On the convergence of normalizing transformations in the presence of symmetries

Journal of Mathematical Analysis and Applications

Volumn 199, Issue 1, 1996, Pages 243-255

  Cicogna, Giampaolo ^a  

^a UNIVERSITY OF PISA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030122291     PISSN: 0022247X     EISSN: None     Source Type: Journal    
DOI: 10.1006/jmaa.1996.0139     Document Type: Article

References (24)

1. 1. D. Arnal, M. Ben Ammar, and G. Pinczon, The Poincaré-Dulac theorem for nonlinear representations of nilpotent Lie algebras, Lett. Math. Phys. 8 (1984), 467-476.
2. 2. V. I. Arnold, "Geometrical Methods in the Theory of Differential Equations," Springer, Berlin, 1988.
3. 3. V. I. Arnold and Yu. S. Il'yashenko, Ordinary differential equations, in "Encyclopaedia of Mathematical Sciences - Vol. I, Dynamical Systems I" (D. V. Anosov and V. I. Arnold, Eds.), pp. 1-148, Springer, Berlin, 1988.
4. 4. A. Bazzani, E. Todesco, G. Turchetti, and G. Servizi, "A Normal Form Approach to the Theory of Nonlinear Betatronic Motion," CERN Report 94-02, Geneva, 1994.
5. 5. H. W. Broer, "Formal Normal Form Theorems for Vector Fields and Some Consequences for Bifurcations in the Volume Preserving Case," Berlin, Springer, 1981.
6. 6. H. W. Broer and F. Takens, Formally symmetric normal forms and genericity, Dynam. Report. 2 (1989), 39-59.
7. 7. A. D. Bruno, Analytic form of differential equations, Trans. Moscow Math. Soc. 25 (1971), 131-288; and "Local Methods in Nonlinear Differential Equations," Springer, Berlin, 1989.
8. 8. A. D. Bruno, Divergence of the real normalizing transformation, Selecta Math. 12 (1993), 13-23.
9. 9. A. D. Bruno, private communication.
10. 10. A. D. Bruno and S. Walcher, Symmetries and convergence of normalizing transformations, J. Math. Anal. Appl. 183 (1994), 571-576.
11. 11. G. Cicogna and G. Gaeta, Lie point-symmetries and Poincaré normal forms for dynamical systems, J. Phys. A 23 (1990), L799-L802.
12. 12. G. Cicogna and G. Gaeta, Poincaré normal forms and Lie point symmetries, J. Phys. A 27 (1994), 461-476.
13. 13. G. Cicogna and G. Gaeta, Normal forms and nonlinear symmetries, J. Phys. A 27 (1994), 7115-7124.
14. 14. G. Cicogna and G. Gaeta, Approximate symmetries in dynamical systems, Nuovo Cimento B 109 (1994), 989-1008.
15. 15. R. Courant and D. Hubert, "Methods of Mathematical Physics," Interscience, New York, 1962.
16. 16. C. Elphick, E. Tirapegui, M. E. Brachet, P. Coullet, and G. Iooss, Simple global characterization for normal forms of singular vector fields, Phys. D 29 (1987), 95-127.
17. 17. H. Ito, Convergence of Birkhoff normal forms for integrable systems, Comment. Math. Helv. 64 (1989), 412-461.
18. 18. H. Ito, Integrability of Hamiltonian systems and Birkhoff normal forms in the simple resonance case, Math. Ann. 292 (1992), 411-444.
19. 19. L. M. Markhashov, On the reduction of differential equations to the normal form by an analytic transformation, J. Appl. Math. Mech. 38 (1974), 788-790.
20. 20. P. J. Olver, "Applications of Lie Groups to Differential Equations," Springer, Berlin, 1986.
21. 21. L. V. Ovsjannikov, "Group Properties of Differential Equations," Novosibirsk, 1962 (English transl, by G. W. Bluman, 1967); "Group Analysis of Differential Equations," Academic Press, New York, 1982.
22. 21. L. V. Ovsjannikov, "Group Properties of Differential Equations," Novosibirsk, 1962 (English transl, by G. W. Bluman, 1967); "Group Analysis of Differential Equations," Academic Press, New York, 1982.
23. 22. J.-C. van der Meer, "The Hamiltonian Hopf Bifurcation," Springer, Berlin, 1985.
24. 23. S. Walcher, On differential equations in normal form, Math. Ann. 291 (1991), 293-314.