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Applications of generic bifurcation. I

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References

  1. Ahmad, S., An existence theorem for periodically perturbed conservative systems. Mich. Math. J. 20, 385–392 (1973).

    Google Scholar 

  2. Ambrosetti, A., & G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces. Annali Mat. Pura Appl. 93, 231–246 (1972).

    Google Scholar 

  3. Ball, J., Saddle point analysis for an ordinary differential equation in a Banach space, and an application to dynamic buckling of a beam. Nonlinear Elasticity, R.W. Dickey, editor. Academic Press 1973.

  4. Bancroft, S., J.K. Hale & D. Sweet, Alternative problems for nonlinear functional equations. J. Diff. Eqns. 4, 40–56 (1968).

    Google Scholar 

  5. Berger, M., On von Karman's equations and the buckling of a thin elastic plate, I, The clamped plate. Comm. Pure Appl. Math. 20, 687–719 (1967).

    Google Scholar 

  6. Berger, M., & P.C. Fife, Von Kármán's equations and the buckling of a thin elastic plate, II, Plate with general edge conditions. Comm. Pure Appl. Math. 21, 227–241 (1968).

    Google Scholar 

  7. Cesari, L., Functional analysis and Galerkin's method. Mich. Math. J. 11, 385–418 (1964).

    Google Scholar 

  8. Chafee, N., The bifurcation of one or more closed orbits from an equilibrium point of an autonomous differential system. J. Diff. Eqns. 4, 661–679 (1968).

    Google Scholar 

  9. Chow, S.N., & J. Mallet-Paret, Averaging and bifurcation from equilibrium and periodic orbits, to appear.

  10. Cohen, D.S., Multiple solutions of nonlinear partial differential equations. Springer-Verlag Lecture Notes in Mathematics 322, 15–77, 1973.

    Google Scholar 

  11. Crandall, M.G., & P.H. Rabinowitz, Bifurcation from simple eigenvalues. J. Funct. Anal. 8, 321–340 (1971).

    Google Scholar 

  12. Golubitsky, M., & V. Guillemin, Stable Mappings and their Singularities. New York: Springer 1973.

    Google Scholar 

  13. Hale, J.K., Applications of alternative problems. CDS Lecture Notes 71-1, Brown University, 1971.

  14. Hopf, E., Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems. Ber. Math.-Phys. Sachsische Akademie der Wissenschaften Leipzig 94, 1–22 (1942).

    Google Scholar 

  15. Kannan, R., Periodically disturbed conservative systems. J. Diff. Eqns. 16, 506–514 (1974).

    Google Scholar 

  16. Keener, J.P., Some modified bifurcation problems with applications to imperfection sensitivity in buckling. Ph.D. Thesis, Cal Inst. Tech., California, 1972.

    Google Scholar 

  17. Keener, J.P., & H.B. Keller, Perturbed bifurcation theory. Arch. Rational Mech. Anal. 50, 159–175 (1973).

    Google Scholar 

  18. Knightly, G., & D. Sather, On nonuniqueness of solutions of the von Kármán equations. Arch. Rational Mech. Anal. 36, 65–78 (1970).

    Google Scholar 

  19. Krasnoselskii, M.A., Topological Methods in the Theory of Nonlinear Integral Equations. New York: MacMillan 1964.

    Google Scholar 

  20. Kuiper, N.H., Cr-functions near non-degenerate critical points. Mimeographed, Warwick University, Coventry, 1966.

    Google Scholar 

  21. Kuo, T.C., On C0-sufficiency of jets of potential functions. Top. 8, 167–172 (1969).

    Google Scholar 

  22. Landesman, E.M., & A. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance. J. Math. Mech. 19, 609–623 (1970).

    Google Scholar 

  23. Lazer, A., Application of a lemma on bilinear forms to a problem in nonlinear oscillations. Proc. Am. Math. Soc. 33, 89–94 (1972).

    Google Scholar 

  24. Lazer, A., & D. Sanchez, On periodically perturbed conservative system. Mich. Math. J. 16, 193–200 (1969).

    Google Scholar 

  25. Lefschetz, S., Algebraic Geometry. Princeton: University Press 1953.

    Google Scholar 

  26. Marsden, J., The Hopf bifurcation for nonlinear semigroups. Bull. Amer. Math. Soc. 79, 537–541 (1973).

    Google Scholar 

  27. Payne, L.E., & I. Stakgold, Nonlinear problems in nuclear reactor analysis. Lecture Notes in Math. 322, pp. 298–309. Berlin-Heidelberg-New York: Springer 1973.

  28. Rabinowitz, P., Periodic solutions of a nonlinear wave equation. Comm. Pure Appl. Math. 20, 145–205 (1969).

    Google Scholar 

  29. Sattinger, D., Topics in stability and bifurcation theory. Lecture Notes in Math. 309, Berlin-Heidelberg-New York: Springer 1973.

    Google Scholar 

  30. de Simon, L., & G. Torelli, Soluzioni periodiche di equazione a derivate parziali di tipo iperbolico non lineare. Rend sem. Mat. Padova 40, 380–401 (1968).

    Google Scholar 

  31. Sotomayor, J., Generic one-parameter families of vector fields in two-dimensional manifolds. Bull. Amer. Math. Soc. 74, 722–726 (1968); complete proofs to be published in Pub. Math. I.H.E.S.

    Google Scholar 

  32. Takens, F., Singularities of functions and vector fields. Nieuw Arch. voor Wisk. (3), XX, 107–130 (1972).

    Google Scholar 

  33. Thom, R., Topological methods in biology. Top. 8, 313–335 (1968).

    Google Scholar 

  34. Vainberg, M.M., & V.A. Trenogin, The method of Liapunov and Schmidt in the theory of nonlinear differential equations and their further development. Russ. Math. Surveys 17, 1–60 (1962).

    Google Scholar 

  35. Vainberg, M.M., & V.A. Trenogin, Theory of Branching of solutions of Nonlinear Equations. Noordhoff, 1974.

  36. Koiter, W.T., The nonlinear buckling problem of a complete spherical shell under uniform external pressure, Report No. 412, Lab. of Eng. Mech. Technological Univ., Delft 1968.

    Google Scholar 

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Communicated by C.L. Dafermos

This research was supported in part by the National Science Foundation under GP-28931 X2, in part by the United States Army, Durham, under DA-ARO-D-31-124-73-G-130, and in part by the Air Force Office of Scientific Research under AF-AFOSR 71-2078C.

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Chow, SN., Hale, J.K. & Mallet-Paret, J. Applications of generic bifurcation. I. Arch. Rational Mech. Anal. 59, 159–188 (1975). https://doi.org/10.1007/BF00249688

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