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Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields
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 |
References:
[1] |
Yu. N. Bibikov, "Local Theory of Nonlinear Analytic Ordinary Differential Equations,", Lecture Notes in Mathematics, 702 (1979).
|
[2] |
D. Birkes, Orbits of linear algebraic groups,, Ann. Math., 93 (1971), 459.
doi: 10.2307/1970884. |
[3] |
A. Borel, "Linear Algebraic Groups,", $2^{nd}$ edition, (1992).
|
[4] |
T. Bröcker and T. tom Dieck, "Representations of Compact Lie Groups,", Springer-Verlag, (1985).
|
[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. |
[6] |
D. Cox, J. Little and D. O'Shea, "Ideals, Varieties and Algorithms,", Springer-Verlag, (1997).
|
[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.
|
[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. |
[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.
|
[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. |
[12] |
K. Gatermann, "Computer Algebra Methods for Equivariant Dynamical Systems,", Lecture Notes in Mathematics, 1728 (2000).
|
[13] |
M. Golubitsky, V. G. LeBlanc and I. Melbourne, Meandering of the spiral tip: An alternative approach,, J. Nonlinear Sci., 7 (1997), 557.
|
[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. |
[15] |
J. E. Humphreys, "Linear Algebraic Groups,", Springer-Verlag, (1975).
|
[16] |
M. Krupa, Bifurcations of relative equilibria,, SIAM J. Math. Anal., 21 (1990), 1453.
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.
|
[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.
|
[19] |
L. Michel, Points critiques des fonctions invariantes sur une $G$-variété,, C.R. Acad. Sc. Paris, 278 (1971), 433.
|
[20] |
D. I. Panyushev, On covariants of reductive algebraic groups,, Indag. Math., 13 (2002), 125.
|
[21] |
V. Poénaru, "Singularités $C^{\infty }$ en Présence de Symétrie,", Lecture Notes in Mathematics, 510 (1976).
|
[22] |
J. Scheurle, Some aspects of successive bifurcations in the Couette-Taylor problem,, Fields Inst. Comm., 5 (1996), 335.
|
[23] |
S. Walcher, On differential equations in normal form,, Math. Ann., 291 (1991), 293.
doi: 10.1007/BF01445209. |
[24] |
S. Walcher, Multi-parameter symmetries of first order ordinary differential equations,, J. Lie Theory, 9 (1999), 249.
|
show all references
References:
[1] |
Yu. N. Bibikov, "Local Theory of Nonlinear Analytic Ordinary Differential Equations,", Lecture Notes in Mathematics, 702 (1979).
|
[2] |
D. Birkes, Orbits of linear algebraic groups,, Ann. Math., 93 (1971), 459.
doi: 10.2307/1970884. |
[3] |
A. Borel, "Linear Algebraic Groups,", $2^{nd}$ edition, (1992).
|
[4] |
T. Bröcker and T. tom Dieck, "Representations of Compact Lie Groups,", Springer-Verlag, (1985).
|
[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. |
[6] |
D. Cox, J. Little and D. O'Shea, "Ideals, Varieties and Algorithms,", Springer-Verlag, (1997).
|
[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.
|
[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. |
[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.
|
[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. |
[12] |
K. Gatermann, "Computer Algebra Methods for Equivariant Dynamical Systems,", Lecture Notes in Mathematics, 1728 (2000).
|
[13] |
M. Golubitsky, V. G. LeBlanc and I. Melbourne, Meandering of the spiral tip: An alternative approach,, J. Nonlinear Sci., 7 (1997), 557.
|
[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. |
[15] |
J. E. Humphreys, "Linear Algebraic Groups,", Springer-Verlag, (1975).
|
[16] |
M. Krupa, Bifurcations of relative equilibria,, SIAM J. Math. Anal., 21 (1990), 1453.
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.
|
[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.
|
[19] |
L. Michel, Points critiques des fonctions invariantes sur une $G$-variété,, C.R. Acad. Sc. Paris, 278 (1971), 433.
|
[20] |
D. I. Panyushev, On covariants of reductive algebraic groups,, Indag. Math., 13 (2002), 125.
|
[21] |
V. Poénaru, "Singularités $C^{\infty }$ en Présence de Symétrie,", Lecture Notes in Mathematics, 510 (1976).
|
[22] |
J. Scheurle, Some aspects of successive bifurcations in the Couette-Taylor problem,, Fields Inst. Comm., 5 (1996), 335.
|
[23] |
S. Walcher, On differential equations in normal form,, Math. Ann., 291 (1991), 293.
doi: 10.1007/BF01445209. |
[24] |
S. Walcher, Multi-parameter symmetries of first order ordinary differential equations,, J. Lie Theory, 9 (1999), 249.
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