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On convergent normal from transformations in presence of symmetries. (English) Zbl 0959.34030

The author proves a convergence criterion for transformations to (Poincaré-Dulac) normal form, which is a generalization of a theorem on the convergence of normalizing transformations in the presence of symmetries due to G. Cicogna [J. Math. Anal. Appl. 199, No. 1, 243-255 (1996; Zbl 0855.34042)]. This last theorem extends results of L. M. Markhashov [J. Appl. Math. Mech. 38, 738-740 (1975; Zbl 0359.34006)] and A. D. Bruno and S. Walcher [J. Math. Anal. Appl. 183, No. 3, 571-576 (1994; Zbl 0804.34040)]. On the other hand, the author also shows that the imposed technical hypotheses allow to carry out a computational verification in several classes of vector fields.

MSC:

34C14 Symmetries, invariants of ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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[1] Bambusi, D.; Cicogna, G.; Gaeta, G.; Marmo, G., Normal forms, symmetry and linearization of dynamical systems, J. Phys. A, 31, 5065-5082 (1998) · Zbl 0969.34031
[2] Bruno, A. D., Analytical form of differential equations, Trans. Moscow Math. Soc., 25, 131-288 (1971) · Zbl 0272.34018
[3] Bruno, A. D., Local Methods in Nonlinear Differential Equations (1989), Springer-Verlag: Springer-Verlag Berlin
[4] Bruno, A. D.; Walcher, S., Symmetries and convergence of normalizing transformations, J. Math. Anal. Appl., 183, 571-576 (1994) · Zbl 0804.34040
[5] Cicogna, G., On the convergence of normalizing transformations in the presence of symmetries, J. Math. Anal. Appl., 199, 243-255 (1996) · Zbl 0855.34042
[6] Cicogna, G., Convergent normal forms of symmetric dynamical systems, J. Phys. A, 30, 6021-6028 (1997) · Zbl 0932.37042
[7] G. Cicogna, and, G. Gaeta, Nonlinear symmetries and normal forms, in, SPT 98—Symmetry and Perturbation Theory II, (, A. Degasperis and G. Gaeta, Eds.), World Scientific, to appear.; G. Cicogna, and, G. Gaeta, Nonlinear symmetries and normal forms, in, SPT 98—Symmetry and Perturbation Theory II, (, A. Degasperis and G. Gaeta, Eds.), World Scientific, to appear. · Zbl 0981.34025
[8] Gramchev, T.; Yoshino, M., Rapidly convergent iteration method for simultaneous normal form of commuting maps, Math. Z., 331, 745-770 (1999) · Zbl 0931.65055
[9] Ito, H., Convergence of Birkhoff normal forms for integrable systems, Comment. Math. Helv., 64, 412-461 (1989) · Zbl 0686.58021
[10] Ito, H., Integrability of Hamiltonian systems and Birkhoff normal forms in the simple resonance case, Math. Ann., 292, 411-444 (1992) · Zbl 0735.58022
[11] Markhashov, L. M., On the reduction of differential equations to the normal form by an analytic transformation, J. Appl. Math. Mech., 38, 788-790 (1974) · Zbl 0359.34006
[12] Pliss, V. A., On the reduction of an analytic system of differential equations to linear form, Differential Equations, 1, 153-161 (1965)
[13] Siegel, C. L., Über die Normalform analytischer Differentialgleichungen in der Nähe einer Gleichgewichtslösung, Nachr. Akad. Wiss. Göttingen, 21-30 (1952) · Zbl 0047.32901
[14] Walcher, S., Algebras and Differential Equations (1991), Hadronic Press: Hadronic Press Palm Harbor · Zbl 0791.17002
[15] Walcher, S., On differential equations in normal form, Math. Ann., 291, 293-314 (1991) · Zbl 0754.34032
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