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

Invariant sets forced by symmetry

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

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.
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.

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