Global symmetry-breaking bifurcations of critical orbits of invariant functionals

Anna Goƚȩbiewska; Norimichi Hirano; Sƚawomir Rybicki

doi:10.3934/dcdss.2019129

Previous Article
Selective Pyragas control of Hamiltonian systems
DCDS-S Home
This Issue
Next Article
Multiple solutions for a critical quasilinear equation with Hardy potential

November 2019, 12(7): 2005-2017. doi: 10.3934/dcdss.2019129

Global symmetry-breaking bifurcations of critical orbits of invariant functionals

Anna Goƚȩbiewska ^1,, , Norimichi Hirano ^2, and Sƚawomir Rybicki ^1,

1.	Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, PL-87-100 Toruń, ul. Chopina 12/18, Poland
2.	Department of Mathematics, Graduate School of Environment and Information Sciences, Yokohama National University, 79-7 Tokiwadai, Hodogayaku, Yokohama, Japan

* Corresponding author

Received October 2017 Revised April 2018 Published December 2018

Fund Project: Partially supported by the National Science Center, Poland, under grant DEC-2012/05/B/ST1/02165.

In this article we present a method of study of a global symmetry-breaking bifurcation of critical orbits of invariant functionals. As a topological tool we use the degree for equivariant gradient maps. We underline that many known results on bifurcations of non-radial solutions of elliptic PDE's from the families of radial ones are consequences of our theory.

Keywords: Equivariant bifurcation theory, global bifurcation, symmetry-breaking bifurcation, degree for equivariant gradient maps, non-radial bifurcations.

Mathematics Subject Classification: Primary: 58E09; Secondary: 35J50.

Citation: Anna Goƚȩbiewska, Norimichi Hirano, Sƚawomir Rybicki. Global symmetry-breaking bifurcations of critical orbits of invariant functionals. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2005-2017. doi: 10.3934/dcdss.2019129

References:

[1]	Z. Balanov, W. Krawcewicz, S. Rybicki and H. Steinlein, A short treatise on the equivariant degree theory and its applications, J. Fixed P. Th. and Appl., 8 (2010), 1-74. doi: 10.1007/s11784-010-0033-9.
[2]	Z. Balanov, W. Krawcewicz and H. Steinlein, Applied Equivariant Degree, AIMS Series on Diff. Equat. & Dyn. Sys., Springfield, 2006.
[3]	P. Bartlomiejczyk, K. Gȩba and M. Izydorek, Otopy classes of equivariant maps, J. Fixed P. Th. and Appl., 7 (2010), 145-160. doi: 10.1007/s11784-010-0013-0.
[4]	G. Cerami, Symmetry breaking for a class of semilinear elliptic problems, Nonl. Anal. TMA, 10 (1986), 1-14. doi: 10.1016/0362-546X(86)90007-6.
[5]	M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.
[6]	E. N. Dancer, On nonradially symmetric bifurcation, J. London Math. Soc., 20 (1979), 287-292. doi: 10.1112/jlms/s2-20.2.287.
[7]	E. N. Dancer, Global breaking of symmetry of positive solutions on two-dimensional annuli, Diff. and Int. Equat., 5 (1992), 903-913.
[8]	T. tom Dieck, Transformation Groups and Representation Theory, Lect. Not. in Math. 766 Springer, Berlin, 1979.
[9]	T. tom Dieck, Transformation Groups, Walter de Gruyter, Berlin, 1987. doi: 10.1515/9783110858372.312.
[10]	J. Fura, A. Ratajczak and S. Rybicki, Existence and continuation of periodic solutions of autonomous Newtonian systems, J. Diff. Equat., 218 (2005), 216-252. doi: 10.1016/j.jde.2005.04.004.
[11]	K. Gȩba, Degree for gradient equivariant maps and equivariant Conley index, Topological Nonlinear Analysis, Degree, Singularity and Variations, Eds. M. Matzeu & A. Vignoli, Progr. in Nonl. Diff. Equat. and Their Appl., 27, Birkhäuser, (1997), 247-272.
[12]	K. Gȩba, M. Izydorek and A. Pruszko, The Conley index in Hilbert spaces and its applications, Stud. Math., 134 (1999), 217-233.
[13]	B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.
[14]	A. Golȩbiewska and S. Rybicki, Global bifurcations of critical orbits of $G$-invariant strongly indefinite functionals, Nonl. Anal. TMA, 74 (2011), 1823-1834. doi: 10.1016/j.na.2010.10.055.
[15]	W. Jäger and K. Schmitt, Symmetry breaking in semilinear elliptic problems, Academic Press, Inc., Analysis, et cetera, Research Papers Published in Honour of Jürgen Moser's 60th Birthday, Eds. P. H. Rabinowitz & E. Zehnder, (1990), 451-470.
[16]	R. Lauterbach and S. Maier, Symmetry-breaking at non-positive solutions of semilinear elliptic equations, Arch. Rational Mech. Anal., 126 (1994), 299-331. doi: 10.1007/BF00380895.
[17]	Y. Y. Li, Existence of many positive solutions of semilinear elliptic equations on annulus, J. Diff. Equat., 83 (1990), 348-367. doi: 10.1016/0022-0396(90)90062-T.
[18]	S. S. Lin, On non-radially symmetric bifurcation in the annulus, J. Diff. Equat., 80 (1989), 251-279. doi: 10.1016/0022-0396(89)90084-3.
[19]	S. S. Lin, Positive radial solutions and non-radial bifurcation for semilinear elliptic equations in annular domains, J. Diff. Equat., 86 (1990), 367-391. doi: 10.1016/0022-0396(90)90035-N.
[20]	S. S. Lin, Existence of positive non-radial solutions for nonlinear elliptic equations in annular domains, Trans. of the Amer. Math. Soc., 332 (1992), 775-791. doi: 10.1090/S0002-9947-1992-1055571-1.
[21]	G. López Garza and S. Rybicki, Equivariant bifurcation index, Nonl. Anal. TMA, 73 (2010), 2779-2791. doi: 10.1016/j.na.2010.06.001.
[22]	K. Nagasaki and T. Suzuki, Radial and nonradial solutions for the nonlinear eigenvalue problem $\Delta u + \lambda e^u = 0$ on annuli in $R^2$, J. Diff. Equat., 87 (1990), 144-168. doi: 10.1016/0022-0396(90)90020-P.
[23]	K. Nagasaki and T. Suzuki, Radial Solutions for $\Delta u + \lambda e^u = 0$ on annuli in higher dimensions, J. Diff. Equat., 100 (1992), 137-161. doi: 10.1016/0022-0396(92)90129-B.
[24]	K. Nagasaki and T. Suzuki, Spectral and related properties about the Emden-Fowler equation $ \Delta u = \lambda e^u$ on circular domains, Math. Ann., 299 (1994), 1-15. doi: 10.1007/BF01459770.
[25]	W. -M. Ni and R. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of $-\Delta u+f(u, r)=0$, Comm. Pure Appl. Math., 38 (1985), 67-108. doi: 10.1002/cpa.3160380105.
[26]	F. Pacard, Radial and non-radial solutions of $-Δ u = λ f(u)$, on an annulus of $\mathbb{R}^n, \:n ≥ 3$, J. Diff. Equat., 101 (1993), 103-138. doi: 10.1006/jdeq.1993.1007.
[27]	M. Ramaswamy and P. N. Srikanth, Symmetry breaking for a class of semilinear elliptic problems, Trans. of the Amer. Math. Soc., 304 (1987), 839-845. doi: 10.1090/S0002-9947-1987-0911098-4.
[28]	S. Rybicki, A degree for $S^1$-equivariant orthogonal maps and its applications to bifurcation theory, Nonl. Anal. TMA, 23 (1994), 83-102. doi: 10.1016/0362-546X(94)90253-4.
[29]	S. Rybicki, Global bifurcations of solutions of Emden-Fowler type equation $- \Delta u(x) = \lambda f(u(x))$ on an annulus in $R^n, \: n \geq 3$, J. Diff. Equat., 183 (2002), 208-223. doi: 10.1006/jdeq.2001.4121.
[30]	S. Rybicki, Degree for equivariant gradient maps, Milan Journal of Mathematics, 73 (2005), 103-144. doi: 10.1007/s00032-005-0040-2.
[31]	S. Rybicki, Global bifurcations of critical orbits via equivariant Conley index, Adv. Nonl. Studies, 11 (2011), 929-940. doi: 10.1515/ans-2011-0410.
[32]	J. Smoller and A. G. Wasserman, Symmetry-breaking for positive solutions of semilinear elliptic equations, Arch. Rational Mech. Anal., 95 (1986), 217-225. doi: 10.1007/BF00251359.
[33]	J. Smoller and A. Wasserman, Bifurcation and symmetry-breaking, Invent. Math., 100 (1990), 63-95. doi: 10.1007/BF01231181.
[34]	P. N. Srikanth, Symmetry breaking for a class of semilinear elliptic problems, Ann. Inst. H. Poincarè Anal. Non Lin., 7 (1990), 107-112. doi: 10.1016/S0294-1449(16)30301-8.
[35]	T. Suzuki, Two-dimensional Emden-Fowler equation with exponential nonlinearity, Nonlinear diffusion equations and their equilibrium states, Proc. of 3rd Conf. Gregynog UK, 1989, Prog. Nonl. Diff. Equat. Appl., 7 (1992), 493-512.

show all references

References:

[1]	Z. Balanov, W. Krawcewicz, S. Rybicki and H. Steinlein, A short treatise on the equivariant degree theory and its applications, J. Fixed P. Th. and Appl., 8 (2010), 1-74. doi: 10.1007/s11784-010-0033-9.
[2]	Z. Balanov, W. Krawcewicz and H. Steinlein, Applied Equivariant Degree, AIMS Series on Diff. Equat. & Dyn. Sys., Springfield, 2006.
[3]	P. Bartlomiejczyk, K. Gȩba and M. Izydorek, Otopy classes of equivariant maps, J. Fixed P. Th. and Appl., 7 (2010), 145-160. doi: 10.1007/s11784-010-0013-0.
[4]	G. Cerami, Symmetry breaking for a class of semilinear elliptic problems, Nonl. Anal. TMA, 10 (1986), 1-14. doi: 10.1016/0362-546X(86)90007-6.
[5]	M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.
[6]	E. N. Dancer, On nonradially symmetric bifurcation, J. London Math. Soc., 20 (1979), 287-292. doi: 10.1112/jlms/s2-20.2.287.
[7]	E. N. Dancer, Global breaking of symmetry of positive solutions on two-dimensional annuli, Diff. and Int. Equat., 5 (1992), 903-913.
[8]	T. tom Dieck, Transformation Groups and Representation Theory, Lect. Not. in Math. 766 Springer, Berlin, 1979.
[9]	T. tom Dieck, Transformation Groups, Walter de Gruyter, Berlin, 1987. doi: 10.1515/9783110858372.312.
[10]	J. Fura, A. Ratajczak and S. Rybicki, Existence and continuation of periodic solutions of autonomous Newtonian systems, J. Diff. Equat., 218 (2005), 216-252. doi: 10.1016/j.jde.2005.04.004.
[11]	K. Gȩba, Degree for gradient equivariant maps and equivariant Conley index, Topological Nonlinear Analysis, Degree, Singularity and Variations, Eds. M. Matzeu & A. Vignoli, Progr. in Nonl. Diff. Equat. and Their Appl., 27, Birkhäuser, (1997), 247-272.
[12]	K. Gȩba, M. Izydorek and A. Pruszko, The Conley index in Hilbert spaces and its applications, Stud. Math., 134 (1999), 217-233.
[13]	B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.
[14]	A. Golȩbiewska and S. Rybicki, Global bifurcations of critical orbits of $G$-invariant strongly indefinite functionals, Nonl. Anal. TMA, 74 (2011), 1823-1834. doi: 10.1016/j.na.2010.10.055.
[15]	W. Jäger and K. Schmitt, Symmetry breaking in semilinear elliptic problems, Academic Press, Inc., Analysis, et cetera, Research Papers Published in Honour of Jürgen Moser's 60th Birthday, Eds. P. H. Rabinowitz & E. Zehnder, (1990), 451-470.
[16]	R. Lauterbach and S. Maier, Symmetry-breaking at non-positive solutions of semilinear elliptic equations, Arch. Rational Mech. Anal., 126 (1994), 299-331. doi: 10.1007/BF00380895.
[17]	Y. Y. Li, Existence of many positive solutions of semilinear elliptic equations on annulus, J. Diff. Equat., 83 (1990), 348-367. doi: 10.1016/0022-0396(90)90062-T.
[18]	S. S. Lin, On non-radially symmetric bifurcation in the annulus, J. Diff. Equat., 80 (1989), 251-279. doi: 10.1016/0022-0396(89)90084-3.
[19]	S. S. Lin, Positive radial solutions and non-radial bifurcation for semilinear elliptic equations in annular domains, J. Diff. Equat., 86 (1990), 367-391. doi: 10.1016/0022-0396(90)90035-N.
[20]	S. S. Lin, Existence of positive non-radial solutions for nonlinear elliptic equations in annular domains, Trans. of the Amer. Math. Soc., 332 (1992), 775-791. doi: 10.1090/S0002-9947-1992-1055571-1.
[21]	G. López Garza and S. Rybicki, Equivariant bifurcation index, Nonl. Anal. TMA, 73 (2010), 2779-2791. doi: 10.1016/j.na.2010.06.001.
[22]	K. Nagasaki and T. Suzuki, Radial and nonradial solutions for the nonlinear eigenvalue problem $\Delta u + \lambda e^u = 0$ on annuli in $R^2$, J. Diff. Equat., 87 (1990), 144-168. doi: 10.1016/0022-0396(90)90020-P.
[23]	K. Nagasaki and T. Suzuki, Radial Solutions for $\Delta u + \lambda e^u = 0$ on annuli in higher dimensions, J. Diff. Equat., 100 (1992), 137-161. doi: 10.1016/0022-0396(92)90129-B.
[24]	K. Nagasaki and T. Suzuki, Spectral and related properties about the Emden-Fowler equation $ \Delta u = \lambda e^u$ on circular domains, Math. Ann., 299 (1994), 1-15. doi: 10.1007/BF01459770.
[25]	W. -M. Ni and R. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of $-\Delta u+f(u, r)=0$, Comm. Pure Appl. Math., 38 (1985), 67-108. doi: 10.1002/cpa.3160380105.
[26]	F. Pacard, Radial and non-radial solutions of $-Δ u = λ f(u)$, on an annulus of $\mathbb{R}^n, \:n ≥ 3$, J. Diff. Equat., 101 (1993), 103-138. doi: 10.1006/jdeq.1993.1007.
[27]	M. Ramaswamy and P. N. Srikanth, Symmetry breaking for a class of semilinear elliptic problems, Trans. of the Amer. Math. Soc., 304 (1987), 839-845. doi: 10.1090/S0002-9947-1987-0911098-4.
[28]	S. Rybicki, A degree for $S^1$-equivariant orthogonal maps and its applications to bifurcation theory, Nonl. Anal. TMA, 23 (1994), 83-102. doi: 10.1016/0362-546X(94)90253-4.
[29]	S. Rybicki, Global bifurcations of solutions of Emden-Fowler type equation $- \Delta u(x) = \lambda f(u(x))$ on an annulus in $R^n, \: n \geq 3$, J. Diff. Equat., 183 (2002), 208-223. doi: 10.1006/jdeq.2001.4121.
[30]	S. Rybicki, Degree for equivariant gradient maps, Milan Journal of Mathematics, 73 (2005), 103-144. doi: 10.1007/s00032-005-0040-2.
[31]	S. Rybicki, Global bifurcations of critical orbits via equivariant Conley index, Adv. Nonl. Studies, 11 (2011), 929-940. doi: 10.1515/ans-2011-0410.
[32]	J. Smoller and A. G. Wasserman, Symmetry-breaking for positive solutions of semilinear elliptic equations, Arch. Rational Mech. Anal., 95 (1986), 217-225. doi: 10.1007/BF00251359.
[33]	J. Smoller and A. Wasserman, Bifurcation and symmetry-breaking, Invent. Math., 100 (1990), 63-95. doi: 10.1007/BF01231181.
[34]	P. N. Srikanth, Symmetry breaking for a class of semilinear elliptic problems, Ann. Inst. H. Poincarè Anal. Non Lin., 7 (1990), 107-112. doi: 10.1016/S0294-1449(16)30301-8.
[35]	T. Suzuki, Two-dimensional Emden-Fowler equation with exponential nonlinearity, Nonlinear diffusion equations and their equilibrium states, Proc. of 3rd Conf. Gregynog UK, 1989, Prog. Nonl. Diff. Equat. Appl., 7 (1992), 493-512.

[1]	Sanjay Dharmavaram, Timothy J. Healey. Direct construction of symmetry-breaking directions in bifurcation problems with spherical symmetry. Discrete and Continuous Dynamical Systems - S, 2019, 12 (6) : 1669-1684. doi: 10.3934/dcdss.2019112
[2]	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
[3]	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
[4]	Anna Lisa Amadori. Global bifurcation for the Hénon problem. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4797-4816. doi: 10.3934/cpaa.2020212
[5]	Juan Sánchez, Marta Net, José M. Vega. Amplitude equations close to a triple-(+1) bifurcation point of D4-symmetric periodic orbits in O(2)-equivariant systems. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1357-1380. doi: 10.3934/dcdsb.2006.6.1357
[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]	J. F. Toland. Path-connectedness in global bifurcation theory. Electronic Research Archive, 2021, 29 (6) : 4199-4213. doi: 10.3934/era.2021079
[8]	Gian-Italo Bischi, Laura Gardini, Fabio Tramontana. Bifurcation curves in discontinuous maps. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 249-267. doi: 10.3934/dcdsb.2010.13.249
[9]	Jeremy L. Marzuola, Michael I. Weinstein. Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1505-1554. doi: 10.3934/dcds.2010.28.1505
[10]	Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032
[11]	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
[12]	Carmen Núñez, Rafael Obaya. A non-autonomous bifurcation theory for deterministic scalar differential equations. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 701-730. doi: 10.3934/dcdsb.2008.9.701
[13]	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
[14]	Fernando Antoneli, Ana Paula S. Dias, Rui Paiva. Coupled cell networks: Hopf bifurcation and interior symmetry. Conference Publications, 2011, 2011 (Special) : 71-78. doi: 10.3934/proc.2011.2011.71
[15]	Guangyu Xu. Emergence of lager densities in chemotaxis system with indirect signal production and non-radial symmetry case. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022096
[16]	Sébastien Biebler. Lattès maps and the interior of the bifurcation locus. Journal of Modern Dynamics, 2019, 15: 95-130. doi: 10.3934/jmd.2019014
[17]	Jingli Ren, Gail S. K. Wolkowicz. Preface: Recent advances in bifurcation theory and application. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : i-ii. doi: 10.3934/dcdss.2020417
[18]	Gunog Seo, Gail S. K. Wolkowicz. Pest control by generalist parasitoids: A bifurcation theory approach. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3157-3187. doi: 10.3934/dcdss.2020163
[19]	Johanna D. García-Saldaña, Armengol Gasull, Hector Giacomini. Bifurcation values for a family of planar vector fields of degree five. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 669-701. doi: 10.3934/dcds.2015.35.669
[20]	Patrick M. Fitzpatrick, Jacobo Pejsachowicz. Branching and bifurcation. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 1955-1975. doi: 10.3934/dcdss.2019127

American Institute of Mathematical Sciences

Global symmetry-breaking bifurcations of critical orbits of invariant functionals

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Global symmetry-breaking bifurcations of critical orbits of invariant functionals

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors