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Morse Theory and Existence of Periodic Solutions of Elliptic Type

  • B. D’Onofrio
  • I. Ekeland
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 4)

Abstract

In this paper we study the problem of finding periodic solutions of elliptic type for Hamiltonian systems of the following form
$$ \left\{ {\begin{array}{*{20}{c}} {\dot x = J(Ax + N'(t,x))}\\ {x(0) = x(T)\quad with\quad T > 0;} \end{array}} \right. $$
(H)
whereJ is the matrix
$$ J \equiv \left( {\begin{array}{*{20}{c}} 0&{{I_n}}\\ { - {I_n}}&0 \end{array}} \right) $$
andI n is the identity of Rn,
$$\begin{array}{*{20}{c}} {A \equiv \left( {\begin{array}{*{20}{c}} {{A_n}}&0\\ 0&{{A_n}} \end{array}} \right),}\\ {{A_n} \equiv \left( {\begin{array}{*{20}{c}} {{a_1}}\\ 0 \end{array} \ddots \begin{array}{*{20}{c}} 0\\ {{a_n}} \end{array}} \right)} \end{array}$$
\( {a_j} \in R,{a_j} \ne 0,j = 1, \ldots ,n,N \in {C^2}(R \times {R^{2n}},R) \) denotes the partial gradient with respect to the second variable.

Keywords

Periodic Solution Hamiltonian System Unit Circle Hermitian Form Morse Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [B]
    R. Bott, On the iteration of closed geodesic and Sturm intersection. theory, Comm. Pure Appl. Math., 9 (1956), 176–206.Google Scholar
  2. [B]
    R. Bott, Lecture on Morse theory, old and new, Bull. Amer. Math. Soc. 7 (1982), 331–358.MathSciNetCrossRefGoogle Scholar
  3. [C]
    F. Clarke, Periodic solutions of Hamiltonian inclusions, J. Differential Equations, 40 (1981), 1–6.MathSciNetCrossRefGoogle Scholar
  4. [C-E]
    F. Clarke and I. Ekeland, Hamiltonian trajectories having prescribed minimal period, Comm. Pure Appl. Math. 33 (1980), 103–116.MATHGoogle Scholar
  5. [C-Z]
    C. Conley and E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture of V. Arnold, Invent. Math. 73 (1983), 33–49.MathSciNetMATHGoogle Scholar
  6. [D-E]
    B. D’Onofrio and I. Ekeland, La théorie de l’index pour certains systèmes hamiltoniens non définis positifs, C.R.A.S. Paris, 305, I, (1987), 249–251.Google Scholar
  7. [D-E]
    B. D’Onofrio and I. Ekeland, Hamiltonian systems with elliptic periodic orbits, preprint (1988).Google Scholar
  8. [E]
    I. Ekeland, Une théorie de Morse pour des systèmes hamiltoniens convexes, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 19–78.Google Scholar
  9. [E]
    I. Ekeland, An index theory for periodic solutions of convex Hamiltonian systems, Proc. of Symp. on Pure Math., 45 (1986).Google Scholar
  10. [E-H]
    I. Ekeland and H. Hofer, Periodic solutions with prescribed minimal period for convex autonomous Hamiltonian systems, Inv. Math., 81 (1985), 155–177.MathSciNetCrossRefMATHGoogle Scholar
  11. [E-H]
    I Ekeland and H. Hofer, Subharmonic solutions for convex nonautonomous Hamiltonian systems,Comm. Pure and Appl. Math. (to appear).Google Scholar
  12. [E-H]
    I. Ekeland and H. Hofer, Convex Hamiltonian energy surfaces and their periodic trajectories, preprint (1987).Google Scholar
  13. [G-M]
    M. Girardi and M. Matzen, Periodic solutions of Convex Autonomous Hamiltonian Systems and a Quadratic Growth at the Origin and Superquadratic at Infinity, Annali di Matematica Pura e Applicata Vol. CXLVII, 1987, 21–72.Google Scholar
  14. [M]
    J. Moser, Proof of a generalized form of a fixed point theorem due to G.D. Birkhoff, Lecture Notes in Math., 597, Springer Verlag, Berlin and New York (1977), 464–494.Google Scholar
  15. [R]
    P. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), 157–184.Google Scholar
  16. [T]
    F. Takens Hamiltonian systems: generic properties of closed orbits and local perturbationsMath. Ann. 188 (1970), 304–312.Google Scholar
  17. [V]1
    C. Viterbo, Thèse de Sème cycle, Université Paris I X, 1985.Google Scholar
  18. [V]2
    C. Viterbo Une théorie de Morse pour les systemès hamitoniens étoilés C.R.A.S. Paris (1), 301 (1985), 487–489.Google Scholar
  19. [Y-S]
    V. Yakubovich and V. Starzhinskii Linear differential equations with periodic coefficients Halstedt Press, Wiley, 1980.Google Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • B. D’Onofrio
    • 1
  • I. Ekeland
    • 1
  1. 1.Université Paris-Dauphine CEREMADEParisFrance

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