August  2013, 6(4): 985-997. doi: 10.3934/dcdss.2013.6.985

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?

Received  October 2011 Revised  April 2012 Published  December 2012

In this article we study the relationship between the degree forinvariant strongly indefinite functionals and the equivariantConley index. We prove that, under certain assumptions, achange of the equivariant Conley indices is equivalent to thechange of the degrees for equivariant gradient maps. Moreover, weformulate easy to verify sufficient conditions for theexistence of a global bifurcation of critical orbits of invariantstrongly indefinite functionals.
Citation: Anna Go??biewska, S?awomir Rybicki. Equivariant Conley index versus degree for equivariant gradient maps. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 985-997. doi: 10.3934/dcdss.2013.6.985
References:
[1]

J. Anal. Math., 76 (1998), 321-335. doi: 10.1007/BF02786940.

[2]

J. Fixed Point Theory App., 8 (2010), 1-74. doi: 10.1007/s11784-010-0033-9.

[3]

Israel Math. Conf. Proc., Conf. Nonlinear Analysis and Optimization, Haifa, Israel, (2008), AMS Contemporary Mathematics, 514 (2010), 45-84. doi: 10.1090/conm/514/10099.

[4]

Kluwer Academic Publishers, 2004.

[5]

J. Fixed P. Th. and Appl., 7 (2010), 145-160. doi: 10.1007/s11784-010-0013-0.

[6]

Lect. Notes in Math., 1560, Springer-Verlag, Berlin, 1993.

[7]

Math. Z., 127 (1972), 105-126.

[8]

Academic Press, Inc. LTD, 1972.

[9]

Nonl. Anal. TMA, 12 (1988), 51-61. doi: 10.1016/0362-546X(88)90012-0.

[10]

CBMS Regional Conference Series in Mathematics, 38, AMS, Providence, R. I., 1978.

[11]

Ann. Inst. H.Poincaré, Ana. Non Lin., 2, (1985), 329-370.

[12]

Fund. Math., 185 (2005), 1-18. doi: 10.4064/fm185-1-1.

[13]

Springer-Verlag, Berlin, 1979.

[14]

Walter de Gruyter, Berlin-New York, 1987. doi: 10.1515/9783110858372.312.

[15]

Springer-Verlag, Berlin Heidelberg New York, 2000. doi: 10.1007/978-3-642-56936-4.

[16]

Nonl. Anal. TMA, 36 (1999), 101-118. doi: 10.1016/S0362-546X(98)00017-0.

[17]

Erg. Th. and Dynam. Sys., 7 (1987), 93-103. doi: 10.1017/S0143385700003825.

[18]

Topological Nonlinear Analysis, Degree, Singularity and Variations, PNDLE 27, Birkhäuser, (1997), 247-272.

[19]

Studia Math., 134, (1999), 217-233.

[20]

Nonl. Anal TMA, 74 (2011), 1823-1834. doi: 10.1016/j.na.2010.10.055.

[21]

Mem. AMS, 174, 1976.

[22]

Topological Nonlinear Analysis, Degree, Singularity and Variations, PNDE, 15, Birkhäuser, (1995), 341-463.

[23]

J. Diff. Equat., 170 (2001), 22-50. doi: 10.1006/jdeq.2000.3818.

[24]

Nonl. Anal. TMA, 51 (2002), 33-66. doi: 10.1016/S0362-546X(01)00811-2.

[25]

Nonl. Anal. TMA, 73 (2010), 2779-2791. doi: 10.1016/j.na.2010.06.001.

[26]

Conf. Sem. Mat. Univ. Bari, 132 (1973).

[27]

Man. Math., 63 (1989), 99-114. doi: 10.1007/BF01173705.

[28]

Handbook of Dynamical Systems, 2, North-Holland, Amsterdam, (2002), 393-460. doi: 10.1016/S1874-575X(02)80030-3.

[29]

Courant Institute of Mathematical Sciences, New York University, 1974.

[30]

Comm. Pure Appl. Math., 23 (1970), 939-961.

[31]

J. Func. Anal., 7 (1971), 487-513.

[32]

Contributions to Nonlinear Functional Analysis, E Academic Press, New York, (1971), 11-36.

[33]

J. Diff. Equat., 202 (2004), 284-305. doi: 10.1016/j.jde.2004.03.037.

[34]

Nonl. Anal. TMA, 68 (2008), 1479-1516. doi: 10.1016/j.na.2006.12.039.

[35]

Trans. Amer. Math. Soc., 269 (1982), 351-382. doi: 10.2307/1998452.

[36]

Nonl. Anal. TMA, 23 (1994), 83-102. doi: 10.1016/0362-546X(94)90253-4.

[37]

Topol. Meth. Nonl. Anal., 9 (1997), 383-417.

[38]

Milan J. Math., 73 (2005), 103-144. doi: 10.1007/s00032-005-0040-2.

[39]

Adv. Nonl. Stud., 11 (2011), 929-940.

[40]

TAMS, 291 (1985), 1-41. doi: 10.2307/1999893.

[41]

Bull. London Math. Soc., 22 (1990), 113-140. doi: 10.1112/blms/22.2.113.

[42]

Invent. Math., 100 (1990), 63-95. doi: 10.1007/BF01231181.

[43]

Fundamental Principles of Mathematical Science, 258, Springer-Verlag, New York-Berlin, 1983.

[44]

Math. Z., 125 (1972), 359-364.

[45]

CMBS Regional Conf. Ser. in Math., 5, AMS, Providence, R.I., 1970.

show all references

References:
[1]

J. Anal. Math., 76 (1998), 321-335. doi: 10.1007/BF02786940.

[2]

J. Fixed Point Theory App., 8 (2010), 1-74. doi: 10.1007/s11784-010-0033-9.

[3]

Israel Math. Conf. Proc., Conf. Nonlinear Analysis and Optimization, Haifa, Israel, (2008), AMS Contemporary Mathematics, 514 (2010), 45-84. doi: 10.1090/conm/514/10099.

[4]

Kluwer Academic Publishers, 2004.

[5]

J. Fixed P. Th. and Appl., 7 (2010), 145-160. doi: 10.1007/s11784-010-0013-0.

[6]

Lect. Notes in Math., 1560, Springer-Verlag, Berlin, 1993.

[7]

Math. Z., 127 (1972), 105-126.

[8]

Academic Press, Inc. LTD, 1972.

[9]

Nonl. Anal. TMA, 12 (1988), 51-61. doi: 10.1016/0362-546X(88)90012-0.

[10]

CBMS Regional Conference Series in Mathematics, 38, AMS, Providence, R. I., 1978.

[11]

Ann. Inst. H.Poincaré, Ana. Non Lin., 2, (1985), 329-370.

[12]

Fund. Math., 185 (2005), 1-18. doi: 10.4064/fm185-1-1.

[13]

Springer-Verlag, Berlin, 1979.

[14]

Walter de Gruyter, Berlin-New York, 1987. doi: 10.1515/9783110858372.312.

[15]

Springer-Verlag, Berlin Heidelberg New York, 2000. doi: 10.1007/978-3-642-56936-4.

[16]

Nonl. Anal. TMA, 36 (1999), 101-118. doi: 10.1016/S0362-546X(98)00017-0.

[17]

Erg. Th. and Dynam. Sys., 7 (1987), 93-103. doi: 10.1017/S0143385700003825.

[18]

Topological Nonlinear Analysis, Degree, Singularity and Variations, PNDLE 27, Birkhäuser, (1997), 247-272.

[19]

Studia Math., 134, (1999), 217-233.

[20]

Nonl. Anal TMA, 74 (2011), 1823-1834. doi: 10.1016/j.na.2010.10.055.

[21]

Mem. AMS, 174, 1976.

[22]

Topological Nonlinear Analysis, Degree, Singularity and Variations, PNDE, 15, Birkhäuser, (1995), 341-463.

[23]

J. Diff. Equat., 170 (2001), 22-50. doi: 10.1006/jdeq.2000.3818.

[24]

Nonl. Anal. TMA, 51 (2002), 33-66. doi: 10.1016/S0362-546X(01)00811-2.

[25]

Nonl. Anal. TMA, 73 (2010), 2779-2791. doi: 10.1016/j.na.2010.06.001.

[26]

Conf. Sem. Mat. Univ. Bari, 132 (1973).

[27]

Man. Math., 63 (1989), 99-114. doi: 10.1007/BF01173705.

[28]

Handbook of Dynamical Systems, 2, North-Holland, Amsterdam, (2002), 393-460. doi: 10.1016/S1874-575X(02)80030-3.

[29]

Courant Institute of Mathematical Sciences, New York University, 1974.

[30]

Comm. Pure Appl. Math., 23 (1970), 939-961.

[31]

J. Func. Anal., 7 (1971), 487-513.

[32]

Contributions to Nonlinear Functional Analysis, E Academic Press, New York, (1971), 11-36.

[33]

J. Diff. Equat., 202 (2004), 284-305. doi: 10.1016/j.jde.2004.03.037.

[34]

Nonl. Anal. TMA, 68 (2008), 1479-1516. doi: 10.1016/j.na.2006.12.039.

[35]

Trans. Amer. Math. Soc., 269 (1982), 351-382. doi: 10.2307/1998452.

[36]

Nonl. Anal. TMA, 23 (1994), 83-102. doi: 10.1016/0362-546X(94)90253-4.

[37]

Topol. Meth. Nonl. Anal., 9 (1997), 383-417.

[38]

Milan J. Math., 73 (2005), 103-144. doi: 10.1007/s00032-005-0040-2.

[39]

Adv. Nonl. Stud., 11 (2011), 929-940.

[40]

TAMS, 291 (1985), 1-41. doi: 10.2307/1999893.

[41]

Bull. London Math. Soc., 22 (1990), 113-140. doi: 10.1112/blms/22.2.113.

[42]

Invent. Math., 100 (1990), 63-95. doi: 10.1007/BF01231181.

[43]

Fundamental Principles of Mathematical Science, 258, Springer-Verlag, New York-Berlin, 1983.

[44]

Math. Z., 125 (1972), 359-364.

[45]

CMBS Regional Conf. Ser. in Math., 5, AMS, Providence, R.I., 1970.

[1]

Zalman Balanov, Wieslaw Krawcewicz, Haibo Ruan. Applied equivariant degree, part I: An axiomatic approach to primary degree. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 983-1016. doi: 10.3934/dcds.2006.15.983

[2]

Jochen Brüning, Franz W. Kamber, Ken Richardson. The equivariant index theorem for transversally elliptic operators and the basic index theorem for Riemannian foliations. Electronic Research Announcements, 2010, 17: 138-154. doi: 10.3934/era.2010.17.138

[3]

Zalman Balanov, Meymanat Farzamirad, Wieslaw Krawcewicz, Haibo Ruan. Applied equivariant degree. part II: Symmetric Hopf bifurcations of functional differential equations. Discrete and Continuous Dynamical Systems, 2006, 16 (4) : 923-960. doi: 10.3934/dcds.2006.16.923

[4]

Aravind Asok, James Parson. Equivariant sheaves on some spherical varieties. Electronic Research Announcements, 2011, 18: 119-130. doi: 10.3934/era.2011.18.119

[5]

Jiaquan Liu, Yuxia Guo, Pingan Zeng. Relationship of the morse index and the $L^\infty$ bound of solutions for a strongly indefinite differential superlinear system. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 107-119. doi: 10.3934/dcds.2006.16.107

[6]

Todd Young. A result in global bifurcation theory using the Conley index. Discrete and Continuous Dynamical Systems, 1996, 2 (3) : 387-396. doi: 10.3934/dcds.1996.2.387

[7]

M. C. Carbinatto, K. Mischaikow. Horseshoes and the Conley index spectrum - II: the theorem is sharp. Discrete and Continuous Dynamical Systems, 1999, 5 (3) : 599-616. doi: 10.3934/dcds.1999.5.599

[8]

Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks and Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617

[9]

Jiaxi Huang, Youde Wang, Lifeng Zhao. Equivariant Schrödinger map flow on two dimensional hyperbolic space. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4379-4425. doi: 10.3934/dcds.2020184

[10]

Pietro-Luciano Buono, V.G. LeBlanc. Equivariant versal unfoldings for linear retarded functional differential equations. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 283-302. doi: 10.3934/dcds.2005.12.283

[11]

Dan-Andrei Geba, Manoussos G. Grillakis. Large data global regularity for the classical equivariant Skyrme model. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5537-5576. doi: 10.3934/dcds.2018244

[12]

Marian Gidea. Leray functor and orbital Conley index for non-invariant sets. Discrete and Continuous Dynamical Systems, 1999, 5 (3) : 617-630. doi: 10.3934/dcds.1999.5.617

[13]

Jintao Wang, Desheng Li, Jinqiao Duan. On the shape Conley index theory of semiflows on complete metric spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1629-1647. doi: 10.3934/dcds.2016.36.1629

[14]

Ketty A. De Rezende, Mariana G. Villapouca. Discrete conley index theory for zero dimensional basic sets. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1359-1387. doi: 10.3934/dcds.2017056

[15]

Silvia Cingolani, Mónica Clapp. Symmetric semiclassical states to a magnetic nonlinear Schrödinger equation via equivariant Morse theory. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1263-1281. doi: 10.3934/cpaa.2010.9.1263

[16]

Yuzhou Tian, Yulin Zhao. Global phase portraits and bifurcation diagrams for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 2941-2956. doi: 10.3934/dcdsb.2020214

[17]

Jianjun Yuan. Global existence and scattering of equivariant defocusing Chern-Simons-Schrödinger system. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5541-5570. doi: 10.3934/dcds.2020237

[18]

Franz W. Kamber and Peter W. Michor. Completing Lie algebra actions to Lie group actions. Electronic Research Announcements, 2004, 10: 1-10.

[19]

Katarzyna Grabowska, Marcin Zając. The Tulczyjew triple in mechanics on a Lie group. Journal of Geometric Mechanics, 2016, 8 (4) : 413-435. doi: 10.3934/jgm.2016014

[20]

Nikolaos Karaliolios. Differentiable Rigidity for quasiperiodic cocycles in compact Lie groups. Journal of Modern Dynamics, 2017, 11: 125-142. doi: 10.3934/jmd.2017006

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (106)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]