This paper is devoted to some of the results in bifurcation theory obtained by topological methods in the last 25 years. The cases of one and several parameters will be reviewed, with “necessary” and sufficient conditions for bifurcation, both local and global, and the structure of the bifurcation set will be studied. The case of equivariant bifurcation will be considered, with a special application to the case of abelian groups.


Fredholm Operator Bifurcation Theory Isotropy Subgroup Degree Theory Global Bifurcation 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Adams, J.F., Prerequisites (on equivariant stable homotopy theory) for Carlsson’s lecture, Lect. Notes in Math. 1091 (1982), 483–532.Google Scholar
  2. Alcaraz, D., Existence Theory for a model of Steady Vortex motion in Ideal Fluids, Oxford Ph.D. thesis, 1983.Google Scholar
  3. Alexander, J.C., Bifurcation of zeros of parametrized functions, J. Funct. Anal. 29 (1978), 37–53.MathSciNetMATHGoogle Scholar
  4. Alexander, J.C., Calculating bifurcation invariants as elements of the homotopy of the general linear group. II, J. Pure and Appl. Algebra, 17 (1980), 117–125.MathSciNetMATHGoogle Scholar
  5. Alexander, J.C., A primer on connectivity, Lect. Notes in Math., Springer-Verlag, 886 (1981), 445–483.Google Scholar
  6. Alexander, J.C. and S.S. Antman, Global and local behavior of bifurcating multidimensional continua of solutions for multiparameter nonlinear eigenvalue problems, Arch, for Rat. Mech. and Anal. 76 (1981), 339–354.MathSciNetMATHGoogle Scholar
  7. Alexander, J.C. and S.S. Antman, Global behavior of solutions of nonlinear equations depending on infinite dimensional parameter, Indiana Univ. Math. J. 32 (1983), 39–62.MathSciNetMATHGoogle Scholar
  8. Alexander, J.C. and J. F.G. Auchmuty, Global branches of waves, Manus. Math. 27 (1979), 208–220.MathSciNetGoogle Scholar
  9. Alexander, J.C. and P.M. Fitzpatrick, The homotopy of certain spaces of nonlinear operators and its relation to global bifurcation of the fixed points of parametrized condensing operators, J. Funct. Anal. 34 (1979), 87–106.MathSciNetMATHGoogle Scholar
  10. Alexander, J.C., I. Massabó and J. Pejsachowicz, On the connectivity properties of the solution set of infinitely parametrized families of vector fields, Boll. Un. Mat. Ital A. (6) 1 (1982), 309–312.Google Scholar
  11. Alexander, J.C. and J.A. Yorke, The implicit function theorem and global methods of cohomology, J. Funct. Anal. 21 (1976), 330–339.MathSciNetMATHGoogle Scholar
  12. Alexander, J.C. and J.A. Yorke, Parametrized functions, bifurcation and vector fields on spheres. Anniversary Volume in Honnor of Mitropolsky, Naukova Dumka, 275 (1977), 15–17.MathSciNetGoogle Scholar
  13. Alexander, J.C. and J. A. Yorke, Global bifurcation of periodic orbits, Amer. J. Math. 100 (1978), 263–292.MathSciNetMATHGoogle Scholar
  14. Alexander, J.C. and Yorke, J.A., Calculating bifurcation invariants as elements of the general linear group I, J. Pure and Appl. Algebra, 13 (1978), 1–8.MathSciNetMATHGoogle Scholar
  15. Alexander, J.C. and J.A. Yorke, On the continuability of periodic orbits of parametrized three-dimensional differential equations, J. of Diff. Eq. 49 (1983), 171–184.MathSciNetMATHGoogle Scholar
  16. Alligood, K.T., Homological indices and homotopy continuation, Ph.D. Thesis, Univ. of Maryland, 1979.Google Scholar
  17. Alligood, K.T., Mallet-Paret, J. and J.A. Yorke, Families of periodic orbits: local continuability does not imply global continuability, J. Diff. Geometry 16 (1981), 483–492.MathSciNetMATHGoogle Scholar
  18. Alligood, K.T. and J. A. Yorke, Hopf bifurcation: the appearance of virtual periods in cases of resonance, J. Diff. Eq. 64 (1986), 375–394.MathSciNetMATHGoogle Scholar
  19. Amann, H. Fixed point equations and Nonlinear eigenvalue problems in ordered Banach spaces, SIAM Reviews, 18 (1976), 620–709.MathSciNetMATHGoogle Scholar
  20. Amann, H., Ambrosetti, A. and G. Mancini, Elliptic equations with noninvertible Fredholm linear part and bounded nonlinearities, Math. Z. 158 (1978), 179–194.MathSciNetMATHGoogle Scholar
  21. Antman, S.S., Buckled states of nonlinearly elastic plates, Archiv. Rat. Mech. and Anal. 67 (1978), 111–149.MathSciNetMATHGoogle Scholar
  22. Arnold, V.I., Lectures on bifurcation and versal systems, Russ. Math. Surveys, 27 (1972), 54–113.Google Scholar
  23. Balonov, Z., Kushkuley, A. and P. Zabrejko, A degree theory for equivariant maps: geometric approach. To appear in Top. Methods in Nonlinear Anal, 1993.Google Scholar
  24. Bartsch. T., Verzweigung in Vektorraumbündels und äquivariante Verzweigung, Ph.D. Thesis, Univ. München, 1986.Google Scholar
  25. Bartsch, T., Global bifurcation from a manifold of trivial solutions, Univ. of Heidelberg Math. Inst. 13 (1987).Google Scholar
  26. Bartsch, T., A global index for bifurcation of fixed points, J. Reine Math. 391 (1988), 181–197.MathSciNetMATHGoogle Scholar
  27. Bartsch, T., The role of the J-homomorphism in multiparameter bifurcation theory, Bull. Sei. Math. 112 (1988), 177–184.MathSciNetMATHGoogle Scholar
  28. Bartsch, T. The global structure of the zero set of a family of semilinear Fredholm maps, Nonlinear Anal. T.M.A. 17 (1991), 313–332.MathSciNetMATHGoogle Scholar
  29. Bartsch, T., A simple proof of the degree formula for ℤ/p-equivariant maps, Univ. of Heidelberg, preprint, to appear in Math. Z., 1991.Google Scholar
  30. Bartsch, T., Topological Methods for variational problems with symmetries, Habilitationsschrift, Heidelberg, 1992.Google Scholar
  31. Bartsch, T. and M. Clapp, Bifurcation theory for symmetric potential operators and the equivariant cup-length, Math. Z. 204 (1990), 341–356.MathSciNetMATHGoogle Scholar
  32. Bazley, N.W., McLeod, J.B., Bifurcation from infinity and singular eigenvalue problems, Proc. London Math. Soc. 34 (1977), 231–244.MathSciNetMATHGoogle Scholar
  33. Benjamin, T.B., Applications of Leray-Schauder degree theory to problems in hydrodynamic stability, Math. Proc. Cambridge Phil. Soc. 79 (1976), 373–392.MathSciNetMATHGoogle Scholar
  34. Berestycki, H., On some nonlinear Sturm Liouville problems, Jour, of Diff. Eq. 26 (1977), 375–390.MathSciNetMATHGoogle Scholar
  35. M. Berger and M. Berger, Perspectives in Nonlinearity. An Introduction to Nonlinear Analysis, Benjamin Inc., 1968.MATHGoogle Scholar
  36. Berger, M.S., A Sturm Liouville theorem for nonlinear elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 3.20 (1966), 543–582.Google Scholar
  37. Berger, M.S., A Bifurcation theory for nonlinear elliptic Partial Differential Equations and related Systems, J.B. Keller and S. Antman, Ed., Benjamin, 113–216, 1969.Google Scholar
  38. Berger, M.S., Applications of global analysis to specific nonlinear eigenvalue problems, Rocky Mountain J. of Math. 3 (1973), 319–354.MATHGoogle Scholar
  39. Berger, M.S., Nonlinearity and Functional Analysis, Academic Press, 1977.MATHGoogle Scholar
  40. Berger, M.S. and Podolak, E., On nonlinear Fredholm operator equations, Bull A.M.S. 80 (1974), 861–864.MathSciNetMATHGoogle Scholar
  41. Berger, M.S. and D. Westreich, A convergent iteration scheme for bifurcation theory on Banach spaces, J. Math. Anal. Appl. 43 (1974), 136–144.MathSciNetGoogle Scholar
  42. Böhme, R., Die Lösung der Verzweigungsgleichungen für nichtlineare Eigenwert-probleme, Math.Z 127 (1972), 105–126.MathSciNetMATHGoogle Scholar
  43. Borisovich, Y.G., Topology and nonlinear functional analysis, Russian Math. Surveys, 37 (1979), 14–23.MathSciNetGoogle Scholar
  44. Borisovich, Y.G., V.G. Zvyagin and Y.I. Sapronov, Nonlinear Fredholm maps and the Leray Schauder theory, Russian Math. Surveys, 32 (1977), 1–54.MATHGoogle Scholar
  45. Bredon, G.E., Introduction to Compact Transformation Groups, Academic Press. New York, 1980.MATHGoogle Scholar
  46. Bröcher, T. and T. tom Dieck, Representations of compact Lie groups, Grad. Texts in Math. 98, Springer-Verlag, New York, 1985.Google Scholar
  47. Browder, F.E., Nonlinear eigenvalue problems and group invariance, Functional Analysis and Related Fields, 1–58, Springer, 1970.Google Scholar
  48. Browder, F.E., Nonlinear operators in Banach spaces, Proc. Symp. P.M. 18, vol. 2, (1970), A.M.S.Google Scholar
  49. Browder, F.E. and R.D. Nussbaum, The Topological degree for non compact nonlinear mappings in Banach spaces, Bull A.M.S. 74 (1968), 671–676.MathSciNetMATHGoogle Scholar
  50. Buchner, M., Marsden, J, and S. Schecher, Applications of the blowing up construction and algebraic geometry to bifurcation problems, J. Diff. Eq. 48 (1983), 404–433.MATHGoogle Scholar
  51. Cantrell, R.S., A homogeneity condition guaranteeing bifurcation in multiparameter eigenvalues problems, Nonlinear Anal. T.M.A. 8 (1984), 159–169.MathSciNetMATHGoogle Scholar
  52. Cantrell, R.S., Multiparameter bifurcation problems and topological degree, J. of Diff. Eq. 52 (1984), 39–51.MathSciNetMATHGoogle Scholar
  53. Cantrell, R.S., A homogeneity condition guaranteeing bifurcation in multiparameter nonlinear eigenvalue problems, Nonlinear Analysis, T.M.A. 8 (1984), 159–169.MathSciNetMATHGoogle Scholar
  54. Cerami, G., Symmetry breaking for a class of semilinear elliptic problems, Nonlinear Anal T.M.A. 10 (1986), 1–14.MathSciNetMATHGoogle Scholar
  55. Cesari, L., Functional Analysis, Nonlinear Differential Equations and the alternative method, Lect. Notes in Pure and Applied Math. Vol. 19 (1976), Marcel Decker, 1–198.Google Scholar
  56. Chang, K.C., Applications of homology theory to some problems in Differential Equations, Proc. of Symp. in Pure Math. 45 (1986), 253–262.Google Scholar
  57. Chen, B., Jorge M.C. and A.A. Minzoni, Bifurcation of solutions for an inverse problem in potential theory, Studies in Applied Math. 86 (1992), 31–51.MathSciNetMATHGoogle Scholar
  58. Chiappinelli, R. and C.A. Stuart, Bifurcation when the linearized problem has no eigenvalues, J. of Diff. Eq. 30 (1978), 296–307.MathSciNetMATHGoogle Scholar
  59. Chossat, P. and G. Iooss, Primary and secondary bifurcation in the Couette-Taylor problem, Japan J. of Appl. Math. 2 (1985), 37–68.MathSciNetMATHGoogle Scholar
  60. Chow, S.N. and J.K. Hale, Methods of bifurcation theory, Grundl. Math. Wiss. 251, Springer Verlag, 1982.Google Scholar
  61. Chow, S.N. and R. Lauterbach, A bifurcation theorem for critical points of variational problems, Nonlinear Anal. T.M.A. 12 (1988), 51–61.MathSciNetMATHGoogle Scholar
  62. Chow, S.N. and J. Mallet-Paret, The Fuller index and global Hopf bifurcation, J. Diff. Eq. 29 (1978), 66–85.MathSciNetMATHGoogle Scholar
  63. Chow, S.N., J. Mallet-Paret and J.A. Yorke, Global Hopf bifurcation from a multiple eigenvalue, Nonlinear Anal TMA, 2 (1978), 753–763.MathSciNetMATHGoogle Scholar
  64. Cicogna, G., Bifurcation from topology and symmetry arguments, Bol. U.M.I. 6 (1984), 131–138.MathSciNetGoogle Scholar
  65. Conley, C., Isolated invariant sets and the Morse index, CBMS regional Conf. Semes in Math. 38, 1978.MATHGoogle Scholar
  66. Cosner, C., Bifurcation from higher eigenvalues in nonlinear elliptic equations: continua that meet infinity. Univ. of Miami, preprint, 1981.Google Scholar
  67. Crandall, M.G. and P.H. Rabinowitz, Bifurcation from simple eigenvalues, J. Func. Anal. 8 (1971), 321–340.MathSciNetMATHGoogle Scholar
  68. Crandall, M.G. and P.H. Rabinowitz, The Hopf bifurcation theorem in infinite dimensions, Arch. Rat. Mech. Anal. 67 (1977), 53–72.MathSciNetMATHGoogle Scholar
  69. J. Cronin, 1964. Fixed points and topological degree in Nonlinear Analysis, Mathematical Surveys, 11 (1964), A.M.S. Providence.Google Scholar
  70. Cronin, J., Eigenvalues of some nonlinear operators, J. of Math. Anal and Appl. 38 (1972), 659–667.MathSciNetMATHGoogle Scholar
  71. Dancer, E.N., Bifurcation theory in real Banach space, Proc. London Math. Soc. 23 (1971), 699–734.MathSciNetMATHGoogle Scholar
  72. Dancer, E.N., Bifurcation theory for analytic operators, Proc. London Math. Soc. 26 (1973), 359–384.MathSciNetMATHGoogle Scholar
  73. Dancer, E.N., Global structure of the solutions of non-linear real analytic eigenvalue problems, Proc. London Math. Soc. 26 (1973), 745–765.MathSciNetGoogle Scholar
  74. Dancer, E.N., On the structure of solutions of non-linear eigenvalue problems, Indiana Univ. Math. J. 23 (1974), 1069–1076.MathSciNetMATHGoogle Scholar
  75. Dancer, E.N., On the existence of bifurcating solutions in the presence of symmetries, Proc. Royal Soc. Edinburg A, 85 (1980), 321–336.MathSciNetMATHGoogle Scholar
  76. Dancer, E.N., Symmetries, degree, homotopy indices and asymptotically homogeneous problems, Nonlinear Anal. T.M.A. 6 (1982), 667–686.MathSciNetMATHGoogle Scholar
  77. Dancer, E.N., On the indices of fixed points of mappings in cones and applications, J. of Math. Anal, and Appl. 91 (1983), 131–151.MathSciNetMATHGoogle Scholar
  78. Dancer, E.N., Perturbation of zeros in the presence of symmetries, J. Austral. Math. Soc. 36 (1984), 106–125.MathSciNetMATHGoogle Scholar
  79. Dancer, E.N., A new degree for S 1-invariant gradient mappings and applications, Annal. Inst. H. Poincaré, Anal. Non. Lin. 2 (1985), 329–370.MathSciNetMATHGoogle Scholar
  80. Dancer, E.N. and J.F. Toland, Degree theory for orbits of prescribed period of flows with a first integral, Proc. London Math. Soc. 60 (1990), 549–580.MathSciNetMATHGoogle Scholar
  81. Dancer, E.N. and J.F. Toland, Equilibrium states in the degree theory of periodic orbits with a first integral, Proc. London Math. Soc. 61 (1991), 564–594.MathSciNetGoogle Scholar
  82. Deimling, K., Nonlinear Functional Analysis, Springer Verlag, 1985.MATHGoogle Scholar
  83. Dellnitz, M., I. Melbourne and J.E. Marsden, Generic bifurcation of Hamiltonian vector fields with symmetry, Nonlinearity 15 (1992), 979–996.MathSciNetGoogle Scholar
  84. Dugundji, J. and A. Granas, Fixed point theory I. Warzawa: PWN-Polish Scientific, 1982.MATHGoogle Scholar
  85. Dylawerski, G., Geba, K., Jodel, J. and W. Marzantowicz, An S 1-equivariant degree and the Fuller index, Ann. Pol. Math. 52 (1991), 243–280.MathSciNetMATHGoogle Scholar
  86. Eells, J., Fredholm structures, Symp. Nonlinear Functional Anal. 18 (1970), 62–85.MathSciNetGoogle Scholar
  87. Elworthy, K.D. and A.J. Tromba, Degree theory on Banach manifolds. “Nonlinear Functional Analysis”, Proc. Symp. Pure Math. Vol. 18/1 (1970), AMS, 86–94.MathSciNetGoogle Scholar
  88. Erbe, L., Geba, K., Krawcewicz, W. and J. Wu, S 1-degree and global Hopf bifurcation theory of functional differential equations, J. Diff. Eq. 98 (1992), 277–298.MathSciNetMATHGoogle Scholar
  89. Esquinas, J. and J. López Gómez, Optimal multiplicity in local bifurcation theory. I. Generalized generic eigenvalues, J. Diff. Eq. 71 (1988), 71–92.Google Scholar
  90. Esquinas, J., Optimal multiplicity in bifurcation theory. II. General Case, J. Diff. Eq. 75 (1988), 206–215.MathSciNetMATHGoogle Scholar
  91. Fadell, E.R. and P.H. Rabinowitz, Generalized cohomological index theories for Lie group actions with application to bifurcation questions for Hamiltonian systems, Invent. Math. 45 (1978), 139–174.MathSciNetMATHGoogle Scholar
  92. Fenske, C., An index for periodic orbits of functional differential equations, Math. Ann. 285 (1989), 381–392.MathSciNetMATHGoogle Scholar
  93. Field, M.J. and R.W. Richardson, Symmetry breaking and branching problems in equivariant bifurcation theory, I, Arch. Mat Mech. Anal. 118 (1992), 297–348.MathSciNetMATHGoogle Scholar
  94. Fiedler, B., An index for global Hopf bifurcation in parabolic systems, J. Reine u. Angew. Math. 359 (1985), 1–36.MathSciNetMATHGoogle Scholar
  95. Fiedler, B., Global Hopf bifurcation of two-parameters flows, Arch. Rat. Mech. Anal. 94 (1986), 59–81.MathSciNetGoogle Scholar
  96. Fiedler, B., Global Bifurcation of periodic solutions with symmetry, Lect. Notes in Math. 1309 (1988), Springer Verlag.MATHGoogle Scholar
  97. Fife, P.C., Branching Phenomena in fluid dynamics and chemical Reaction-diffusion theory, CIME Lect. Notes (1974).Google Scholar
  98. Fitzpatrick, P.M., A generalized degree for uniform limits of A-proper mappings, J. Math. Anal Appl. 35 (1970), 536–552.MathSciNetGoogle Scholar
  99. Fitzpatrick, P.M., A-proper mappings and their uniform limits, Bull. AMS 78 (1972), 806–809.MathSciNetMATHGoogle Scholar
  100. Fitzpatrick, P.M., On the structure of the set of solutions of equations involving A-proper mappings, Trans. AMS 189 (1974), 107–131.MathSciNetMATHGoogle Scholar
  101. Fitzpatrick, P.M., Homotopy, linearization and bifurcation, Nonlinear Anal T.M.A. 12 (1988), 171–184.MathSciNetMATHGoogle Scholar
  102. Fitzpatrick, P.M., The stability of parity and global bifurcation via Galerkin Approximation, J. London Math. Soc. 38 (1988), 153–165.MathSciNetMATHGoogle Scholar
  103. Fitzpatrick, P.M., Massabó, I. and Pejsachowicz, J., Complementing maps, continuation and global bifurcation, Bull A.M.S. 9 (1983), 79–81.MATHGoogle Scholar
  104. Fitzpatrick, P.M., Massabó I. and Pejsachowicz, J., Global several parameters bifurcation and continuation theorems, a unified approach via complementing maps, Math. Ann. 263 (1983), 61–73.MathSciNetMATHGoogle Scholar
  105. Fitzpatrick, P.M., I. Massabó and J. Pejsachowicz, On the covering dimension of the set of solutions of some nonlinear equations, Trans. A.M.S. 296 (1986), 777–798.MATHGoogle Scholar
  106. Fitzpatrick, P.M. and J. Pejsachowicz, The fundamental group of the space of linear Fredholm operators and the global analysis of semilinear equations, Contemp. Math. 72 (1988), 47–87.MathSciNetGoogle Scholar
  107. Fitzpatrick, P.M. and J. Pejsachowicz, A local bifurcation theorem for C 1- Fredholm maps, Proc. Amer. Math. Soc. 109 (1990), 995–1002.MathSciNetMATHGoogle Scholar
  108. Fitzpatrick, P.M. and J. Pejsachowicz, Parity and generalized multiplicity, Trans. A.M.S. 326 (1991), 281–305.MathSciNetMATHGoogle Scholar
  109. Fitzpatrick, P.M. and J. Pejsachowicz, Nonorientability of the index bundle and several parameter bifurcation, J. of Funct. Anal. 98 (1991), 42–58.MATHGoogle Scholar
  110. Fitzpatrick, P.M. and J. Pejsachowicz, The Leray-Schauder theory and fully non-linear elliptic boundary value problems, Memoirs A.M.S., Vol. 101, No. 483 (1993).Google Scholar
  111. Fitzpatrick, P.M., J. Pejsachowicz and P.J. Rabier, Topological degree for nonlinear Fredholm operators, C.R. Acad. Sci. Paris, 311 (1990), 711–716.MathSciNetMATHGoogle Scholar
  112. Fucik, S., Necas, J., and Soucek, V., Spectral Analysis of Nonlinear Operators, Lect. Notes in Math. 343 (1973), Springer-Verlag.Google Scholar
  113. Fuller, F.B., An index of fixed point type for periodic orbits, Am. J. Math. 89 (1967), 133–148.MathSciNetMATHGoogle Scholar
  114. Furi, M., Martelli, M. and A. Vignoli, On the Solvability of nonlinear operator equations in normed spaces, Ann. Mat. Pura Appl. 124 (1980), 321–343.MathSciNetMATHGoogle Scholar
  115. Furi, M. and M.P. Per a, A continuation principle for forced oscillations on differentiable manifolds, Pacific J. of Math. 121 (1986), 321–338.MathSciNetMATHGoogle Scholar
  116. Gaines, R.E. and J.L. Mawhin, Coincidence degree and nonlinear differential equations, Lect. Notes in Math. 568 (1977), Springer Verlag.Google Scholar
  117. Gavalas, G.R., Nonlinear differential equations of chemically reacting systems, Springer Verlag Tracts in Nat. Phil. 17, 1968.Google Scholar
  118. Geba, K. and A. Granas, Infinite dimensional cohomology theories, J. Math. Pures et Appl. 52 (1973), 147–270.MathSciNetGoogle Scholar
  119. Geba, K., Krawcewicz, W. and J. Wu, An equivariant degree with applications to symmetric bifurcation problems, Preprint, 1993.Google Scholar
  120. Geba, K., Marzantowicz, W., Global bifurcation of periodic orbits, Topological Methods in Nonlinear Analysis, 1 (1993), 67–93.MathSciNetMATHGoogle Scholar
  121. Geba, K., Massabó, I. and Vignoli, A., Generalized topological degree and bifurcation, Nonlinear Functional Analysis and its Applications, Reidel, (1986), 55–73Google Scholar
  122. Geba, K., Massabó, I, and Vignoli, A., On the Euler characteristic of equivariant vector fields, Boll UMI. 4A (1990), 243–251.Google Scholar
  123. Golubitsky, M. and J.M. Gukenheimer (eds), Multiparameter bifurcation theory, Cont. Math. 56 (1986), AMSGoogle Scholar
  124. Golubitsky, M. and D.G. Schaeffer, Singularities and groups in bifurcation theory I, Appl Math. Sc. 51 (1986), Springer Verlag.Google Scholar
  125. Golubitsky, M., D.G. Schaeffer and I.N. Stewart, Singularities and Groups in Bifurcation Theory II, Springer Verlag, 1988.MATHGoogle Scholar
  126. Granas, A., The theory of compact vector fields and some of its applications to topology of functional spaces, Rozprawy Mat. 30 (1962), Warzawa.Google Scholar
  127. Granas, A., The Leray-Schauder index and the fixed point theory for arbitrary ANRs, Bull. Soc. Math. France, 100 (1972), 209–228.MathSciNetMATHGoogle Scholar
  128. Guckenheimer, J., Holmes, P., Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields, Springer Verlag, 1983.Google Scholar
  129. Gurel, O., Rössler, O.E., Bifurcation theory and applications in scientific disciplines, Annals New York Acad. Sci. 316 (1979).Google Scholar
  130. Hassard, B.J., Kazarinoff, N.D., Wan, Y-H., Theory and Applications of Hopf Bifurcation, Cambridge University Press, 1981.MATHGoogle Scholar
  131. Hauschild, H., Aquivariante homotopie I, Arch. Math. 29 (1977a), 158–167.MathSciNetMATHGoogle Scholar
  132. Hauschild, H., Zerspaltung äquivarianter Homotopiemengen, Math. Ann. 230 (1977b), 279–292.MathSciNetMATHGoogle Scholar
  133. Healey, T.J. and K. Kielhöfer, Symmetry and nodal properties in the global bifurcation analysis of quasi-linear elliptic equations, Arch. Rat. Mech. Anal. 113 (1991), 299–311.MATHGoogle Scholar
  134. Heinz, G., Lösungsverzweigung bei analytischen gleichungen mit Fredholmoperator vom Index null, Math. Nachr. 128 (1986), 243–254.MathSciNetMATHGoogle Scholar
  135. Hetzer, G., Stallbohm, V., Global behaviour of bifurcation branches and the essential spectrum, Math. Nachr. 86 (1978), 347–360.MathSciNetMATHGoogle Scholar
  136. Hoyle, S.C., Local solutions manifolds for nonlinear equations, Nonlinear Anal. T.M.A. 4 (1980), 285–295.MathSciNetGoogle Scholar
  137. Hoyle, S.C., Hopf bifurcation for ordinary differential equations with a zero eigenvalue, J. Math. Anal. Appl. 74 (1980), 212–232.MathSciNetMATHGoogle Scholar
  138. Hernández, J., Bifurcación y soluciones positivas para algunos problemas de tipo unilateral. Tesis doctoral, Univ. Aut. de Madrid, 1977.Google Scholar
  139. Iooss, G., Bifurcation of Maps and Applications, North Holland, 1979.MATHGoogle Scholar
  140. Iooss, G. and D.D. Joseph, Elementary Stability and Bifurcation Theory, Springer Verlag, 1980.MATHGoogle Scholar
  141. Ize, J., Bifurcation theory for Fredholm operators, Memoirs A.M.S. 174 (1976).Google Scholar
  142. Ize, J., Periodic solutions for nonlinear parabolic equations, Comm. in P.D.E. 12 (1979), 1299–1387.MathSciNetGoogle Scholar
  143. Ize, J. Introduction to bifurcation theory, Springer Verlag, Lect. Notes in Math. 957 (1982), 145–202.MathSciNetGoogle Scholar
  144. Ize, J., Obstruction theory and multiparameter Hopf bifurcation, Trans. A.M.S. 289 (1985), 757–792.MathSciNetMATHGoogle Scholar
  145. Ize, J., Massabó, I., Pejsachowicz, J. and A. Vignoli, Structure and dimension of global branches of solutions to multiparameter nonlinear equations, Trans. A.M.S. 291 (1985), 383–435.MATHGoogle Scholar
  146. Ize, J., Massabó, I. and A. Vignoli, Global results on continuation and bifurcation for equivariant maps, NATO-ASI, 173 (1986), 75–111.Google Scholar
  147. Ize, J., Necessary and sufficient conditions for multiparameter bifurcation, Rocky Mountain J. of Math. 18 (1988), 305–337.MathSciNetMATHGoogle Scholar
  148. Ize, J., Massabó, I. and A. Vignoli, Degree theory for equivariant maps I, Trans. A.M.S. 315 (1989), 433–510.MATHGoogle Scholar
  149. Ize, J., Massabó and A. Vignoli, Degree theory for equivariant maps: The general S 1-action, Memoirs A.M.S. 481 (1992).Google Scholar
  150. Ize, J. and A. Vignoli, Equivariant Degree for abelian groups. Part I: Equivariant Homotopy groups. Preprint, 1993.Google Scholar
  151. James, I.M., On the suspension sequence, Annals of Math. 65 (1957), 74–107.MATHGoogle Scholar
  152. Jorge, M.C. and A.A. Minzoni, Examples of bifurcation from a continuum of eigenvalues and from the continuous spectrum, Quart, of Appl. Math. 51 (1993), 37–42.MathSciNetMATHGoogle Scholar
  153. J.B. Keller and S. Antman, eds., Bifurcation Theory and Nonlinear Eigenvalue Problems, Benjamin, 1969.MATHGoogle Scholar
  154. Kielhöfer, H., Hopf bifurcation at multiple eigenvalues, Arch. Rat. Mech. Anal. 69 (1979), 53–83.MATHGoogle Scholar
  155. Kielhöfer, H., Multiple eigenvalue bifurcation for potential operators, J. Reine Angew. Math. 358 (1985), 104–124.MathSciNetMATHGoogle Scholar
  156. Kielhöfer, H., Interaction of periodic and stationary bifurcation from multiple eigenvalues, Math Z. 192 (1986), 159–166.MathSciNetMATHGoogle Scholar
  157. Kielhöfer, H., A bifurcation theorem for potential operators, J. Funct. Anal. 77 (1988), 1–8.MathSciNetMATHGoogle Scholar
  158. Kielhöfer, H., Hopf bifurcation from a differentiate viewpoint, J. Diff. Eq. 97 (1992), 189–232.MATHGoogle Scholar
  159. Kirchgässner, K. and P. Sorger, Branching analysis for the Taylor problem, Quart. J. Mech. Appl. Math. 22 (1969), 183–210.MathSciNetMATHGoogle Scholar
  160. Kirchgässner, K., Bifurcation in Nonlinear hydrodynamic stability, SIAM Review, 17 (1975), 652–683.MathSciNetMATHGoogle Scholar
  161. Komiya, K., Fixed point indices of equivariant maps and Moebius inversion, Invent. Math. 91 (1988), 129–135.MathSciNetMATHGoogle Scholar
  162. Kosniowski C., Equivariant cohomology and stable cohomotopy, Math. Ann. 210 (1974), 83–104.MathSciNetMATHGoogle Scholar
  163. Kötzner, P., Calculating homotopy classes and bifurcation, part I. Univ. of Augsburg, preprint, 1990.Google Scholar
  164. Krasnosel’skii, M.A., Positive Solutions of Operator Equations, Noordhoff, 1964.Google Scholar
  165. Krasnosel’skii, M.A., Topological Methods in the Theory of Nonlinear Inte- gral Equations, Pergamon Press, MacMillan, New York, 1964.Google Scholar
  166. Krasnosel’skii, M.A. and P.P. Zabreiko, Geometrical Methods in Nonlinear Analysis, Grund Math. Wiss. 263 (1984), Springer Verlag.Google Scholar
  167. Küpper, T., Riemer, D., Necessary and sufficient conditions for bifurcation from the continuous spectrum, Nonlinear Anal. 3 (1979), 555–561.MathSciNetMATHGoogle Scholar
  168. Laloux, B. and J. Mawhin, Multiplicity, Leray-Schauder formula and bifurcation, J. Diff. Eq. 24 (1977), 309–322.MathSciNetMATHGoogle Scholar
  169. Landman, K.A. and S. Rosenblat, Bifurcation from a multiple eigenvalue and stability of solutions, SIAM J. of Appl Math. 34 (1978), 743–759.MathSciNetMATHGoogle Scholar
  170. Leray, J. and Schauder, J., Topologie et equations fonctionnelles, Ann. Sci. Ecole Normale Sup. 51 (1934), 45–78.MathSciNetMATHGoogle Scholar
  171. Ljusternik, L. and L. Schnirelmann, Methodes Topologiques dans les problèmes variationels. Herman, Paris, 1934.Google Scholar
  172. Lloyd, N.G., Degree theory, Cambridge tracts in Math. 73 (1978), Cambridge Univ. Press.MATHGoogle Scholar
  173. Lopez Gómez, J., Hopf bifurcation at multiple eigenvalues with zero eigenvalue, Proc. Roy. Soc. Edin. 101 (1985), 335–352.MATHGoogle Scholar
  174. Lopez Gómez, J., Multiparameter local bifurcation, Nonlinear Anal T.M.A. 10 (1986), 1249–1259.MATHGoogle Scholar
  175. López Gómez, J., Multiparameter local bifurcation based on the linear part, J. Math. Anal and Appl. 138 (1989), 358–370.MathSciNetMATHGoogle Scholar
  176. Ma. T, Topological degrees of set-valued compact fields in locally convex spaces, Rozprawy Mat. 42 (1972), Warszawa.Google Scholar
  177. MacBain, J.A., Local and global bifurcation from normal eigenvalues II, Pacific J. Math. 74 (1978), 143–152.MathSciNetMATHGoogle Scholar
  178. Magnus, R.J., A generalization of multiplicity and the problem of bifurcation, Proc. London Math. Soc. 32 (1976), 251–278.MathSciNetMATHGoogle Scholar
  179. Magnus, R.J., On the local structure of the zero set of a Banach space valued mapping, J. Funct. Anal. 22 (1976), 58–72.MathSciNetMATHGoogle Scholar
  180. Mallet-Paret, J. and J. A. Yorke, Snakes: oriented families of periodic orbits, their sources, sinks and continuation, J. Dif. Eq. 43 (1982), 419–450.MathSciNetMATHGoogle Scholar
  181. Mallet-Paret, J. and R. Nussbaum, Boundary layer phenomena for differential-delay equations with state dependent time delays, Arch. Rat Math. Anal. 120 (1992), 99–146.MathSciNetMATHGoogle Scholar
  182. Marino A, La biforcazione nel caso variazionale, Conf. Sem. Mat. Univ. Bari, 132 (1977).Google Scholar
  183. Marsden, J.E., McCracken, M., The Hopf Bifurcation and Its Applications, Springer Verlag, 1976.MATHGoogle Scholar
  184. Marzantowicz, W., On the nonlinear elliptic equations with symmetries, J. Math. Anal. Appl. 81 (1981), 156–181.MathSciNetMATHGoogle Scholar
  185. Massabó, I. and J. Pejsachowicz, On the connectivity properties of the solutions set of parametrized families of compact vector fields, J. Funct. Anal. 59 (1984), 151–166.MathSciNetMATHGoogle Scholar
  186. Matsuoka, T., Equivariant function spaces and bifurcation points, J. Math. Soc. Japan 35 (1983), 43–52.MathSciNetMATHGoogle Scholar
  187. Mawhin, J., Topological degree methods in nonlinear boundary value problems, CBMS 40 (1977), AMS.Google Scholar
  188. McLeod, J.B. and Turner, R.E.L., Bifurcation for Lipschitz operators with an application to elasticity, Arch. R. Mech. and Anal. 63 (1977), 1–45.MathSciNetGoogle Scholar
  189. Milojevic, P.S., On the index and the covering dimension of the solution set of semilinear equations, Proc. Symp. Pure Math. AMS 45 (1986), 2, 183–205.MathSciNetGoogle Scholar
  190. Morse, M., The Calculus of Variations in the Large, A.M.S., 1934.MATHGoogle Scholar
  191. Namboodiri, V., Equivariant vector fields on spheres, Trans. A.M.S. 278 (1983), 431–460.MathSciNetMATHGoogle Scholar
  192. Nirenberg, L., An application of generalized degree to a class of nonlinear problems, 3rd. Colloq. Anal. Funct., Liege, Centre Beige de Recherches Math., (1971), 57–73.Google Scholar
  193. Nirenberg. L., Topics in nonlinear functional analysis, Led. Notes Courant Institute, New York Univ., 1974.MATHGoogle Scholar
  194. Nirenberg. L., Variational and topological methods in nonlinear problems, Bull. AMS 4 (1981), 267–302.MathSciNetMATHGoogle Scholar
  195. Nirenberg, L., Comments on Nonlinear Problems, Le Matimatiche 36 (1981), 109–119.MathSciNetMATHGoogle Scholar
  196. Nishimura, T. Fukuda T. and K. Aoki, An algebraic formula for the topological types of one parameter bifurcation diagrams, Arch. Rat. Mech. 108 (1989), 247–266.MathSciNetMATHGoogle Scholar
  197. Nitkura, Y., Existence and bifurcation of solutions for Fredholm operators with nonlinear perturbations, Nagoya Math. J. 86 (1982), 249–271.MathSciNetGoogle Scholar
  198. Nussbaum, R.D., Degree theory for local condensing maps, J. Math. Anal. Appl. 37 (1972), 741–766.MathSciNetMATHGoogle Scholar
  199. Nussbaum, R. D., A global bifurcation theorem with applications to functional differential equations, J. Funct. Anal. 19 (1975), 319–338.MathSciNetMATHGoogle Scholar
  200. Nussbaum, R.D., Some generalizations of the Borsuk-Ulam theorem, Proc. London Math. Soc. 35 (1977), 136–158.MathSciNetMATHGoogle Scholar
  201. Nussbaum, R.D., A Hopf bifurcation theorem for retarded functional differential equations, Trans. A.M.S. 238 (1978), 139–163.MathSciNetMATHGoogle Scholar
  202. Nussbaum, R.D., Differential-delay equations with two time lags, Memoirs A.M.S. 205 (1978).Google Scholar
  203. Peitgen, H.O., Topologische Perturbationen bein globalen numerischen Studium nichtlinearer Eigenvert - und Verzweigungsprobleme, Jber. d. Dt Math. Verein. 84 (1982), 107–162.MathSciNetMATHGoogle Scholar
  204. Peitgen, H.O., Walter, H.O., (eds.): Functional differential equations and approximation of fixed points, Lecture Notes in Mathematics, 730 (1982), Springer Verlag.Google Scholar
  205. Pejsachowicz, J., K-theoretic methods in bifurcation theory, Contemporary Math. 72 (1988), 193–205.MathSciNetGoogle Scholar
  206. Petryshyn, W.V., Nonlinear equations involving noncompact operators. “Nonlinear Functional Analysis”, Proc. Symp. Pure Math. 18/1 (1970), AMS, 206–223.MathSciNetGoogle Scholar
  207. Petryshyn, W.V., On the approximation solvability of equations involving A-proper and pseudo-A-proper mappings, Bull. AMS 81 (1975), 223–312.MathSciNetMATHGoogle Scholar
  208. Petryshyn, W.V., Bifurcation and asymptotic bifurcation for equations involving A-proper mappings with applications to differential equations, J. Diff. Eq. 28 (1978), 124–154.MathSciNetMATHGoogle Scholar
  209. Petryshyn, W.V., Approximation-solvability of nonlinear functional and differential equation, M. Dekker, 1992.Google Scholar
  210. Petryshyn, W.V., Fitzpatrick, P.M., A degree theory, fixed point theorems and mapping theorems for multivalued noncompact mappings, Trans. AMS 194 (1974), 1–25.MathSciNetMATHGoogle Scholar
  211. Pimbley, G., Eigenfunction branches of nonlinear operators, and their bifurcation, Lectures Notes in Math. 104 (1969), Springer Verlag.Google Scholar
  212. Poincaré, H., Les figures equilibrium, Acta Math. 7 (1885), 259–302.MathSciNetGoogle Scholar
  213. Rabier, P.J., Generalized Jordan chains and two bifurcation theorems of Krasnoselskii, Nonlinear Anal. TMA. 13, 8 (1989), 903–934.MathSciNetMATHGoogle Scholar
  214. Rabier, P.J., Topological degree and the theorem of Borsuk for general covariant mappings with applications, Nonlinear Anal. T.M.A. 16 (1991), 393–420.MathSciNetGoogle Scholar
  215. Rabinowitz, P.H., Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487–513.MathSciNetMATHGoogle Scholar
  216. Rabinowitz, P.H., Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math. 3 (1973), 161–202.MathSciNetMATHGoogle Scholar
  217. Rabinowitz, P.H., On bifurcation from infinity, J. Diff. Eq. 14 (1973), 462–475.MathSciNetMATHGoogle Scholar
  218. Rabinowitz, P.H., Theorie du Degree Topologique et Applications (Lectures Notes), 1975.Google Scholar
  219. Rabinowitz, P.H., A bifurcation theorem for potential operators, J. of Funct. Anal. 25 (1977), 412–424.MathSciNetMATHGoogle Scholar
  220. Romero Ruiz del Portal, F., Teoría del grado topológico generalizado y aplicaciones, Ph. Thesis, Madrid, 1990.Google Scholar
  221. Rubinstein, R.L., On the equivariant homotopy of spheres, Sissertationes Math. 134 (1976), 1–48.Google Scholar
  222. Sadovskii, B.N., Limit-compact and condensing operators, Russian Math. Surveys 27 (1972), 85–155.Google Scholar
  223. Sather, D., Branching of solutions of nonlinear equations, Rocky Mountain J. Math. 3 (1973), 203–250.MathSciNetMATHGoogle Scholar
  224. Sattinger, D.H., Stability of bifurcating solutions by Leray-Schauder degree, Arch. Rational Mech. Anal. 43 (1971), 154–166.MathSciNetMATHGoogle Scholar
  225. Sattinger, D.H., Topics in Stability and Bifurcation Theory, Lect, Notes in Math. 309, (1973), Springer-Verlag.MATHGoogle Scholar
  226. Sattinger, D.H., Group representation theory, bifurcation theory and Pattern formation, J. of Fund. Anal. 28 (1978), 58–101.MathSciNetMATHGoogle Scholar
  227. Sattinger, D.H., Group theoretic methods in bifurcation theory, Lecture Notes in Math. 762 (1979), Springer VerlagMATHGoogle Scholar
  228. Sattinger, D.H., Branching in the Presence of Symmetry, Wiley, 1983.Google Scholar
  229. Schaaf, R., Global behavior of solutions branches for some Neumann problems depending on one or several parameters, J. für die Reine und Ang. Math. 346 (1984), 1–31.MathSciNetMATHGoogle Scholar
  230. Schmidt, E., Zur theorie der linearen und nichtlinearen integralgleichungen, III, Math. Ann. 65 (1908), 370–399.MathSciNetMATHGoogle Scholar
  231. Schmitt, K., A study of eigenvalue and bifurcation problems for nonlinear elliptic partial differential equations via topological continuation methods, Inst. Math. Pure et Appl. Louvain-la-Neuve, 1982.Google Scholar
  232. Schmitt, K. and Z.Q. Wang, On bifurcation from infinity for potential operators, Diff. Int. Equations 4 (1991), 933–943.MathSciNetMATHGoogle Scholar
  233. Sijbrand, J., Studies in nonlinear stability and bifurcation theory, Ph.D. Thesis, Utrecht, 1981.Google Scholar
  234. Schwartz, J.T., Nonlinear Functional Analysis, Gordon and Breach, New York, 1969.MATHGoogle Scholar
  235. Stakgold, I., Branching solutions of nonlinear equations, SIAM Rev. 13 (1971), 289–332.MathSciNetMATHGoogle Scholar
  236. Steinlein, H., Borsuk’s antipodal theorem and its generalizations and applications: a survey, Topol. en Anal. Non Lineaire. Press de l’univ. de Montreal, (1985), 166–235.Google Scholar
  237. Stuart, C.A., Some bifurcation theory for k-set contractions, Proc. London Math. Soc. 27 (1973), 531–550.MathSciNetMATHGoogle Scholar
  238. Stuart, C.A., Bifurcation from the essential spectrum, Lect. Notes in Math. 1017 (1983), Springer, 575–596.Google Scholar
  239. Stuart, C.A., Bifurcation from the continuous spectrum in L p(R), Inter Sem. Numer. Math. 79 (1987), Birkhäuser, 307–318.MathSciNetGoogle Scholar
  240. Stuart, C.A., Guidance properties of nonlinear planar waveguides. Preprint, 1992.Google Scholar
  241. Takens, F., Some remarks on the Böhme-Berger bifurcation theorem, Math Z. 125 (1972), 359–364.MathSciNetGoogle Scholar
  242. Toland, J., Global bifurcation for k-set contractions without multiplicity assumptions, Quart. J. Math. 27 (1976), 199–216.MathSciNetMATHGoogle Scholar
  243. Toland, J. Global bifurcation for Neumann problems without eigenvalues, J. Dig. Eq. 44 (1982), 82–110.MathSciNetMATHGoogle Scholar
  244. tom Dieck, T., Transformation groups and representation theory, Lect. Notes in Math. 766 (1979), Springer Verlag.MATHGoogle Scholar
  245. Turner, R.E.L., Transversality in nonlinear eigenvalue problems. “Contributions to nonlinear functional analysis”, Zarantonello, E. H. ed. Acad. Press, (1971), 37–68.Google Scholar
  246. Vainberg, M.M., Variational method and method of monotone operators in the theory of nonlinear equations, Wiley, 1973.MATHGoogle Scholar
  247. Vainberg, M.M., Trenogin, V.A., Theory of branching of solutions of nonlinear equations, Noordhoff, Leyden, 1974.Google Scholar
  248. Vanderbauwhede, A., Local bifurcation and symmetry, Pitman Research Notes in Math. 75 (1982).MATHGoogle Scholar
  249. Vignoli, A., L’Analisi Nonlineare nella teoria della biforcazione, Enciclopedia delle Scienze Fiziche dell’ Inst. dell’Enciclopedia Italiana, Vol. 1, (1992), 134–144.Google Scholar
  250. Wang, Z.Q., Symmetries and the calculation of degree, Chin. Ann. of Math. 10 B (1989), 520–536.Google Scholar
  251. Webb, J.R.L. and S.C. Welsh, Topological degree and global bifurcation, Proc. Symp. Pure Math. 45/II. A.M.S., (1986), 527–531.MathSciNetGoogle Scholar
  252. Welsh, S.C., Global results concerning bifurcation for Fredholm maps of index zero with a transversality condition, Nonlinear Anal. T.M.A. 12 (1988), 1137–1148.MathSciNetMATHGoogle Scholar
  253. Werner, B., Eigenvalue problems with the symmetry of a group and bifurcation, NATO-ASI Series 313 (1990), 71–88.Google Scholar
  254. Westreich, D., Banach space bifurcation theory, Trans. A.M.S. 171 (1972), 135–156.MathSciNetMATHGoogle Scholar
  255. Westreich, D., Bifurcation at eigenvalues of odd multiplicity, Proc. A.M.S. 41 (1973), 609–614.MathSciNetMATHGoogle Scholar
  256. Westreich, D., Bifurcation at double characteristic values, J. London Math. Soc. (2), 15 (1977), 345–350.MathSciNetMATHGoogle Scholar
  257. Whitehead, G.W., On the homotopy groups of spheres and rotation groups, Annals of Math. 43 (1942), 634–640.MathSciNetMATHGoogle Scholar
  258. Whitehead, G.W., Elements of homotopy theory, Graduate texts in Math. 61 (1978), Springer Verlag.MATHGoogle Scholar
  259. Whyburn, G.T., Topological Analysis, 2nd. ed. Univ. Press, 1964.MATHGoogle Scholar
  260. Wolkowisky, J.H., A geometric theory of bifurcation, Proc. Symp. Pure Maths. 45/2 (1986), 553–564.MathSciNetGoogle Scholar
  261. Zarantonello, E.H., Contributions to Nonlinear Functional Analysis, Acad. Press, 1971.MATHGoogle Scholar
  262. Zeidler, E., Nonlinear Functional Analysis and its Applications, Vol. III, Springer Verlag, 1984.Google Scholar
  263. Zeidler, E., Nonlinear Functional Analysis and its Applications, Vol. IV. Springer Verlag, 1985.MATHGoogle Scholar

Copyright information

© Birkhäuser Boston 1995

Authors and Affiliations

  • Jorge Ize
    • 1
  1. 1.Departamento de Matemáticas y MecánicaITMAS-UNAMMexicoUSA

Personalised recommendations