Invariant sets forced by symmetry

Previous Article
Dual pairs in resonances
JGM Home
This Issue
Next Article
Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields

September 2012, 4(3): 271-296. doi: 10.3934/jgm.2012.4.271

Invariant sets forced by symmetry

Frank D. Grosshans ^1, , Jürgen Scheurle ^2, and Sebastian Walcher ^3,

1.	Department of Mathematics, West Chester University, West Chester, PA 19383, United States
2.	Zentrum Mathematik, TU München, Boltzmannstr. 3, 85747 Garching, Germany
3.	Lehrstuhl A für Mathematik, RWTH Aachen, 52056 Aachen

Received May 2011 Revised May 2012 Published October 2012

Abstract
Related Papers

Given a linear (algebraic) group $G$ acting on real or complex $n$-space, we determine all the common invariant sets of $G$-symmetric vector fields. It turns out that the investigation of certain algebraic varieties is sufficient to characterize these invariant sets forced by symmetry. Toral, compact and reductive groups are discussed in some detail, and examples, including a Couette-Taylor system, are presented.

Keywords: invariant sets, symmetries, reduction of systems., Equivariant dynamical systems, non-compact symmetry groups.

Mathematics Subject Classification: Primary: 37C80, 34C14; Secondary: 34C20, 58J7.

Citation: Frank D. Grosshans, Jürgen Scheurle, Sebastian Walcher. Invariant sets forced by symmetry. Journal of Geometric Mechanics, 2012, 4 (3) : 271-296. doi: 10.3934/jgm.2012.4.271

References:

[1]	Yu. N. Bibikov, "Local Theory of Nonlinear Analytic Ordinary Differential Equations," Lecture Notes in Mathematics, 702, Springer-Verlag, Berlin, 1979.
[2]	D. Birkes, Orbits of linear algebraic groups, Ann. Math., 93 (1971), 459-475. doi: 10.2307/1970884.
[3]	A. Borel, "Linear Algebraic Groups," $2^{nd}$ edition, Springer-Verlag, New York - Berlin, 1992.
[4]	T. Bröcker and T. tom Dieck, "Representations of Compact Lie Groups," Springer-Verlag, New York - Berlin, 1985.
[5]	P. Chossat, The reduction of equivariant dynamics to the orbit space of compact group actions, Acta Appl. Math., 70 (2002), 71-94. doi: 10.1023/A:1013970014204.
[6]	D. Cox, J. Little and D. O'Shea, "Ideals, Varieties and Algorithms," Springer-Verlag, New York, 1997.
[7]	R. Cushman and J. Sanders, A survey of invariant theory applied to normal forms of vector fields with nilpotent linear part, in "Invariant Theory and Tableaux", Minneapolis, MN 1988. IMA Vol. Math. Appl. 19, Springer-Verlag, New York (1990), 82-106.
[8]	E. B. Elliot, "An Introduction to the Algebra of Binary Quantics," $2^{nd}$ edition, Reprinted. Chelsea Publ. Co., New York, 1964.
[9]	M. J. Field, Equivariant dynamical systems, Trans. Amer. Math. Soc., 259 (1980), 185-205. doi: 10.1090/S0002-9947-1980-0561832-4.
[10]	B. Fiedler, B. Sandstede, A. Scheel and C. Wulff, Bifurcation from relative equilibria of noncompact group actions: Skew products, meanders, and drifts, Documenta Math., 1 (1996), 479-505.
[11]	G. Gaeta, F. D. Grosshans, J. Scheurle and S. Walcher, Reduction and reconstruction for symmetric ordinary differential equations, J. Differential Equations, 244 (2008), 1810-1839. doi: 10.1016/j.jde.2008.01.009.
[12]	K. Gatermann, "Computer Algebra Methods for Equivariant Dynamical Systems," Lecture Notes in Mathematics, 1728, Springer-Verlag, Berlin, 2000.
[13]	M. Golubitsky, V. G. LeBlanc and I. Melbourne, Meandering of the spiral tip: An alternative approach, J. Nonlinear Sci., 7 (1997), 557-586.
[14]	V. G. Guillemin and S. Sternberg, Remarks on a paper of Hermann, Trans. Amer. Math. Soc., 130 (1968), 110-116. doi: 10.1090/S0002-9947-1968-0217226-9.
[15]	J. E. Humphreys, "Linear Algebraic Groups," Springer-Verlag, New York - Heidelberg, 1975.
[16]	M. Krupa, Bifurcations of relative equilibria, SIAM J. Math. Anal., 21 (1990), 1453-1486. doi: 10.1137/0521081.
[17]	A. G. Kushnirenko, An analytic action of a semisimple Lie group in a neighborhood of a fixed point is equivalent to a linear one, Funct. Anal. Appl., 1 (1967), 273-274.
[18]	G. I. Lehrer and T. A. Springer, A note concerning fixed points of parabolic subgroups of unitary reflection groups, Indag. Math., N. S., 10 (1999), 549-553.
[19]	L. Michel, Points critiques des fonctions invariantes sur une $G$-variété, C.R. Acad. Sc. Paris, 278 (1971), 433-436.
[20]	D. I. Panyushev, On covariants of reductive algebraic groups, Indag. Math., N. S., 13 (2002), 125-129.
[21]	V. Poénaru, "Singularités $C^{\infty }$ en Présence de Symétrie," Lecture Notes in Mathematics, 510. Springer-Verlag, Berlin-New York, 1976.
[22]	J. Scheurle, Some aspects of successive bifurcations in the Couette-Taylor problem, Fields Inst. Comm., 5 (1996), 335-345.
[23]	S. Walcher, On differential equations in normal form, Math. Ann., 291 (1991), 293-314. doi: 10.1007/BF01445209.
[24]	S. Walcher, Multi-parameter symmetries of first order ordinary differential equations, J. Lie Theory, 9 (1999), 249-289.

show all references

References:

[1]	Yu. N. Bibikov, "Local Theory of Nonlinear Analytic Ordinary Differential Equations," Lecture Notes in Mathematics, 702, Springer-Verlag, Berlin, 1979.
[2]	D. Birkes, Orbits of linear algebraic groups, Ann. Math., 93 (1971), 459-475. doi: 10.2307/1970884.
[3]	A. Borel, "Linear Algebraic Groups," $2^{nd}$ edition, Springer-Verlag, New York - Berlin, 1992.
[4]	T. Bröcker and T. tom Dieck, "Representations of Compact Lie Groups," Springer-Verlag, New York - Berlin, 1985.
[5]	P. Chossat, The reduction of equivariant dynamics to the orbit space of compact group actions, Acta Appl. Math., 70 (2002), 71-94. doi: 10.1023/A:1013970014204.
[6]	D. Cox, J. Little and D. O'Shea, "Ideals, Varieties and Algorithms," Springer-Verlag, New York, 1997.
[7]	R. Cushman and J. Sanders, A survey of invariant theory applied to normal forms of vector fields with nilpotent linear part, in "Invariant Theory and Tableaux", Minneapolis, MN 1988. IMA Vol. Math. Appl. 19, Springer-Verlag, New York (1990), 82-106.
[8]	E. B. Elliot, "An Introduction to the Algebra of Binary Quantics," $2^{nd}$ edition, Reprinted. Chelsea Publ. Co., New York, 1964.
[9]	M. J. Field, Equivariant dynamical systems, Trans. Amer. Math. Soc., 259 (1980), 185-205. doi: 10.1090/S0002-9947-1980-0561832-4.
[10]	B. Fiedler, B. Sandstede, A. Scheel and C. Wulff, Bifurcation from relative equilibria of noncompact group actions: Skew products, meanders, and drifts, Documenta Math., 1 (1996), 479-505.
[11]	G. Gaeta, F. D. Grosshans, J. Scheurle and S. Walcher, Reduction and reconstruction for symmetric ordinary differential equations, J. Differential Equations, 244 (2008), 1810-1839. doi: 10.1016/j.jde.2008.01.009.
[12]	K. Gatermann, "Computer Algebra Methods for Equivariant Dynamical Systems," Lecture Notes in Mathematics, 1728, Springer-Verlag, Berlin, 2000.
[13]	M. Golubitsky, V. G. LeBlanc and I. Melbourne, Meandering of the spiral tip: An alternative approach, J. Nonlinear Sci., 7 (1997), 557-586.
[14]	V. G. Guillemin and S. Sternberg, Remarks on a paper of Hermann, Trans. Amer. Math. Soc., 130 (1968), 110-116. doi: 10.1090/S0002-9947-1968-0217226-9.
[15]	J. E. Humphreys, "Linear Algebraic Groups," Springer-Verlag, New York - Heidelberg, 1975.
[16]	M. Krupa, Bifurcations of relative equilibria, SIAM J. Math. Anal., 21 (1990), 1453-1486. doi: 10.1137/0521081.
[17]	A. G. Kushnirenko, An analytic action of a semisimple Lie group in a neighborhood of a fixed point is equivalent to a linear one, Funct. Anal. Appl., 1 (1967), 273-274.
[18]	G. I. Lehrer and T. A. Springer, A note concerning fixed points of parabolic subgroups of unitary reflection groups, Indag. Math., N. S., 10 (1999), 549-553.
[19]	L. Michel, Points critiques des fonctions invariantes sur une $G$-variété, C.R. Acad. Sc. Paris, 278 (1971), 433-436.
[20]	D. I. Panyushev, On covariants of reductive algebraic groups, Indag. Math., N. S., 13 (2002), 125-129.
[21]	V. Poénaru, "Singularités $C^{\infty }$ en Présence de Symétrie," Lecture Notes in Mathematics, 510. Springer-Verlag, Berlin-New York, 1976.
[22]	J. Scheurle, Some aspects of successive bifurcations in the Couette-Taylor problem, Fields Inst. Comm., 5 (1996), 335-345.
[23]	S. Walcher, On differential equations in normal form, Math. Ann., 291 (1991), 293-314. doi: 10.1007/BF01445209.
[24]	S. Walcher, Multi-parameter symmetries of first order ordinary differential equations, J. Lie Theory, 9 (1999), 249-289.

[1]	Morched Boughariou. Closed orbits of Hamiltonian systems on non-compact prescribed energy surfaces. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 603-616. doi: 10.3934/dcds.2003.9.603
[2]	Alain Bensoussan, Jens Frehse, Jens Vogelgesang. Systems of Bellman equations to stochastic differential games with non-compact coupling. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1375-1389. doi: 10.3934/dcds.2010.27.1375
[3]	Byungik Kahng, Miguel Mendes. The characterization of maximal invariant sets of non-linear discrete-time control dynamical systems. Conference Publications, 2013, 2013 (special) : 393-406. doi: 10.3934/proc.2013.2013.393
[4]	Denis de Carvalho Braga, Luis Fernando Mello, Carmen Rocşoreanu, Mihaela Sterpu. Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 785-803. doi: 10.3934/dcdsb.2009.11.785
[5]	Gary Froyland, Philip K. Pollett, Robyn M. Stuart. A closing scheme for finding almost-invariant sets in open dynamical systems. Journal of Computational Dynamics, 2014, 1 (1) : 135-162. doi: 10.3934/jcd.2014.1.135
[6]	Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087
[7]	L. Yu. Glebsky and E. I. Gordon. On approximation of locally compact groups by finite algebraic systems. Electronic Research Announcements, 2004, 10: 21-28.
[8]	Grzegorz Łukaszewicz, James C. Robinson. Invariant measures for non-autonomous dissipative dynamical systems. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4211-4222. doi: 10.3934/dcds.2014.34.4211
[9]	E. García-Toraño Andrés, Bavo Langerock, Frans Cantrijn. Aspects of reduction and transformation of Lagrangian systems with symmetry. Journal of Geometric Mechanics, 2014, 6 (1) : 1-23. doi: 10.3934/jgm.2014.6.1
[10]	Michael Hochman. Smooth symmetries of $\times a$-invariant sets. Journal of Modern Dynamics, 2018, 13: 187-197. doi: 10.3934/jmd.2018017
[11]	Mike Crampin, Tom Mestdag. Reduction of invariant constrained systems using anholonomic frames. Journal of Geometric Mechanics, 2011, 3 (1) : 23-40. doi: 10.3934/jgm.2011.3.23
[12]	Ji Li, Kening Lu, Peter W. Bates. Invariant foliations for random dynamical systems. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3639-3666. doi: 10.3934/dcds.2014.34.3639
[13]	Chunyan Zhao, Chengkui Zhong, Xiangming Zhu. Existence of compact $ \varphi $-attracting sets and estimate of their attractive velocity for infinite-dimensional dynamical systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (12) : 7493-7520. doi: 10.3934/dcdsb.2022051
[14]	Carsten Collon, Joachim Rudolph, Frank Woittennek. Invariant feedback design for control systems with lie symmetries - A kinematic car example. Conference Publications, 2011, 2011 (Special) : 312-321. doi: 10.3934/proc.2011.2011.312
[15]	Sebastian Hage-Packhäuser, Michael Dellnitz. Stabilization via symmetry switching in hybrid dynamical systems. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 239-263. doi: 10.3934/dcdsb.2011.16.239
[16]	Qihuai Liu, Pedro J. Torres. Orbital dynamics on invariant sets of contact Hamiltonian systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (10) : 5821-5844. doi: 10.3934/dcdsb.2021297
[17]	Ítalo Melo, Sergio Romaña. Contributions to the study of Anosov geodesic flows in non-compact manifolds. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5149-5171. doi: 10.3934/dcds.2020223
[18]	Mohamed Boulanouar. On a Mathematical model with non-compact boundary conditions describing bacterial population (Ⅱ). Evolution Equations and Control Theory, 2019, 8 (2) : 247-271. doi: 10.3934/eect.2019014
[19]	Eva Stadler, Johannes Müller. Analyzing plasmid segregation: Existence and stability of the eigensolution in a non-compact case. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4127-4164. doi: 10.3934/dcdsb.2020091
[20]	Jacques Demongeot, Dan Istrate, Hajer Khlaifi, Lucile Mégret, Carla Taramasco, René Thomas. From conservative to dissipative non-linear differential systems. An application to the cardio-respiratory regulation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2121-2134. doi: 10.3934/dcdss.2020181

2021 Impact Factor: 0.737

Tools

Metrics

Other articles
by authors

on AIMS
Frank D. Grosshans Jürgen Scheurle Sebastian Walcher
on Google Scholar
Frank D. Grosshans Jürgen Scheurle Sebastian Walcher

[Back to Top]