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 (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

[1]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[2]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381

[3]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012

[4]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[5]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[6]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[7]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375

[8]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[9]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121

[10]

Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020379

[11]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[12]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215

[13]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103

[14]

Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020

[15]

Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011

[16]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

[17]

Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020349

[18]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

[19]

Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020460

[20]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378

2019 Impact Factor: 0.649

Metrics

  • PDF downloads (30)
  • HTML views (0)
  • Cited by (4)

[Back to Top]