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Bifurcation of periodic solutions from a ring configuration of discrete nonlinear oscillators
Equivariant Conley index versus degree for equivariant gradient maps
1. | Faculty of Mathematics and Computer Science |
2. | Nicolaus Copernicus University |
3. | ul. Chopina 12/18, PL-87-100 Toru? |
References:
[1] |
A. Ambrosetti, Branching points for a class of variational operators,, J. Anal. Math., 76 (1998), 321.
doi: 10.1007/BF02786940. |
[2] |
Z. Balanov, W. Krawcewicz, S. Rybicki and H. Steinlein, A short treatise on the equivariant degree theory and its applications,, J. Fixed Point Theory App., 8 (2010), 1.
doi: 10.1007/s11784-010-0033-9. |
[3] |
Z. Balanov, W. Krawcewicz and H. Ruan, Periodic solutions to $O(2)$-symmetric variational problems: $O(2)\times S^1$-equivariant gradient degree approach,, Israel Math. Conf. Proc., 514 (2008), 45.
doi: 10.1090/conm/514/10099. |
[4] |
A. Banyaga and D. Hurtubise, "Lectures on Morse Homology,", Kluwer Academic Publishers, (2004).
|
[5] |
P. Bartłomiejczyk. K. Gęba and M. Izydorek, Otopy classes of equivariant maps,, J. Fixed P. Th. and Appl., 7 (2010), 145.
doi: 10.1007/s11784-010-0013-0. |
[6] |
T. Bartsch, "Topological Methods for Variational Problems with Symmetries,", Lect. Notes in Math., 1560 (1993).
|
[7] |
R. Böhme, Die lösung der versweigungsgleichungen für nichtlineare eigenwert-probleme,, Math. Z., 127 (1972), 105.
|
[8] |
G. Bredon, "Introduction to Compact Transformation Groups,", Academic Press, (1972).
|
[9] |
S. N. Chow and R. Lauterbach, A bifurcation theorem for critical points of variational problems,, Nonl. Anal. TMA, 12 (1988), 51.
doi: 10.1016/0362-546X(88)90012-0. |
[10] |
Ch. Conley, "Isolated Invariant Sets and the Morse Index,", CBMS Regional Conference Series in Mathematics, 38 (1978).
|
[11] |
E. N. Dancer, A new degree for $S^1$-invariant mappings and applications,, Ann. Inst. H.Poincaré, 2 (1985), 329.
|
[12] |
E. N. Dancer, K. Gęba and S. Rybicki, Classification of homotopy classes of equivariant gradient maps,, Fund. Math., 185 (2005), 1.
doi: 10.4064/fm185-1-1. |
[13] |
T. tom Dieck, "Transformation Groups and Representation Theory,", Springer-Verlag, (1979).
|
[14] |
T. tom Dieck, "Transformation Groups,", Walter de Gruyter, (1987).
doi: 10.1515/9783110858372.312. |
[15] |
J. J. Duistermaat and J. A. C. Kolk, "Lie Groups,", Springer-Verlag, (2000).
doi: 10.1007/978-3-642-56936-4. |
[16] |
G. Fang, Morse indices of degenerate critical orbits and applications - perturbation methods in equivariant cases,, Nonl. Anal. TMA, 36 (1999), 101.
doi: 10.1016/S0362-546X(98)00017-0. |
[17] |
A. Floer, A refinement of the Conley index and an application to the stability of hyperbolic invariant sets,, Erg. Th. and Dynam. Sys., 7 (1987), 93.
doi: 10.1017/S0143385700003825. |
[18] |
K. Gęba, Degree for gradient equivariant maps and equivariant Conley index,, Topological Nonlinear Analysis, 27 (1997), 247.
|
[19] |
K. Gęba, M. Izydorek and A. Pruszko, The Conley index in Hilbert spaces and its applications,, Studia Math., 134 (1999), 217.
|
[20] |
A. Gołębiewska and S. Rybicki, Global bifurcations of critical orbits of $G$-invariant strongly indefinite functionals,, Nonl. Anal TMA, 74 (2011), 1823.
doi: 10.1016/j.na.2010.10.055. |
[21] |
J. Ize, "Bifurcation Theory for Fredholm Operators,", Mem. AMS, 174 (1976).
|
[22] |
J. Ize, Topological bifurcation,, Topological Nonlinear Analysis, 15 (1995), 341.
|
[23] |
M. Izydorek, A Cohomological Conley index in Hilbert spaces and applications to strongly indefinite problems,, J. Diff. Equat., 170 (2001), 22.
doi: 10.1006/jdeq.2000.3818. |
[24] |
M. Izydorek, Equivariant Conley index in Hilbert spaces and applications to strongly indefinite problems,, Nonl. Anal. TMA, 51 (2002), 33.
doi: 10.1016/S0362-546X(01)00811-2. |
[25] |
G. López Garza and S. Rybicki, Equivariant bifurcation index,, Nonl. Anal. TMA, 73 (2010), 2779.
doi: 10.1016/j.na.2010.06.001. |
[26] |
A. Marino, La biforcazione nel caso variazionale,, Conf. Sem. Mat. Univ. Bari, 132 (1973).
|
[27] |
K. H. Mayer, G-invariante Morse-funktionen,, Man. Math., 63 (1989), 99.
doi: 10.1007/BF01173705. |
[28] |
K. Mischaikow and M. Mrozek, Conley index,, Handbook of Dynamical Systems, 2 (2002), 393.
doi: 10.1016/S1874-575X(02)80030-3. |
[29] |
L. Nirenberg, "Topics in Nonlinear Functional Analysis,", Courant Institute of Mathematical Sciences, (1974).
|
[30] |
P. H. Rabinowitz, Nonlinear Sturm-Liouville problems for second order ordinary differential equations,, Comm. Pure Appl. Math., 23 (1970), 939.
|
[31] |
P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems,, J. Func. Anal., 7 (1971), 487.
|
[32] |
P. H. Rabinowitz, A global theorem for nonlinear eigenvalue problems and applications,, Contributions to Nonlinear Functional Analysis, (1971), 11.
|
[33] |
W. Radzki and S. Rybicki, Degenerate bifurcation points of periodic solutions of autonomous Hamiltonian systems,, J. Diff. Equat., 202 (2004), 284.
doi: 10.1016/j.jde.2004.03.037. |
[34] |
H. Ruan and S. Rybicki, Applications of equivariant degree for gradient maps to symmetric Newtonian systems,, Nonl. Anal. TMA, 68 (2008), 1479.
doi: 10.1016/j.na.2006.12.039. |
[35] |
K. Rybakowski, On the homotopy index for infinite dimensional semiflows,, Trans. Amer. Math. Soc., 269 (1982), 351.
doi: 10.2307/1998452. |
[36] |
S. Rybicki, A degree for $S^1$-equivariant orthogonal maps and its applications to bifurcation theory,, Nonl. Anal. TMA, 23 (1994), 83.
doi: 10.1016/0362-546X(94)90253-4. |
[37] |
S. Rybicki, Applications of degree for $S^1$-equivariant gradient maps to variational nonlinear problems with $S^1$-symmetries,, Topol. Meth. Nonl. Anal., 9 (1997), 383.
|
[38] |
S. Rybicki, Degree for equivariant gradient maps,, Milan J. Math., 73 (2005), 103.
doi: 10.1007/s00032-005-0040-2. |
[39] |
S. Rybicki, Global bifurcations of critical orbits via equivariant Conley index,, Adv. Nonl. Stud., 11 (2011), 929.
|
[40] |
D. Salamon, Connected simple systems and the Conley index of isolated invariant sets,, TAMS, 291 (1985), 1.
doi: 10.2307/1999893. |
[41] |
D. Salamon, Morse theory, the Conley index and Floer homology,, Bull. London Math. Soc., 22 (1990), 113.
doi: 10.1112/blms/22.2.113. |
[42] |
J. Smoller and A. Wasserman, Bifurcation and symmetry-breaking,, Invent. Math., 100 (1990), 63.
doi: 10.1007/BF01231181. |
[43] |
J. Smoller, "Shock Waves and Reaction-diffusion Equations,", Fundamental Principles of Mathematical Science, 258 (1983).
|
[44] |
F. Takens, Some remarks on the Böhme-Berger bifurcation theorem,, Math. Z., 125 (1972), 359.
|
[45] |
G. W. Whitehead, "Recent Advances in Homotopy Theory,", CMBS Regional Conf. Ser. in Math., 5 (1970).
|
show all references
References:
[1] |
A. Ambrosetti, Branching points for a class of variational operators,, J. Anal. Math., 76 (1998), 321.
doi: 10.1007/BF02786940. |
[2] |
Z. Balanov, W. Krawcewicz, S. Rybicki and H. Steinlein, A short treatise on the equivariant degree theory and its applications,, J. Fixed Point Theory App., 8 (2010), 1.
doi: 10.1007/s11784-010-0033-9. |
[3] |
Z. Balanov, W. Krawcewicz and H. Ruan, Periodic solutions to $O(2)$-symmetric variational problems: $O(2)\times S^1$-equivariant gradient degree approach,, Israel Math. Conf. Proc., 514 (2008), 45.
doi: 10.1090/conm/514/10099. |
[4] |
A. Banyaga and D. Hurtubise, "Lectures on Morse Homology,", Kluwer Academic Publishers, (2004).
|
[5] |
P. Bartłomiejczyk. K. Gęba and M. Izydorek, Otopy classes of equivariant maps,, J. Fixed P. Th. and Appl., 7 (2010), 145.
doi: 10.1007/s11784-010-0013-0. |
[6] |
T. Bartsch, "Topological Methods for Variational Problems with Symmetries,", Lect. Notes in Math., 1560 (1993).
|
[7] |
R. Böhme, Die lösung der versweigungsgleichungen für nichtlineare eigenwert-probleme,, Math. Z., 127 (1972), 105.
|
[8] |
G. Bredon, "Introduction to Compact Transformation Groups,", Academic Press, (1972).
|
[9] |
S. N. Chow and R. Lauterbach, A bifurcation theorem for critical points of variational problems,, Nonl. Anal. TMA, 12 (1988), 51.
doi: 10.1016/0362-546X(88)90012-0. |
[10] |
Ch. Conley, "Isolated Invariant Sets and the Morse Index,", CBMS Regional Conference Series in Mathematics, 38 (1978).
|
[11] |
E. N. Dancer, A new degree for $S^1$-invariant mappings and applications,, Ann. Inst. H.Poincaré, 2 (1985), 329.
|
[12] |
E. N. Dancer, K. Gęba and S. Rybicki, Classification of homotopy classes of equivariant gradient maps,, Fund. Math., 185 (2005), 1.
doi: 10.4064/fm185-1-1. |
[13] |
T. tom Dieck, "Transformation Groups and Representation Theory,", Springer-Verlag, (1979).
|
[14] |
T. tom Dieck, "Transformation Groups,", Walter de Gruyter, (1987).
doi: 10.1515/9783110858372.312. |
[15] |
J. J. Duistermaat and J. A. C. Kolk, "Lie Groups,", Springer-Verlag, (2000).
doi: 10.1007/978-3-642-56936-4. |
[16] |
G. Fang, Morse indices of degenerate critical orbits and applications - perturbation methods in equivariant cases,, Nonl. Anal. TMA, 36 (1999), 101.
doi: 10.1016/S0362-546X(98)00017-0. |
[17] |
A. Floer, A refinement of the Conley index and an application to the stability of hyperbolic invariant sets,, Erg. Th. and Dynam. Sys., 7 (1987), 93.
doi: 10.1017/S0143385700003825. |
[18] |
K. Gęba, Degree for gradient equivariant maps and equivariant Conley index,, Topological Nonlinear Analysis, 27 (1997), 247.
|
[19] |
K. Gęba, M. Izydorek and A. Pruszko, The Conley index in Hilbert spaces and its applications,, Studia Math., 134 (1999), 217.
|
[20] |
A. Gołębiewska and S. Rybicki, Global bifurcations of critical orbits of $G$-invariant strongly indefinite functionals,, Nonl. Anal TMA, 74 (2011), 1823.
doi: 10.1016/j.na.2010.10.055. |
[21] |
J. Ize, "Bifurcation Theory for Fredholm Operators,", Mem. AMS, 174 (1976).
|
[22] |
J. Ize, Topological bifurcation,, Topological Nonlinear Analysis, 15 (1995), 341.
|
[23] |
M. Izydorek, A Cohomological Conley index in Hilbert spaces and applications to strongly indefinite problems,, J. Diff. Equat., 170 (2001), 22.
doi: 10.1006/jdeq.2000.3818. |
[24] |
M. Izydorek, Equivariant Conley index in Hilbert spaces and applications to strongly indefinite problems,, Nonl. Anal. TMA, 51 (2002), 33.
doi: 10.1016/S0362-546X(01)00811-2. |
[25] |
G. López Garza and S. Rybicki, Equivariant bifurcation index,, Nonl. Anal. TMA, 73 (2010), 2779.
doi: 10.1016/j.na.2010.06.001. |
[26] |
A. Marino, La biforcazione nel caso variazionale,, Conf. Sem. Mat. Univ. Bari, 132 (1973).
|
[27] |
K. H. Mayer, G-invariante Morse-funktionen,, Man. Math., 63 (1989), 99.
doi: 10.1007/BF01173705. |
[28] |
K. Mischaikow and M. Mrozek, Conley index,, Handbook of Dynamical Systems, 2 (2002), 393.
doi: 10.1016/S1874-575X(02)80030-3. |
[29] |
L. Nirenberg, "Topics in Nonlinear Functional Analysis,", Courant Institute of Mathematical Sciences, (1974).
|
[30] |
P. H. Rabinowitz, Nonlinear Sturm-Liouville problems for second order ordinary differential equations,, Comm. Pure Appl. Math., 23 (1970), 939.
|
[31] |
P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems,, J. Func. Anal., 7 (1971), 487.
|
[32] |
P. H. Rabinowitz, A global theorem for nonlinear eigenvalue problems and applications,, Contributions to Nonlinear Functional Analysis, (1971), 11.
|
[33] |
W. Radzki and S. Rybicki, Degenerate bifurcation points of periodic solutions of autonomous Hamiltonian systems,, J. Diff. Equat., 202 (2004), 284.
doi: 10.1016/j.jde.2004.03.037. |
[34] |
H. Ruan and S. Rybicki, Applications of equivariant degree for gradient maps to symmetric Newtonian systems,, Nonl. Anal. TMA, 68 (2008), 1479.
doi: 10.1016/j.na.2006.12.039. |
[35] |
K. Rybakowski, On the homotopy index for infinite dimensional semiflows,, Trans. Amer. Math. Soc., 269 (1982), 351.
doi: 10.2307/1998452. |
[36] |
S. Rybicki, A degree for $S^1$-equivariant orthogonal maps and its applications to bifurcation theory,, Nonl. Anal. TMA, 23 (1994), 83.
doi: 10.1016/0362-546X(94)90253-4. |
[37] |
S. Rybicki, Applications of degree for $S^1$-equivariant gradient maps to variational nonlinear problems with $S^1$-symmetries,, Topol. Meth. Nonl. Anal., 9 (1997), 383.
|
[38] |
S. Rybicki, Degree for equivariant gradient maps,, Milan J. Math., 73 (2005), 103.
doi: 10.1007/s00032-005-0040-2. |
[39] |
S. Rybicki, Global bifurcations of critical orbits via equivariant Conley index,, Adv. Nonl. Stud., 11 (2011), 929.
|
[40] |
D. Salamon, Connected simple systems and the Conley index of isolated invariant sets,, TAMS, 291 (1985), 1.
doi: 10.2307/1999893. |
[41] |
D. Salamon, Morse theory, the Conley index and Floer homology,, Bull. London Math. Soc., 22 (1990), 113.
doi: 10.1112/blms/22.2.113. |
[42] |
J. Smoller and A. Wasserman, Bifurcation and symmetry-breaking,, Invent. Math., 100 (1990), 63.
doi: 10.1007/BF01231181. |
[43] |
J. Smoller, "Shock Waves and Reaction-diffusion Equations,", Fundamental Principles of Mathematical Science, 258 (1983).
|
[44] |
F. Takens, Some remarks on the Böhme-Berger bifurcation theorem,, Math. Z., 125 (1972), 359.
|
[45] |
G. W. Whitehead, "Recent Advances in Homotopy Theory,", CMBS Regional Conf. Ser. in Math., 5 (1970).
|
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