 (1974), 386–400

    Article  MathSciNet  MATH  Google Scholar 

  84. T.W. Wilson: "On the minimal sets of nonsingular vectorfields", Annals of Math. 84 (1966), 529–536

    Article  Google Scholar 

  85. D. Clark: "On periodic solutions of autonomous Hamiltonian systems of ordinary differential equations", Proc. AMS 39 (1973), 579–584

    Article  MathSciNet  MATH  Google Scholar 

  86. A. Bahri / H. Berestycki: "Forced vibrations of super-quadratic Hamiltonian systems", Preprint, Univ. Pierre et Marie Curie (1982)

    Google Scholar 

  87. I. Ekeland: "A perturbation theory near convex Hamiltonian systems", Technical Report 82-1 (1982)

    Google Scholar 

  88. C.B. Croke / A. Weinstein: "Closed Curves on Convex Hypersurfaces and Periods of Nonlinear Oscillations", Preprint

    Google Scholar 

  89. A. Weinstein: "Symplectic V-Manifolds, Periodic orbits of Hamiltonian Systems, and the Volume of certain Riemannian Manifolds", Comm. on Pure and Appl. Math. 30 (1977), 265–271

    Article  MathSciNet  MATH  Google Scholar 

  90. J. Moser: "Addendum to ‘Periodic orbits near an Equilibrium and a Theorem by A. Weinstein'", Comm. Pure and Appl. Math. 31 (1978), 529–530

    Article  MathSciNet  Google Scholar 

  91. K.P. Rybakowski / E. Zehnder: "A Morse-Equation in Conley's Index theory for semiflows on metric spaces", to be published in Dynamical Systems and Ergodic Theory

    Google Scholar 

  92. P.H. Carter: "An Improvement of the Poincaré-Birkhoff Fixed point theorem", Transactions AMS 269 (1982), 285–299

    MATH  Google Scholar 

  93. A. Chenciner: "Sur un eńoncé dissipatif du théorème geómétrique de Poincaré-Birkhoff", C.R. Acad. Sc. Paris 294 I (1982), 243–245

    MathSciNet  MATH  Google Scholar 

  94. E. Zehnder: "Homoclinic Points near elliptic Fixed Points", Comm. Pure and Appl. Math. XXVI (1973) 131–182

    Article  MathSciNet  MATH  Google Scholar 

  95. E. Zehnder: "Generalized Implicit Function Theorems with Applications to Some Small Divisor Problems I, and II", Comm. Pure and Appl. Math. XXVIII (1975) 91–140 and XXIX (1976) 49–111

    MathSciNet  MATH  Google Scholar 

  96. A. Hofer: "A new proof for a result of Ekeland and Lasry concerning the Number of periodic Hamiltonian trajectories on a prescribed energy surface", Boll. U.M.I. (6), 1-B (1982) 931–942

    MathSciNet  MATH  Google Scholar 

  97. C.L. Siegel / J. Moser: "Lectures on Celestial Mechanics", Springer Grundlehren Bd. 187 (1971)

    Google Scholar 

  98. J. Moser: "Stable and Random Motions in Dynamical Systems", Annals of Math. Studies, Vol. 77, Princeton University Press (1973)

    Google Scholar 

  99. C.C. Pugh / R.C. Robinson: "The C1 Closing Lemma, including Hamiltonians", Preprint

    Google Scholar 

  100. J.V. Ralston: "On the construction of quasimodes associated with stable periodic orbits", Comm. Math. Phys. 51 (1976) 219–242

    Article  ADS  MathSciNet  MATH  Google Scholar 

  101. Y. Colin de Verdiere: Compositio Math. 27 (1973) 83–106 and 159–184

    MathSciNet  Google Scholar 

  102. J. Mather: "Existence of Quasiperiodic Orbits for Twist Homeomorphisms of the Annulus", to appear in Topology

    Google Scholar 

  103. J. Mather: "A Criterion for the Nonexistence of invariant Circles", Preliminary draft (1982)

    Google Scholar 

  104. M. Herman: "Contre examples de classes C3−ε et à nombre de rotation fixé au théorème des courbes invariantes", (1979), to be published

    Google Scholar 

  105. A. Katok: "Some Remarks on Birkhoff and Mather twist map theorems" (1982) to be published in Dynamical systems and ergodic theory

    Google Scholar 

  106. S. Albeverio / P. Blanchard / R. Höegh-Krohn: "Feynman path integrals and the trace formula for Schrödinger operators", Commun. math. Phys. 83 (1982) 49–76

    Article  ADS  MATH  Google Scholar 

  107. P. Hartman: "On boundary value problems for superlinear second order differential equations", Jour. Diff. Eq. 26 (1977) 37–53

    Article  ADS  MathSciNet  MATH  Google Scholar 

  108. V. Benci: "A geometrical index for the group S1 and some applications to the research of periodic solutions of ordinary differential equations", to appear

    Google Scholar 

  109. V. Benci: "On the critical point theory for indefinite functionals in the presence of symmetries", to be published.

    Google Scholar 

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Zehnder, E. (1983). Periodic solutions of hamiltonian equations. In: Blanchard, P., Streit, L. (eds) Dynamics and Processes. Lecture Notes in Mathematics, vol 1031. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072117

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