American Institute of Mathematical Sciences

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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

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 (1979). Google Scholar

[2]

D. Birkes, Orbits of linear algebraic groups,, Ann. Math., 93 (1971), 459. doi: 10.2307/1970884. Google Scholar

[3]

A. Borel, "Linear Algebraic Groups,", $2^{nd}$ edition, (1992). Google Scholar

[4]

T. Bröcker and T. tom Dieck, "Representations of Compact Lie Groups,", Springer-Verlag, (1985). Google Scholar

[5]

P. Chossat, The reduction of equivariant dynamics to the orbit space of compact group actions,, Acta Appl. Math., 70 (2002), 71. doi: 10.1023/A:1013970014204. Google Scholar

[6]

D. Cox, J. Little and D. O'Shea, "Ideals, Varieties and Algorithms,", Springer-Verlag, (1997). Google Scholar

[7]

R. Cushman and J. Sanders, A survey of invariant theory applied to normal forms of vector fields with nilpotent linear part,, in, 19 (1990), 82. Google Scholar

[8]

E. B. Elliot, "An Introduction to the Algebra of Binary Quantics,", $2^{nd}$ edition, (1964). Google Scholar

[9]

M. J. Field, Equivariant dynamical systems,, Trans. Amer. Math. Soc., 259 (1980), 185. doi: 10.1090/S0002-9947-1980-0561832-4. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/j.jde.2008.01.009. Google Scholar

[12]

K. Gatermann, "Computer Algebra Methods for Equivariant Dynamical Systems,", Lecture Notes in Mathematics, 1728 (2000). Google Scholar

[13]

M. Golubitsky, V. G. LeBlanc and I. Melbourne, Meandering of the spiral tip: An alternative approach,, J. Nonlinear Sci., 7 (1997), 557. Google Scholar

[14]

V. G. Guillemin and S. Sternberg, Remarks on a paper of Hermann,, Trans. Amer. Math. Soc., 130 (1968), 110. doi: 10.1090/S0002-9947-1968-0217226-9. Google Scholar

[15]

J. E. Humphreys, "Linear Algebraic Groups,", Springer-Verlag, (1975). Google Scholar

[16]

M. Krupa, Bifurcations of relative equilibria,, SIAM J. Math. Anal., 21 (1990), 1453. doi: 10.1137/0521081. Google Scholar

[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. Google Scholar

[18]

G. I. Lehrer and T. A. Springer, A note concerning fixed points of parabolic subgroups of unitary reflection groups,, Indag. Math., 10 (1999), 549. Google Scholar

[19]

L. Michel, Points critiques des fonctions invariantes sur une $G$-variété,, C.R. Acad. Sc. Paris, 278 (1971), 433. Google Scholar

[20]

D. I. Panyushev, On covariants of reductive algebraic groups,, Indag. Math., 13 (2002), 125. Google Scholar

[21]

V. Poénaru, "Singularités $C^{\infty }$ en Présence de Symétrie,", Lecture Notes in Mathematics, 510 (1976). Google Scholar

[22]

J. Scheurle, Some aspects of successive bifurcations in the Couette-Taylor problem,, Fields Inst. Comm., 5 (1996), 335. Google Scholar

[23]

S. Walcher, On differential equations in normal form,, Math. Ann., 291 (1991), 293. doi: 10.1007/BF01445209. Google Scholar

[24]

S. Walcher, Multi-parameter symmetries of first order ordinary differential equations,, J. Lie Theory, 9 (1999), 249. Google Scholar

show all references

References:
[1]

Yu. N. Bibikov, "Local Theory of Nonlinear Analytic Ordinary Differential Equations,", Lecture Notes in Mathematics, 702 (1979). Google Scholar

[2]

D. Birkes, Orbits of linear algebraic groups,, Ann. Math., 93 (1971), 459. doi: 10.2307/1970884. Google Scholar

[3]

A. Borel, "Linear Algebraic Groups,", $2^{nd}$ edition, (1992). Google Scholar

[4]

T. Bröcker and T. tom Dieck, "Representations of Compact Lie Groups,", Springer-Verlag, (1985). Google Scholar

[5]

P. Chossat, The reduction of equivariant dynamics to the orbit space of compact group actions,, Acta Appl. Math., 70 (2002), 71. doi: 10.1023/A:1013970014204. Google Scholar

[6]

D. Cox, J. Little and D. O'Shea, "Ideals, Varieties and Algorithms,", Springer-Verlag, (1997). Google Scholar

[7]

R. Cushman and J. Sanders, A survey of invariant theory applied to normal forms of vector fields with nilpotent linear part,, in, 19 (1990), 82. Google Scholar

[8]

E. B. Elliot, "An Introduction to the Algebra of Binary Quantics,", $2^{nd}$ edition, (1964). Google Scholar

[9]

M. J. Field, Equivariant dynamical systems,, Trans. Amer. Math. Soc., 259 (1980), 185. doi: 10.1090/S0002-9947-1980-0561832-4. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/j.jde.2008.01.009. Google Scholar

[12]

K. Gatermann, "Computer Algebra Methods for Equivariant Dynamical Systems,", Lecture Notes in Mathematics, 1728 (2000). Google Scholar

[13]

M. Golubitsky, V. G. LeBlanc and I. Melbourne, Meandering of the spiral tip: An alternative approach,, J. Nonlinear Sci., 7 (1997), 557. Google Scholar

[14]

V. G. Guillemin and S. Sternberg, Remarks on a paper of Hermann,, Trans. Amer. Math. Soc., 130 (1968), 110. doi: 10.1090/S0002-9947-1968-0217226-9. Google Scholar

[15]

J. E. Humphreys, "Linear Algebraic Groups,", Springer-Verlag, (1975). Google Scholar

[16]

M. Krupa, Bifurcations of relative equilibria,, SIAM J. Math. Anal., 21 (1990), 1453. doi: 10.1137/0521081. Google Scholar

[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. Google Scholar

[18]

G. I. Lehrer and T. A. Springer, A note concerning fixed points of parabolic subgroups of unitary reflection groups,, Indag. Math., 10 (1999), 549. Google Scholar

[19]

L. Michel, Points critiques des fonctions invariantes sur une $G$-variété,, C.R. Acad. Sc. Paris, 278 (1971), 433. Google Scholar

[20]

D. I. Panyushev, On covariants of reductive algebraic groups,, Indag. Math., 13 (2002), 125. Google Scholar

[21]

V. Poénaru, "Singularités $C^{\infty }$ en Présence de Symétrie,", Lecture Notes in Mathematics, 510 (1976). Google Scholar

[22]

J. Scheurle, Some aspects of successive bifurcations in the Couette-Taylor problem,, Fields Inst. Comm., 5 (1996), 335. Google Scholar

[23]

S. Walcher, On differential equations in normal form,, Math. Ann., 291 (1991), 293. doi: 10.1007/BF01445209. Google Scholar

[24]

S. Walcher, Multi-parameter symmetries of first order ordinary differential equations,, J. Lie Theory, 9 (1999), 249. Google Scholar

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