Homoclinic Solutions of Differential Equations

  • Maria do Rosário Grossinho
  • Stepan Agop Tersian
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 52)


In recent years, starting with works of Bolotin [Bol], Coti-Zelati, Ekeland and Séré [CZES], Coti-Zelati & Rabinowitz [CZR1], [CZR2], Rabinowitz [Ra4], variational methods have been applied to study the existence of homoclinic and heteroclinic solutions of second-order equations and Hamiltonian systems. The search for homoclinic and heteroclinic solutions is a classical problem, originated from the work of Poincaré and has been developed from several points of view. Existence of homoclinic solutions can be obtained by analyzing the intersection properties of the stable and unstable manifolds of the fixed points. There is a standard method to find infinitely nearby homoclinics provided that the stable and unstable manifolds of a fixed point intersect transversally. For this approach we refer the reader to Moser [Mo], Guckenheimer & Holmes [GH], Wiggins [Wig].


Hamiltonian System Vortex Ring Nontrivial Solution Unstable Manifold Homoclinic Orbit 
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  1. [AB]
    Ambrosetti A, Bertotti M. Homoclinics for second-order conservative systems. Scuola Norm. Sup. Preprint N 107, 1991.Google Scholar
  2. [ASt]
    Ambrosetti A, Struwe M. Existence of steady vortex rings of an ideal fluid. Scuola Norm. Sup. Preprint N 38, 1989.Google Scholar
  3. [AT]
    Amick CJ, Toland JF. Homoclinic orbits in the dynamic phase space analogy of an elastic strut. Eur. J. Appl. Math., 1991;3:97–114.MathSciNetCrossRefGoogle Scholar
  4. [ASz]
    Arioli G, Szulkin A. Homoclinic solutions for a class of systems of second-order equations. Reports, Dept. Math., Univ. Stockholm. 5, 1995.Google Scholar
  5. [BW]
    Bartsch T, Willem M. Infinitely many radial solutions of semilinear elliptic problem on Rn. Arch. Rational Mech. Anal. 1993;124:261–276.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [BS]
    Berezin, Felix and Schubin, Michail. The Schrödinger Equation. Moscow: Univ. Moscow, 1983. (In Russian)zbMATHGoogle Scholar
  7. [BMO]
    Biroli M, Mosco U. Sobolev and isoperimetric inequalities for Dirichlet forms on homogeneous spaces. Rend. Mat. Acc. Lincei. 1995;VI(1):37–44.MathSciNetGoogle Scholar
  8. [BM1]
    Biroli M, Mosco U. Formes de Dirichlet et estimations structurelles dans les milieux discontinus. C.R. Acád. Sc. Paris, 1991;313:593–597.MathSciNetzbMATHGoogle Scholar
  9. [BM2]
    Biroli M, Mosco U. Sobolev inequalities on homogeneous spaces. Proceedings “Potential theory and degenerate P.D. Operators”. Parma Feb. 1994, Kluwer, 1995.Google Scholar
  10. [BM3]
    Biroli M, Mosco U. A Saint-Venant principle for Dirichlet forms on discontinuous media. Ann. Mat. Pura e Appl., 1995;169:125–181.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [BT]
    Biroli M, Tchou T. Asymptotic behavior of relaxed Dirichlet problems involving a Dirichlet Poincaré form. ZAA, 1997.Google Scholar
  12. [BMT]
    Biroli M, Mosco U, Tchou N. Homogenization by the Heisenberg group. Publications du Laboratoire D’Analyse Numerique. R 94032, Univ. Pierre et Marie Curie.Google Scholar
  13. [BTer]
    Biroli M, Tersian S. On the existence of nontrivial solutions to semilinear equations relative to a Dirichlet form. Rendiconti Istituto Lombardo, to appear.Google Scholar
  14. [Buf]
    Buffoni B. Periodic and homoclinic orbits for Lorentz-Lagrangian systems via variational method. Nonlin. Anal., 1996; 26, 3: 443–462.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [Bur]
    Burton G. Semilinear Elliptic Equations on Unbounded Domains. Math. Z. 1985;190:519–525.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [CL]
    Coddington EA, Levinson N. Theory of Ordinary Differetial Equations. N.Y.: McGraw-Hill, 1955.Google Scholar
  17. [CZES]
    Coti-Zelati V., Ekeland I., Séré E. A variational approach to homoclinic orbits in Hamiltonian systems. Math. Ann., 1990; 288: 133–160.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [CZR1]
    Coti-Zelati V, Rabinowitz P. Homoclinic type solutions for a semilinear elliptic PDE on Rn. Ref. SISSA 58, 91,M, 1991.Google Scholar
  19. [CZR2]
    Coti-Zelati V, Rabinowitz P. Homoclinic orbits for secondorder Hamiltonian systems possessing superquadratic potentials. J.A.M.S., 1991, 4,4: 693–727.MathSciNetzbMATHGoogle Scholar
  20. [Dev]
    Devaney R. Homoclinic Orbits in Hamiltonian Systems. J. Diff. Eq., 1976; 21:431–438.MathSciNetCrossRefGoogle Scholar
  21. [DN]
    Ding WY, Ni WM. On the existence of positive entire solutions of a semilinear elliptic equation. Arch. Rational Mech. Anal. 1986;91:283–308.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [Est]
    Esteban M. Nonlinear elliptic problems in strip-like domains: Symmetry of positive vortex rings. Nonlinear Anal., 1983;7,4:365–379.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [FKS]
    Fabes E, Kenig C, Serapioni R. The local regularity of solutions of degenerate elliptic equations Comm. Part. Diff. Eq., 1982;7:77–116.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [Fuk]
    Fukushima. Dirichlet Forms and Markov Processes. Amsterdam: North Holland, 1980.zbMATHGoogle Scholar
  25. [GNN]
    Gidas B, Ni WM and Nirenberg L. Symmetry and related properties via the maximum principle, Commun. Math. Phys., 1979;68:209–243.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [GMT3]
    Grossinho MR, Minhos F, Tersian S. Positive Homoclinic Solutions for a Class of Second Order Differential Equations. J. Math. Anal. Appl., 1999;240:163–173.MathSciNetzbMATHCrossRefGoogle Scholar
  27. [GH]
    Guckenheimer J, Holmes P. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. N.Y.: Springer-Verlag, 1983.zbMATHGoogle Scholar
  28. [Je]
    Jerison J. The Poincaré inequality for vector fields satisfying Hörmander’s condition. Duke Math. J., 1986;53:503–523.MathSciNetzbMATHCrossRefGoogle Scholar
  29. [KL]
    Korman P, Lazer A. Homoclinic orbits for a class of symmetric hamiltonian systems, Electr. Journal Diff. Eq. 1994;1:1–10.MathSciNetGoogle Scholar
  30. [KO]
    Korman P, Ouyang T. Exact multiplicity results for two classes of boundary value problems. Differential and Integral Equations. 1993;6,6:1507–1517.MathSciNetzbMATHGoogle Scholar
  31. [PLL]
    Lions PL. Symmétrie et compacité dans les espaces de Sobolev. J. Funct. Anal., 1982;49:315–334.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [Mel]
    Melnikov VK. On the stability of the centre for time periodic perturbations. Trans.Moscow.Math.Soc., 1963;12:1–57.Google Scholar
  33. [Mo]
    Mosco U. Variational metrics on selfsimilar fractals. C. R. Acád. Sci. Paris, 1995;321:715–720.MathSciNetzbMATHGoogle Scholar
  34. [Mos]
    Moser J. Stable and Random Motions in Dynamical Systems. Princeton University Press: Princeton, 1973.zbMATHGoogle Scholar
  35. [Nit]
    Nitecki, Zbignev. Differentiable Dynamics. M.I.T. Press: Cambridge, 1971.Google Scholar
  36. [OW]
    Omana O, Willem M. Homoclinic orbits for a class of Hamiltonian systems. Diff. Int. Eq, 1992;5:5.MathSciNetGoogle Scholar
  37. [PTV]
    Peletier LA, Troy WC, Van der Vorst RCAM. Stationary solutions of a forth-order nonlinear diffusion equation. Differential Equations, 1995; 31,2:301–314.MathSciNetzbMATHGoogle Scholar
  38. [Ra3]
    Rabinowitz P. A note on a semilinear elliptic equations on Rn. Scuola Norm. Sup. Pisa, 1991.Google Scholar
  39. [Ra4]
    Rabinowitz P. Homoclinic orbits for a class of Hamiltonian systems. Proc. Roy. Soc. Edinbourgh. 1990;114A:33–38.MathSciNetzbMATHCrossRefGoogle Scholar
  40. [RT]
    Rabinowitz P, Tanaka, K. Some results on connection orbits for a class of Hamiltonian systems. Math. Z. 1991;206:473–499.MathSciNetzbMATHCrossRefGoogle Scholar
  41. [RS]
    Reed, Michael & Simon, Barry. Methods of Modern Mathematical Physics, v. IV. Moskow: Mir, 1983. (In Russian).Google Scholar
  42. [Sal]
    Sanchez L. Boundary Value Problems for Some Fourth Order Ordinary Differential Equations. Appl. Anal., 1990;38:161–177.MathSciNetzbMATHCrossRefGoogle Scholar
  43. [Str]
    Strauss W. Existence of solitary waves in higher dimensions. Comm. Math. Ph., 1977;55:149–162.CrossRefGoogle Scholar
  44. [Ter8]
    Tersian S. On nontrivial solutions of semilinear Schrödinger equations on Rn. Rendiconti Istituto Lombardo (Rend.Sc.) 1995;A 129:97–109.MathSciNetGoogle Scholar
  45. [Ter9]
    Tersian S. On a resonance problem for semilinear Schrödinger equations on R n. Rendiconti Istituto Lombardo (Rend.Sc.) 1995;A 129:135–145.MathSciNetGoogle Scholar
  46. [Ter10]
    Tersian S. Nontrivial solutions to the semilinear Kohn—Laplace equation on R 3. Electronic J. Diff.Eq. 1999:v.1999,12;1–12.MathSciNetGoogle Scholar
  47. [Ter11]
    Tersian S. On the solvablility of semilinear Schrödinger equations in strip-like domains. C. R. Acad. Sci. Bulg., 1998;51,6.MathSciNetGoogle Scholar
  48. [Wig]
    Wiggins S. Global Bifurcations and Chaos. N.Y.: Springer-Verlag, 1988.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Maria do Rosário Grossinho
    • 1
    • 2
  • Stepan Agop Tersian
    • 3
  1. 1.ISEGUniversidade Técnica de LisboaPortugal
  2. 2.CMAFUniversidade de LisboaPortugal
  3. 3.University of RousseBulgaria

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