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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, 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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