Advertisement

Keywords

Periodic Solution Periodic Orbit Hopf Bifurcation Erential Equation Nonlinear Anal 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AMO]
    R. P. Agarwal, M. Meehan and D. O'Regan, Fixed point theory and applications, Cambridge Tracts in Mathematics, vol. 141, Cambridge University Press, 2001.Google Scholar
  2. [AKPR]
    R. R. Akhmerov, M. I. Kamenskiĭ, A. S. Potapov and A. E. Rodkina, Measures of Noncompactness and Condensing Operators, Operator Theory: Advances and Applications, vol. 55, Birkhäuser—Verlag, Basel, 1992.Google Scholar
  3. [A1]
    J. C. Alexander, Bifurcation of zeros of parametrized functions, J. Funct. Anal. 29 (1978), 37–53.zbMATHMathSciNetCrossRefGoogle Scholar
  4. [A2]
    _____, Calculating bifurcation invariants as elements of the homotopy of the general linear group, Illinois J. Pure Appl. Algebra 17 (1980), 117–125.zbMATHMathSciNetCrossRefGoogle Scholar
  5. [A3]
    _____, A primer on connectivity, Lecture Notes in Math. 886 (1981), Springer-Verlag, 445–483.Google Scholar
  6. [AA1]
    J. C. Alexander and S. S. Antman, Global and local behavior of bifurcating multidimensional continua of solutions for multiparameter nonlinear eigenvalue problems, Arch. Rational. Mech. Anal. 76 (1981), 339–354.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [AA2]
    _____, Global behavior of solutions of nonlinear equations depending on infinite dimensional parameter, Indiana Univ. Math. J. 32 (1983), 39–62.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [AF]
    J. C. Alexander 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.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [AMP]
    J. C. Alexander, 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.MathSciNetzbMATHGoogle Scholar
  10. [AY1]
    J. C. Alexander and J. A. Yorke, The implicit function theorem and global methods of cohomology, J. Funct. Anal. 21 (1976), 330–339.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [AY2]
    _____, Parametrized functions, bifurcation and vector fields on spheres, Anniversary Volume in Honor of Mitropolsky, vol. 275, Naukova Dumka, 1977, pp. 15–17.MathSciNetGoogle Scholar
  12. [AY3]
    _____, Global bifurcation of periodic orbits, Amer. J. Math. 100 (1978), 263–292.MathSciNetzbMATHGoogle Scholar
  13. [AY4]
    _____, Calculating bifurcation invariants as elements of the general linear group I, J. Pure Appl. Algebra 13 (1978), 1–8.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [AMY]
    K. T. Alligood, J. Mallet-Paret and J. A. Yorke, Families of periodic orbits: local continuability does not imply global continuability, J. Differential Geometry 16 (1981), 483–492.MathSciNetzbMATHGoogle Scholar
  15. [AlY]
    K. T. Alligood and J. A. Yorke, Hopf bifurcation: the appearance of virtual periods in cases of resonance, J. Differential Equations 64 (1986), 375–394.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [A]
    H. Amann, Fixed point equations and Nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620–709.zbMATHMathSciNetCrossRefGoogle Scholar
  17. [Ap]
    J. Appell (Ed), Recent trends in nonlinear analysis, Progress in Nonlinear Differential Equations and their Applications, vol. 40, Birkhäuser, 2000.Google Scholar
  18. [Ar]
    V. I. Arnold, Geometrical methods in the theory of differential equations, Springer—Verlag, New York, 1983.zbMATHGoogle Scholar
  19. [Ba]
    Z. Balanov, Equivariant Hopf theorem, Nonlinear Anal. 30 (1997), 3463–3474.zbMATHMathSciNetCrossRefGoogle Scholar
  20. [BK]
    Z. Balanov and W. Krawcewicz, Remarks on the equivariant degree theory, Topol. Methods Nonlinear Anal. 13 (1999), 91–103.MathSciNetzbMATHGoogle Scholar
  21. [BKK]
    Z. Balanov, W. Krawcewicz and A. Kushkuley, Brouwer degree, equivariant maps and tensor powers, Abstr. Appl. Anal. 3 (1998), 401–409.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [BKR]
    Z. Balanov, W. Krawcewicz and B. Rai, Taylor-Couette problem and related topics, Nonlinear Analysis: Real World Applications 4 (2003), 541–559.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [BKS1]
    Z. Balanov, W. Krawcewicz and H. Steinlein, Reduced SO(3) × S 1-equivariant degree with applications to symmetric bifurcation problems: the case of one free parameter, Nonlinear Anal. 47 (2001), 1617–1628.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [BKS2]
    _____, SO(3) × S 1-equivariant degree with applications to symmetric bifurcation problems: the case of one free parameter, Topol. Methods Nonlinear Anal. 20 (2002), 335–374.MathSciNetzbMATHGoogle Scholar
  25. [BaK]
    Z. Balanov and A. Kushkuley, On the problem of equivariant homotopic classification, Arch. Math. 65 (1995), 546–552.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [B1]
    T. Bartsch, A global index for bifurcation of fixed points, J. Reine Math. 391 (1988), 181–197.zbMATHMathSciNetGoogle Scholar
  27. [B2]
    _____, The role of the J-homomorphism in multiparameter bifurcation theory, Bull. Sci. Math. 112 (1988), 177–184.zbMATHMathSciNetGoogle Scholar
  28. [B3]
    _____, A simple proof of the degree formula for ℤ/p-equivariant maps, Comment. Math. Helv. 65 (1990), 85–95.zbMATHMathSciNetGoogle Scholar
  29. [B4]
    _____, The global structure of the zero set of a family of semilinear Fredholm maps, Nonlinear Anal. 17 (1991), 313–332.zbMATHMathSciNetCrossRefGoogle Scholar
  30. [B5]
    _____, The Conley index over a space, Math. Z 209 (1992), 167–177.zbMATHMathSciNetGoogle Scholar
  31. [B]
    _____, Topological Methods for variational problems with symmetries, Lect. Notes 1560 (1993), Springer-Verlag.Google Scholar
  32. [BC]
    T. Bartsch and M. Clapp, Bifurcation theory for symmetric potential operators and the equivariant cup-length, Math. Z. 204 (1990), 341–356.MathSciNetzbMATHGoogle Scholar
  33. [Be1]
    V. Benci, A geometrical index for the group S 1 and some applications to the study of periodic solutions of ordinary equations, Comm. Pure Appl. Math. 34 (1981), 393–432.zbMATHMathSciNetGoogle Scholar
  34. [Be2]
    _____, Introduction to Morse theory: a new approach, Topological Nonlinear Analysis (M. Matzeu and A. Vignoli, eds.), Progress in Nonlinear Differential Equations and Their Applications, vol. 15, Birkhaüser, 1994, pp. 37–177.Google Scholar
  35. [Ber]
    M. S. Berger, Nonlinearity and Functional Analysis, Academic Press, 1977.Google Scholar
  36. [BB]
    M.S. Berger and M. Berger, Perspectives in nonlinearity. An introduction to nonlinear analysis, Benjamin Inc., 1968.Google Scholar
  37. [Bo1]
    Y. G. Borisovich, Topology and nonlinear functional analysis, Russian Math. Surveys 37 (1979), 14–23.MathSciNetCrossRefGoogle Scholar
  38. [Bo2]
    _____, Topological characteristics of infinite dimensional mappings and the solvability of nonlinear boundary value problems, Proc. Steklov Inst. Math. 3 (1992), 43–50.MathSciNetGoogle Scholar
  39. [BZS]
    Y. G. Borisovich, V. G. Zvyagin and Y. I. Sapronov, Non linear Fredholm maps and the Leray Schauder theory, Russian Math. Surveys 32 (1977), 1–54.CrossRefzbMATHGoogle Scholar
  40. [Br]
    G. E. Bredon, Introduction to Compact Transformation Groups, Academic Press, New York, 1980.zbMATHGoogle Scholar
  41. [Bre]
    H. Brezis, Degree theory: old and new, Topological Nonlinear Analysis II (M. Matzeu and A. Vignoli, eds.), Progress in Nonlinear Differential Equations and Their Applications, vol. 27, Birkhaüser, 1996, pp. 87–108.Google Scholar
  42. [BD]
    T. Bröcker and T. tom Dieck, Representations of compact Lie groups, Grad. Texts in Math. 98 (1985).Google Scholar
  43. [Bro]
    F. E. Browder, Nonlinear operators in Banach spaces, Proc. Sympos. Pure Math 18 (1976).Google Scholar
  44. [Ch]
    K. C. Chang, Infinite Dimensional Morse Theory, Birkhaüser, 1993.Google Scholar
  45. [CI]
    P. Chossat and G. Iooss, The Couette-Taylor problem, Appl. Math. Sci. 102 (1994).Google Scholar
  46. [CL]
    P. Chossat and R. Lauterbach, Methods in equivariant bifurcations and dynamical systems, Adv. Ser. Nonlinear Dynam. 15 (2000).Google Scholar
  47. [CH]
    S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Grundlehren Math. Wiss., vol. 251, Springer-Verlag, 1982.Google Scholar
  48. [CMP]
    S. N. Chow and J. Mallet-Paret, The Fuller index and global Hopf bifurcation, J. Differential Equations 29 (1978), 66–85.MathSciNetCrossRefzbMATHGoogle Scholar
  49. [CV]
    M. A. Ciocci and A. Vanderbauwhede, Bifurcation of periodic orbits for symplectic mappings, J. Differential Equations Appl. 3 (1998), 485–500.MathSciNetzbMATHGoogle Scholar
  50. [CM]
    M. Clapp and W. Marzantowicz, Essential equivariant maps and Borsuk-Ulam theorems, J. London Math. Soc. 61 (2000), 950–960.MathSciNetCrossRefzbMATHGoogle Scholar
  51. [Co]
    C. Conley, Isolated invariant sets and the Morse index, vol. 38, CBMS regional Conf. Series in Math., 1978.Google Scholar
  52. [D]
    J. Damon, Applications of singularity theory to the solutions of nonlinear equations, Topological Nonlinear Analysis (M. Matzeu and A. Vignoli, eds.), Birkhäuser; Progr. Nonlinear Differential Equations Appl. 15 (1994), 178–302.Google Scholar
  53. [Da1]
    E. N. Dancer, On the existence of bifurcating solution in the presence of symmetries, Proc. Roy. Soc. Edinburg A 85 (1980), 321–336.zbMATHMathSciNetGoogle Scholar
  54. [Da2]
    _____, Symmetries, degree, homotopy indices and asymptotically homogeneous problems, Nonlinear Anal. 6 (1982), 667–686.zbMATHMathSciNetCrossRefGoogle Scholar
  55. [Da3]
    _____, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl. 91 (1983), 131–151.zbMATHMathSciNetCrossRefGoogle Scholar
  56. [Da4]
    _____, Perturbation of zeros in the presence of symmetries, J. Austral. Math. Soc. 36 (1984), 106–125.zbMATHMathSciNetCrossRefGoogle Scholar
  57. [Da]
    _____, A new degree for S 1-invariant gradient mappings and applications, Ann. Inst. H. Poincaré, Anal. Non. Linéaire 2 (1985), 329–370.zbMATHMathSciNetGoogle Scholar
  58. [Da5]
    _____, Fixed point index calculations and applications, Topological Nonlinear Analysis (M. Matzeu and A. Vignoli, eds.), Birkhaüser; Progr. Nonlinear Differential Equations Appl. 15 (1994), 303–340.Google Scholar
  59. [DT1]
    E. N. Dancer 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.MathSciNetzbMATHGoogle Scholar
  60. [DT2]
    _____, Equilibrium states in the degree theory of periodic orbits with a first integral, Proc. London Math. Soc. 61 (1991), 564–594.MathSciNetGoogle Scholar
  61. [De]
    K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, 1985.Google Scholar
  62. [DMM]
    M. Dellnitz, I. Melbourne and J. E. Marsden, Generic bifurcation of Hamiltonian vector fields with symmetry, Nonlinearity 15 (1992), 979–996.MathSciNetCrossRefGoogle Scholar
  63. [Di]
    W. Y. Ding, Generalizations of the Borsuk theorem, J. Math. Anal. Appl. 110 (1985), 553–567.zbMATHMathSciNetCrossRefGoogle Scholar
  64. [Do]
    A. Dold, Fixpunkttheorie, Kurseinheiten 1–7. Hagen: Fernuniversität-Gesamthochschule, Fachbereich Mathematik und Informatik, 1983.Google Scholar
  65. [DG]
    J. Dugundji and A. Granas, Fixed Point Theory I, PWN (Polish Scientific Publications), Warszawa, 1982.zbMATHGoogle Scholar
  66. [DGJW]
    G. Dylawerski, K. Gęba, J. Jodel and W. Marzantowicz, An S 1-equivariant degree and the Fuller index, Ann. Pol. Math. 52 (1991), 243–280.zbMATHGoogle Scholar
  67. [EGKW]
    L. Erbe, K. Gęba, W. Krawcewicz and J. Wu, S1-degree and global Hopf bifurcation theory of functional differential equations, J. Differential Equations 98 (1992), 277–298.MathSciNetCrossRefzbMATHGoogle Scholar
  68. [FR]
    E. R. Fadell 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.MathSciNetCrossRefzbMATHGoogle Scholar
  69. [FHR]
    E. R. Fadell, S. Husseini and P. Rabinowitz, Borsuk-Ulam theorems for arbitrary S 1-action and applications, Trans. Amer. Math. Soc. 274 (1982), 345–360.MathSciNetCrossRefzbMATHGoogle Scholar
  70. [FiM1]
    M. Field, Symmetry breaking for compact Lie groups, vol. 574, Mem. Amer. Math. Soc., 1996.Google Scholar
  71. [FiM2]
    _____, Symmetry breaking for equivariant maps, Algebraic groups and Lie groups (G. I. Lehrer, ed.), Cambridge Univ. Press, 1997, pp. 219–253.Google Scholar
  72. [FiM3]
    _____, Symmetry breaking for equivariant maps, Austral. Math. Soc. Lect. Ser. 9 (1997), 219–253.zbMATHMathSciNetGoogle Scholar
  73. [FR]
    M. Field and R. W. Richardson, Symmetry breaking and branching problems in equivariant bifurcation theory, I. Arch. Mat. Mech. Anal. 118 (1992), 297–348.MathSciNetCrossRefzbMATHGoogle Scholar
  74. [Fi]
    B. Fiedler, Global Bifurcation of periodic solutions with symmetry, Lecture Notes in Math., vol. 1309, Springer-Verlag, 1988.Google Scholar
  75. [FH]
    B. Fiedler and S. Heinze, Homotopy invariants of time reversible periodic orbits I. Theory, J. Differential Equations 126 (1996), 184–203.MathSciNetCrossRefzbMATHGoogle Scholar
  76. [FPM]
    P. M. Fitzpatrick, Homotopy, linearization and bifurcation, Nonlinear Anal. 12 (1988), 171–184.zbMATHMathSciNetCrossRefGoogle Scholar
  77. [FMP]
    P. M. Fitzpatrick, I. Massabó and J. Pejsachowicz, Complementing maps, continuation and global bifurcation, Bull. Amer. Math. Soc. 9 (1983), 79–81.MathSciNetzbMATHGoogle Scholar
  78. [FP1]
    P. M. Fitzpatrick 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
  79. [FP2]
    _____, Parity and generalized multiplicity, Trans. Amer. Math. Soc. 326 (1991), 281–305.MathSciNetCrossRefzbMATHGoogle Scholar
  80. [FP3]
    _____, The Leray-Schauder theory and fully non-linear elliptic boundary value problems, vol. 483, Mem. Amer. Math. Soc. Vol. 101, 1993.Google Scholar
  81. [FPR]
    P. M. Fitzpatrick, J. Pejsachowicz and P. J. Rabier, The degree of proper C 2 Fredholm maps: covariant theory, Topol. Methods Nonlinear Anal. 3 (1993), 325–367.MathSciNetGoogle Scholar
  82. [FPRe]
    P. M. Fitzpatrick, J. Pejsachowicz, and L. Recht, Spectral flow and bifurcation of critical points of strongly indefinite functionals I. General theory, J. Funct. Anal. 162 (1999), 52–95.MathSciNetCrossRefzbMATHGoogle Scholar
  83. [FG]
    I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications, Oxford Lecture Series in Mathematics and its Applications 2., Clarendon Press, 1995.Google Scholar
  84. [F]
    F. B. Fuller, An index of fixed point type for periodic orbits, Amer. J. Math. 89 (1967), 133–148.zbMATHMathSciNetGoogle Scholar
  85. [FMV]
    M. Furi, M. Martelli and A. Vignoli, On the Solvability of nonlinear operator equations in normed spaces, Ann. Mat. Pura Appl. 124 (1980), 321–343.MathSciNetCrossRefzbMATHGoogle Scholar
  86. [G]
    K. Gęba, Degree for gradient equivariant maps and equivariant Conley index, Topological Nonlinear Analysis II, Progress in Nonlinear Differential Equations and their Applications (M. Matzeu and A. Vignoli, eds.), vol. 27, Birkhaüser, 1997, pp. 247–272.Google Scholar
  87. [GKW]
    K. Gęba, W. Krawcewicz and J. Wu, An equivariant degree with applications to symmetric bifurcation problems, Part I, Construction of the degree, Bull. London Math. Soc. 69 (1994), 377–398.zbMATHGoogle Scholar
  88. [GW]
    K. Gęba and W. Marzantowicz, Global bifurcation of periodic orbits, Topol. Methods Nonlinear Anal. 1 (1993), 67–93.MathSciNetzbMATHGoogle Scholar
  89. [GMV]
    K. Gęba, I. Massabó and A. Vignoli, On the Euler characteristic of equivariant vector fields, Boll. Un. Mat. Ital. A 4 (1990), 243–251.MathSciNetzbMATHGoogle Scholar
  90. [GS]
    M. Golubitsky and D. G. Schaeffer, Appl. Math. Sci. 51 (1986), Springer-Verlag.Google Scholar
  91. [GSS]
    M. Golubitsky, D. G. Schaeffer and I. N. Stewart, Singularities and groups in bifurcation theory II, Springer-Verlag, 1988.Google Scholar
  92. [Gr]
    A. Granas, The Leray-Schauder index and the fixed point theory for arbitrary ANRs, Bull. Soc. Math. France 100 (1972), 209–228.zbMATHMathSciNetGoogle Scholar
  93. [HO]
    J. K. Hale and J. C. F. De Oliviera, Hopf bifurcation for functional equations, J. Math. Anal. Appl. 74 (1980), 41–59.MathSciNetCrossRefzbMATHGoogle Scholar
  94. [H1]
    H. Hauschild, Äquivariante homotopie I., Arch. Math. 29 (1977), 158–167.zbMATHMathSciNetCrossRefGoogle Scholar
  95. [H2]
    _____, Zerspaltung äquivarianter Homotopiemengen, Math. Ann. 230 (1977), 279–292.zbMATHMathSciNetCrossRefGoogle Scholar
  96. [HIR]
    D. Hyers, G. Isac and T. M. Rassias, Topics in nonlinear analysis and applications, World Scientific, 1997.Google Scholar
  97. [I1]
    J. Ize, Obstruction theory and multiparameter Hopf bifurcation, Trans. Amer. Math. Soc. 289 (1985), 757–792.zbMATHMathSciNetCrossRefGoogle Scholar
  98. [I2]
    _____, Necessary and sufficient conditions for multiparameter bifurcation, Rocky Mountain J. Math. 18 (1988), 305–337.zbMATHMathSciNetCrossRefGoogle Scholar
  99. [I3]
    _____, Topological bifurcation, Topological Nonlinear Analysis (M. Matzeu and A. Vignoli, eds.), Birkhaüser; Progr. Nonlinear Differential Equations Appl. 15 (1994), 341–463.Google Scholar
  100. [I4]
    _____, Two mechanical systems and equivariant degree theory, Progr. Nonlinear Differential Equations Appl. 40 (2000), Birkhaüser, 177–190.zbMATHMathSciNetGoogle Scholar
  101. [IMPV]
    J. Ize, I. Massabó, J. Pejsachowicz and A. Vignoli, Structure and dimension of global branches of solutions to multiparameter nonlinear equations, Trans. Amer. Math. Soc. 291 (1985), 383–435.MathSciNetCrossRefzbMATHGoogle Scholar
  102. [IMV1]
    J. Ize, I. Massabó and A. Vignoli, Global results on continuation and bifurcation for equivariant maps, NATO Adv. Sci. Inst. 173 (1986), 75–111.Google Scholar
  103. [IMV2]
    _____, Degree theory for equivariant maps I, Trans. Amer. Math. Soc. 315 (1989), 433–510.MathSciNetCrossRefzbMATHGoogle Scholar
  104. [IMV3]
    _____, Degree theory for equivariant maps: The general S 1-action., vol. 481, Mem. Amer. Math. Soc., 1992.Google Scholar
  105. [IV1]
    J. Ize and A. Vignoli, Equivariant degree for abelian action, Part I. Equivariant homotopy groups, Topol. Methods Nonlinear Anal. 2 (1993), 367–413.MathSciNetzbMATHGoogle Scholar
  106. [IV2]
    _____, Equivariant degree for abelian actions, Part II. Index computations, Topol. Methods Nonlinear Anal. 7 (1996), 369–430.MathSciNetzbMATHGoogle Scholar
  107. [IV3]
    _____, Equivariant degree for abelian actions, Part III. Orthogonal maps, Topol. Methods Nonlinear Anal. 13 (1999), 105–146.MathSciNetzbMATHGoogle Scholar
  108. [IV]
    _____, Equivariant degree theory, de Gruyter Ser. Nonlinear Anal. Appl. 8 (2003).Google Scholar
  109. [J]
    J. Jaworowski, Extensions of G-maps and Euclidean G-retracts, Math. Z. 146 (1976), 143–148.zbMATHMathSciNetCrossRefGoogle Scholar
  110. [K1]
    H. Kielhöfer, Multiple eigenvalue bifurcation for potential operators, J. Reine Angew. Math. 358 (1985), 104–124.zbMATHMathSciNetGoogle Scholar
  111. [K2]
    _____, A bifurcation theorem for potential operators, J. Funct. Anal. 77 (1988), 1–8.zbMATHMathSciNetCrossRefGoogle Scholar
  112. [K3]
    _____, Hopf bifurcation from a differentiable viewpoint, J. Diff. Eq. 97 (1992), 189–232.zbMATHCrossRefGoogle Scholar
  113. [KK]
    K. Komiya, Fixed point indices of equivariant maps and Moebius inversion, Invent. Math. 91 (1988), 129–135.zbMATHMathSciNetCrossRefGoogle Scholar
  114. [Ko]
    C. Kosniowski, Equivariant cohomology and stable cohomotopy, Math. Ann. 210 (1974), 83–104.zbMATHMathSciNetCrossRefGoogle Scholar
  115. [Kr]
    M. A. Krasnosel'skiĭ, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, MacMillan, New York, 1964.Google Scholar
  116. [KZ]
    M. A. Krasnosel'skiĭ and P. P. Zabreĭko, Geometrical Methods in Nonlinear Analysis, Grund Math. Wiss, vol. 263, Springer-Verlag, 1984.Google Scholar
  117. [KV]
    W. Krawcewicz and P. Vivi, Equivariant degree and normal bifurcations, B. N. Prasad birth centenary commemoration volume II, Indian J. Math. 42 (2000), 55–67.MathSciNetzbMATHGoogle Scholar
  118. [KVW]
    W. Krawcewicz, P. Vivi and J. Wu, Hopf bifurcations of functional differential equations with dihedral symmetries, J. Differential Equations 146 (1998), 157–184.MathSciNetCrossRefzbMATHGoogle Scholar
  119. [KX]
    W. Krawcewicz and H. Xia, Analytic definition of an equivariant degree, Russian Math. 40 (1996), 34–49.MathSciNetzbMATHGoogle Scholar
  120. [KW]
    W. Krawcewicz and J. Wu, Theory of Degrees with Applications to Bifurcations and Differential Equations, Canad. Math. Soc. Ser. Monogr. Adv. Texts, John Wiley & Sons, Inc., New York, 1997.Google Scholar
  121. [KS]
    W. Kryszewski and A. Szulkin, An infinite dimensional Morse theory with applications, Trans. Amer. Math. Soc. 349 (1997), 3181–3234.MathSciNetCrossRefzbMATHGoogle Scholar
  122. [KB]
    A. Kushkuley and Z. Balanov, Geometric Methods in Degree Theory for Equivariant Maps, Lecture Notes in Math., vol. 1632, Springer—Verlag, Berlin, 1996.Google Scholar
  123. [LLH]
    A. Lari-Lavassani, W. F. Langford and K. Huseyin, Symmetry breaking bifurcations on multidimensional fixed point subspaces, Dynam. Stability Systems 9 (1994), 345–373.MathSciNetzbMATHGoogle Scholar
  124. [LMSM]
    L. G. Lewis, J. P. May, M. Steinberger and J. E. McClure, Equivariant stable homotopy theory, vol. 1213, Springer Lect. Notes in Math., 1986.Google Scholar
  125. [MR]
    A. Maciejewski and S. Rybicki, Global bifurcations of periodic solutions of Hénon-Heiles system via degree for S 1-equivariant orthogonal maps, Rev. Math. Phys. 10 (1998), 1125–1146.MathSciNetCrossRefzbMATHGoogle Scholar
  126. [MPN]
    J. Mallet-Paret and R. Nussbaum, Boundary layer phenomena for differential-delay equations with state dependent time delays, Arch. Rational Mech. Anal. 120 (1992), 99–146.MathSciNetzbMATHGoogle Scholar
  127. [MPY]
    J. Mallet-Paret and J. A. Yorke, Snakes: oriented families of periodic orbits, their sources, sinks and continuation, J. Diferential Equations 43 (1982), 419–450.MathSciNetCrossRefzbMATHGoogle Scholar
  128. [MM]
    J. E. Marsden and M. McCracken, The Hopf Bifurcation and its Applications, Springer-Verlag, 1976.Google Scholar
  129. [M1]
    W. Marzantowicz, On the nonlinear elliptic equations with symmetries, J. Math. Anal. Appl. 81 (1981), 156–181.zbMATHMathSciNetCrossRefGoogle Scholar
  130. [M2]
    _____, Borsuk-Ulam theorem for any compact Lie group, J. London Math. Soc. 49 (1994), 195–208.zbMATHMathSciNetGoogle Scholar
  131. [M]
    T. Matsuoka, Equivariant function spaces and bifurcation points, J. Math. Soc. Japan 35 (1983), 43–52.zbMATHMathSciNetCrossRefGoogle Scholar
  132. [MV1]
    M. Matzeu and A. Vignoli (Eds), Topological Nonlinear Analysis, Progr. Nonlinear Differential Equations Appl., vol. 15, Birkhaüser, 1994.Google Scholar
  133. [MV2]
    _____, Topological Nonlinear Analysis II, Progr. Nonlinear Differential Equations Appl., vol. 27, Birkhaüser, 1996.Google Scholar
  134. [MW]
    J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, New York, 1989.zbMATHGoogle Scholar
  135. [Maw]
    J. Mawhin, Leray-Schauder degree: a half century of extensions and applications, Topol. Methods Nonlinear Anal. 14 (1999), 195–228.zbMATHMathSciNetGoogle Scholar
  136. [MRS]
    J. Montaldi, M. Roberts and I. Stewart, Existence of nonlinear normal modes of symmetric Hamiltonian systems, Nonlinearity 3 (1990), 695–730.MathSciNetCrossRefzbMATHGoogle Scholar
  137. [N]
    V. Namboodiri, Equivariant vector fields on spheres, Trans. Amer. Math. Soc. 278 (1983), 431–460.zbMATHMathSciNetCrossRefGoogle Scholar
  138. [Ni1]
    L. Nirenberg, Topics in nonlinear functional analysis, Lect. Notes Courant Institute, New York Univ., 1974.Google Scholar
  139. [Ni2]
    _____, Variational and topological methods in nonlinear problems, Bull. Amer. Math. Soc. 4 (1981), 267–302.zbMATHMathSciNetCrossRefGoogle Scholar
  140. [Nu]
    R. D. Nussbaum, Differential-delay equations with two time lags, vol. 205, Memoirs Amer. Math. Soc., 1978.Google Scholar
  141. [Pa]
    A. Parusinski, Gradient homotopies of gradient vector fields, Studia Mathematica 46 (1990), 73–80.MathSciNetGoogle Scholar
  142. [P]
    J. Pejsachowicz, K-theoretic methods in bifurcation theory, Contem. Math. 72 (1988), 193–205.zbMATHMathSciNetGoogle Scholar
  143. [PR]
    J. Pejsachowicz and P. Rabier, Degree theory for C 1 Fredholm mappings of index 0, J. Anal. Math. 76 (1998), 289–319.MathSciNetzbMATHGoogle Scholar
  144. [Pe]
    G. Peschke, Degree of certain equivariant maps into a representation sphere, Topology Appl. 59 (1994), 137–156.zbMATHMathSciNetCrossRefGoogle Scholar
  145. [Pe1]
    W. V. Petryshyn, Approximation-Solvability of Nonlinear Functional and Differential Equation, M. Dekker, 1992.Google Scholar
  146. [Pe2]
    _____, Generalized Topological Degree and Semilinear Equations, Cambridge Tracts in Math., vol. 117, Cambridge Univ. Press..Google Scholar
  147. [PU]
    C. Prieto and H. Ulrich, Equivariant fixed point index and fixed point transfer in nonzero dimension, Trans. Amer. Math. Soc. 328 (1991), 731–745.MathSciNetCrossRefzbMATHGoogle Scholar
  148. [Rab1]
    P. J. Rabier, Generalized Jordan chains and two bifurcation theorems of Krasnosel'skiĭ, Nonlinear Anal. 13 (1989), 903–934.zbMATHMathSciNetCrossRefGoogle Scholar
  149. [Rab2]
    _____, Topological degree and the theorem of Borsuk for general covariant mappings with applications, Nonlinear Anal. 16 (1991), 393–420.MathSciNetCrossRefGoogle Scholar
  150. [Ra1]
    P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487–513.zbMATHMathSciNetCrossRefGoogle Scholar
  151. [Ra2]
    _____, Theorie du Degree Topologique et Applications (1975), (Lectures Notes).Google Scholar
  152. [Ra3]
    _____, A bifurcation theorem for potential operators, J. Funct. Anal. 25 (1977), 412–424.zbMATHMathSciNetCrossRefGoogle Scholar
  153. [Ra4]
    _____, Critical point theory, Topological Nonlinear Analysis (M. Matzeu and A. Vignoli, eds.), Birkhaüser; Progr. Nonlinear Differential Equations Appl. 15 (1994), 464–513.Google Scholar
  154. [Ro]
    E. H. Rothe, Introduction to Various Aspects of Degree Theory in Banach Spaces, Math. Surveys Monogr., vol. 23, Amer. Math. Soc., Providence, 1986.Google Scholar
  155. [Ru]
    R. L. Rubinstein, On the equivariant homotopy of spheres, Dissertationes Math. 134 (1976), 1–48.Google Scholar
  156. [R1]
    S. Rybicki, S 1-degree for orthogonal maps and its applications to bifurcation theory, Nonlinear Anal. 23 (1994), 83–102.zbMATHMathSciNetCrossRefGoogle Scholar
  157. [R2]
    _____, Applications of degree for S 1-equivariant gradient maps to variational problems with S 1-symmetries, Topol. Methods Nonlinear Anal. 9 (1997), 383–417.zbMATHMathSciNetGoogle Scholar
  158. [R3]
    _____, On periodic solutions of autonomous Hamiltonian systems via degree for S 1-equivariant gradient maps, Nonlinear Anal. 34 (1998), 537–569.zbMATHMathSciNetCrossRefGoogle Scholar
  159. [R4]
    _____, On bifurcation from infinity for S 1-equivariant potential operators, Nonlinear Anal. 31 (1998), 343–361.zbMATHMathSciNetCrossRefGoogle Scholar
  160. [R5]
    _____, Degree for S 1-equivariant strongly indefinite functionals, Nonlinear Anal. 43 (2001), 1001–1017.zbMATHMathSciNetCrossRefGoogle Scholar
  161. [S1]
    D. H. Sattinger, Group Theoretic Methods in Bifurcation Theory, Lecture Notes in Math., vol. 762, Springer-Verlag, 1979.Google Scholar
  162. [S2]
    _____, Branching in the Presence of Symmetry, Wiley, 1983.Google Scholar
  163. [ScWa]
    K. Schmitt and Z. Q. Wang, On bifurcation from infinity for potential operators, Differential Integral Equations 4 (1991), 933–943.MathSciNetzbMATHGoogle Scholar
  164. [SW]
    J. Smoller and A. G. Wasserman, Bifurcation and symmetry-breaking, Inv. Math. 100 (1990), 63–95.MathSciNetCrossRefzbMATHGoogle Scholar
  165. [S]
    H. Steinlein, Borsuk's antipodal theorem and its generalizations and applications: a survey, Topol. Anal. Non Linéaire (1985), Press de l'Univ. de Montreal, 166–235.Google Scholar
  166. [St]
    C. A. Stuart, Bifurcation from the essential spectrum, Topological Nonlinear Analysis II (M. Matzeu and A. Vignoli, eds.), Birkhaüser; Progr. Nonlinear Differential Equations Appl. 27 (1996), 397–444.Google Scholar
  167. [Sz]
    A. Szulkin, Index theories for indefinite functionals and applications, Pitman Res. Notes Math. Ser. 365 (1997), 89–121.zbMATHMathSciNetGoogle Scholar
  168. [SZ]
    A. Szulkin and W. Zou, Infinite dimensional cohomology groups and periodic of assymptotically linear Hamiltonian systems, J. Differential Equations 174 (2001), 369–391.MathSciNetCrossRefzbMATHGoogle Scholar
  169. [tD1]
    T. tom Dieck, Transformation groups and representation theory, Lecture Notes in Math., vol. 766, Springer-Verlag, 1979.Google Scholar
  170. [tD2]
    _____, Transformation Groups, de Gruyter, Berlin, 1987.Google Scholar
  171. [T]
    J. Tornehave, Equivariant maps of spheres with conjugate orthogonal actions, Algebraic Topology, Proc. Conf. London Ont. 1981,, vol. 2, Canadian Math. Soc. Conf. Proc., 1982, pp. 275–301.Google Scholar
  172. [U]
    H. Ulrich, Fixed point theory of parametrized equivariant maps, Lecture Notes in Math., vol. 1343, Springer-Verlag, 1988.Google Scholar
  173. [V]
    A. Vanderbauwhede, Local bifurcation and symmetry, vol. 75, Pitman Research Notes in Math., 1982.Google Scholar
  174. [W]
    Z. Q. Wang, Symmetries and the calculation of degree, Chinesw Ann. Math. Ser. B 10 (1989), 520–536.zbMATHGoogle Scholar
  175. [W1]
    G. W. Whitehead, On the homotopy groups of spheres and rotation groups, Ann. of Math. 43 (1942), 634–640.zbMATHMathSciNetCrossRefGoogle Scholar
  176. [W2]
    _____, Elements of homotopy theory, Grad. Texts in Math., vol. 61, Springer-Verlag, 1978.Google Scholar
  177. [Z1]
    E. Zeidler, Nonlinear Functional Analysis and its Applications, vol. IV, Springer Verlag, 1985.Google Scholar
  178. [Z2]
    _____, Nonlinear functional analysis and its applications, I. Fixed-Point Theorems, Springer-Verlag, 1993.Google Scholar
  179. [Z]
    P. P. Zabreĭko, Rotation of vector fields: definition, basic properties and calculations, Topological Nonlinear Analysis II (M. Matzeu and A. Vignoli, eds.), Birkhaüser; Progr. Nonlinear Differential Equations Appl. 27 (1996), 449–601.Google Scholar
  180. [Zv]
    V. G. Zvyagin, On a degree theory for equivariant Φ0 C I BH-mappings, Dokl. Akad. Nauk 364 (1999), 155–157. (Russian)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Jorge Ize

There are no affiliations available

Personalised recommendations