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Normal forms in a cyclically graded Lie algebra. (English) Zbl 1132.37020

An extension of those Lie series methods developed by the present authors is used. Those extended methods make use of spectral sequences and are developed in considerable generality, i.e., their applicability is by no means to the Hamiltonian context of this paper. Normal forms are computed directly from spectral sequences. Also, the relationship of the present authors’ approach to the work of V. I. Arnol’d [Sel. Math. Sov. 1, 3–18 (1981); translation from Zadachy Mekh. Mat. Fiz., Moscow, 7–20 (1976; Zbl 0472.58006)], J. Sanders [J. Differ. Equations 192, No. 2, 536–552 (2003; Zbl 1039.34032) and Acta Appl. Math. 87, No. 1–3, 165–189 (2005; Zbl 1077.34043)] and J. Murdock [J. Differ. Equations 205, No. 2, 424–465 (2004; Zbl 1062.34042)] is discussed. In particular, the results can be derived using more classical methods, although one must still extend A. Baider’s ideas [J. Differ. Equations 78, No. 1, 33–52 (1989; Zbl 0689.70005)]. A specific example is worked out in detail.

MSC:

37G05 Normal forms for dynamical systems
17B80 Applications of Lie algebras and superalgebras to integrable systems
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
70K45 Normal forms for nonlinear problems in mechanics
17B81 Applications of Lie (super)algebras to physics, etc.
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References:

[1] Arnold, V. I., (Spectral Sequences for the Reduction of Functions to Normal Form. Spectral Sequences for the Reduction of Functions to Normal Form, Problems in Mechanics and Mathematical Physics, vol. 297 (1976), Izdat. Nauka: Izdat. Nauka Moscow), 7-20, (in Russian)
[2] Baider, A., Unique normal forms for vector fields and Hamiltonians, J. Differential Equations, 78, 33-52 (1989) · Zbl 0689.70005
[3] Bendersky, M., Churchill, R.C., A spectral sequence approach to normal forms. In: Adem, A., Pastor, G., Gonzalez, J. (Eds.), Recent Developments in Algebraic Topology. In: Contemporary Mathematics. Amer. Math. Soc., Providence, RI (in press); Bendersky, M., Churchill, R.C., A spectral sequence approach to normal forms. In: Adem, A., Pastor, G., Gonzalez, J. (Eds.), Recent Developments in Algebraic Topology. In: Contemporary Mathematics. Amer. Math. Soc., Providence, RI (in press) · Zbl 1112.55015
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[8] Murdock, J., Hypernormal form theory: Foundations and algorithms, J. Differential Equations, 205, 424-465 (2004) · Zbl 1062.34042
[9] Sanders, J., Normal form theory and spectral sequences, J. Differential Equations, 192, 536-552 (2003) · Zbl 1039.34032
[10] Sanders, J., Normal form in filtered Lie algebra representations, Acta Appl. Math., 87, 165-189 (2005) · Zbl 1077.34043
[11] Williamson, J., On the algebraic problem concerning the normal form of linear dynamical systems, Amer. J. Math., 58, 141-163 (1936) · JFM 63.1290.01
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