Normal forms

Mathematics and Computers in Simulation

Volumn 45, Issue 5-6, 1998, Pages 413-427

  Bruno, A D ^a  

^a KELDYSH INSTITUTE OF APPLIED MATHEMATICS   (Russian Federation)

Author keywords

Belitskii normal form; Computation; Ordinary differential equations; Resonant normal form

Indexed keywords

CALCULATIONS; MATRIX ALGEBRA; PROBLEM SOLVING; SET THEORY; VECTORS;

BELITSKII NORMAL FORM; HENON-HEILES SYSTEM; NORMAL FORMS; ORDINARY DIFFERENTIAL EQUATIONS; RESONANT NORMAL FORM;

DIFFERENTIAL EQUATIONS;

EID: 0032021814     PISSN: 03784754     EISSN: None     Source Type: Journal    
DOI: 10.1016/s0378-4754(97)00120-1     Document Type: Article

References (56)

1. V.I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer, Berlin, 1988.
2. G.R. Beltskii, The normal form of local mappings, Uspekhi Matem. Nauk 30(1) (1975) 223 (Russian).
3. G.R. Belitskii, Equivalence and normal form of germs of smooth mappings, Uspekhi Matem. Nauk 33(1) (1978) 95-155 (Russian); Russian Math. Surveys 33(1) (1978) 107-177 (English).
4. G.R. Belitskii, Equivalence and normal form of germs of smooth mappings, Uspekhi Matem. Nauk 33(1) (1978) 95-155 (Russian); Russian Math. Surveys 33(1) (1978) 107-177 (English).
5. G.R. Belitskii, Normal Forms, Invariants and Local Mappings, Naukova Dumka, Kiev, 1979 (Russian).
6. G.R. Belitskii, Normal forms relative to a filtering action of a group, Trudy Mosk. Matem. Ob-va 40 (1979) 3-46 (Russian); Trans. Mosc. Math. Soc. 40 (1981) 1-39 (English).
7. G.R. Belitskii, Normal forms relative to a filtering action of a group, Trudy Mosk. Matem. Ob-va 40 (1979) 3-46 (Russian); Trans. Mosc. Math. Soc. 40 (1981) 1-39 (English).
8. R.I. Bogdanov, Reduction to orbital normal form of a vector field on the plane, Funkts. Anal. i Prilozh. 10(1) (1976) 73-74 (Russian); Funct. Anal. Appl. 10 (1976) 61-62 (English).
9. R.I. Bogdanov, Reduction to orbital normal form of a vector field on the plane, Funkts. Anal. i Prilozh. 10(1) (1976) 73- 74 (Russian); Funct. Anal. Appl. 10 (1976) 61-62 (English).
10. A.D. Bruno, Normal form of differential equations, Dokl. Akad. Nauk SSSR 157(6) (1964) 1276-1279 (Russian); Soviet Math. Dokl. 5 (1964) 1105-1108 (English).
11. A.D. Bruno, Normal form of differential equations, Dokl. Akad. Nauk SSSR 157(6) (1964) 1276-1279 (Russian); Soviet Math. Dokl. 5 (1964) 1105-1108 (English).
12. A.D. Bruno, Analytical form of differential equations, Trudy Mosk. Matem. Obsc. 25 (1971) 119-262; 26 (1972) 199- 239 (Russian); Trans. Moscow Math. Soc. 25 (1971) 131-288; 26 (1972) 199-239 (English).
13. A.D. Bruno, Analytical form of differential equations, Trudy Mosk. Matem. Obsc. 25 (1971) 119-262; 26 (1972) 199-239 (Russian); Trans. Moscow Math. Soc. 25 (1971) 131-288; 26 (1972) 199-239 (English).
14. A.D. Bruno, Analytical form of differential equations, Trudy Mosk. Matem. Obsc. 25 (1971) 119-262; 26 (1972) 199- 239 (Russian); Trans. Moscow Math. Soc. 25 (1971) 131-288; 26 (1972) 199-239 (English).
15. A.D. Bruno, Analytical form of differential equations, Trudy Mosk. Matem. Obsc. 25 (1971) 119-262; 26 (1972) 199- 239 (Russian); Trans. Moscow Math. Soc. 25 (1971) 131-288; 26 (1972) 199-239 (English).
16. A.D. Bruno, Local Method of Nonlinear Analysis of Differential Equations, Nauka, Moscow, 1979 (Russian); Local Methods in Nonlinear Differential Equations, Springer, Berlin, 1989 (English).
17. A.D. Bruno, Local Method of Nonlinear Analysis of Differential Equations, Nauka, Moscow, 1979 (Russian); Local Methods in Nonlinear Differential Equations, Springer, Berlin, 1989 (English).
18. A.D. Burno, The normal form of a Hamiltonian system, Uspehi Mat. Nauk 43(1) (1988) 23-56 (Russian); Russian Math. Surveys 43(1) (1988) 25-66 (English).
19. A.D. Burno, The normal form of a Hamiltonian system, Uspehi Mat. Nauk 43(1) (1988) 23-56 (Russian); Russian Math. Surveys 43(1) (1988) 25-66 (English).
20. A.D. Burno, Normalizations of a Hamiltonian system near an invariant cycle or torus, Uspehi Mat. Nauk 44(2) (1989) 49-78 (Russian); Russian Math. Surveys 44(2) (1989) 53-89 (English).
21. A.D. Burno, Normalizations of a Hamiltonian system near an invariant cycle or torus, Uspehi Mat. Nauk 44(2) (1989) 49-78 (Russian); Russian Math. Surveys 44(2) (1989) 53-89 (English).
22. A.D. Burno, The Restricted 3-Body Problem, Nauka, Moscow, 1990 (Russian); Walter de Gruyter, Berlin, 1994 (English).
23. A.D. Burno, The Restricted 3-Body Problem, Nauka, Moscow, 1990 (Russian); Walter de Gruyter, Berlin, 1994 (English).
24. A.D. Burno, System similar to a normal form, Mat. Zametki 48(3) (1990) 20-31 (Russian); Math. Notes 48(3) (1991) 885-893 (English).
25. A.D. Burno, System similar to a normal form, Mat. Zametki 48(3) (1990) 20-31 (Russian); Math. Notes 48(3) (1991) 885-893 (English).
26. A.D. Burno, The normal form of a system, close to a Hamiltonian system, Ibid. 48(5) (1990) 35-46 (Russian); Ibid. 48(5/ 6) (1991) 1100-1108 (English).
27. A.D. Burno, The normal form of a system, close to a Hamiltonian system, Ibid. 48(5) (1990) 35-46 (Russian); Ibid. 48(5/ 6) (1991) 1100-1108 (English).
28. A.D. Burno, Bifurcation of the periodic solutions in the symmetric case of a multiple pair of imaginary eigenvalues, in: S.S. Filippov (Ed.), Numerical Solution of Ordinary Differential Equations, pp. 161-176, Inst. Appl. Math., Moscow, 1988 (Russian); Selecta Math. Formerly Sovietica 12(1) (1993) 1-12 (English).
29. A.D. Burno, Bifurcation of the periodic solutions in the symmetric case of a multiple pair of imaginary eigenvalues, in: S.S. Filippov (Ed.), Numerical Solution of Ordinary Differential Equations, pp. 161-176, Inst. Appl. Math., Moscow, 1988 (Russian); Selecta Math. Formerly Sovietica 12(1) (1993) 1-12 (English).
30. A.D. Burno, Ways of computing a normal form, Dokl. Akad. Nauk 344(3) (1995) 298-300 (Russian); Russian Ac. Sci. Doklady. Math. 52 (1995) 200-202 (English).
31. A.D. Burno, Ways of computing a normal form, Dokl. Akad. Nauk 344(3) (1995) 298-300 (Russian); Russian Ac. Sci. Doklady. Math. 52 (1995) 200-202 (English).
32. A.D. Bruno, S.Yu Sadov, Formal integral of a divergence free system, Matem. Zametki 57(6) (1995) 803-813 (Russian); Math. Notes 57(3) (1995) 565-572 (English).
33. A.D. Bruno, S.Yu Sadov, Formal integral of a divergence free system, Matem. Zametki 57(6) (1995) 803-813 (Russian); Math. Notes 57(3) (1995) 565-572 (English).
34. A.D. Bruno, A. Soleev, Local analysis of a singularity of a reversible system of ordinary differential equations, Preprints 40, 47, 54, Inst. of Applied Mathematics, Moscow, 1995 (Russian).
35. A.D. Bruno, A. Soleev, Local analysis of singularities of a reversible system of ordinary differential equations, Uspehi Mat. Nauk 50(6) (1995) 169-170 (Russian); Russian Math. Surveys 50(6) (1995) 1258-1259 (English).
36. A.D. Bruno, A. Soleev, Local analysis of singularities of a reversible system of ordinary differential equations, Uspehi Mat. Nauk 50(6) (1995) 169-170 (Russian); Russian Math. Surveys 50(6) (1995) 1258-1259 (English).
37. A.D. Bruno, A. Soleev, Bifurcations of solutions in a reversible system of ordinary differential equations, Doklady Akademii Nauk 345(5) (1995) 590-592 (Russian); Russian Acad. Sci. Doklady. Mathematics 52(3) (1995) 419-421 (English).
38. A.D. Bruno, A. Soleev, Bifurcations of solutions in a reversible system of ordinary differential equations, Doklady Akademii Nauk 345(5) (1995) 590-592 (Russian); Russian Acad. Sci. Doklady. Mathematics 52(3) (1995) 419-421 (English).
39. R. Cushman, J.A. Sanders, Nilpotent normal form and representation theory of s1(2,R), Rapport No. 301, Vrije Universiteit, Amsterdam, 1985.
40. R. Cushman, J.A. Sanders, A survey of invariant theory applied to normal forms of vector fields with nilpotent linear part, in: D. Stanton (Ed.), Invariant Theory and Tableaux, IMA 19, Springer, New York, 1990, 82-106.
41. H. Dulac, Solution d'une systeme d'equations differentielles dans le voisinage des valeurs singulieres, Bull. Soc. Math. France 40 (1912) 324-384.
42. C. Elphick, E. Tirapegui, M.E. Brachet, P. Coullet, G. Iooss, A simple global characterization for normal forms of singular vector fields, Physica 29D (1987) 95-127.
43. G. Iooss, M. Adelmeyer, Topics in Bifurcation Theory and Applications, World Scientific, Singapore, 1992.
44. G. Iooss, A codimension 2 bifurcation for reversible vector fields, Fields Institute Communication 4 (1995) 201-217.
45. J.E. Marsden, M. Mccracken, The Hopf Bifurcation and Its Applications, Springer, New York, 1976 (English); Mir, Moscow, 1980 (Russian).
46. H. Poincare, Sur les proprietes des fonctions definies par les equations aux differences partielles, These, Paris 1879; reprinted in: Oeuvres de Henri Poincare, Vol. 1, Gauthier-Villars, Paris, 1928, 49-129.
47. H. Poincare, Sur les proprietes des fonctions definies par les equations aux differences partielles, These, Paris 1879; reprinted in: Oeuvres de Henri Poincare, Vol. 1, Gauthier-Villars, Paris, 1928, 49-129.
48. S. Yu. Sadov, On a dynamic system arising from a finite-dimensional approximation of the Schrödinger equation, Matem. Zametki 56(3) (1994) 118-133 (Russian); Mathematical Notes 56(3) (1994) 960-971 (English).
49. S. Yu. Sadov, On a dynamic system arising from a finite-dimensional approximation of the Schrödinger equation, Matem. Zametki 56(3) (1994) 118-133 (Russian); Mathematical Notes 56(3) (1994) 960-971 (English).
50. A.P. Sadovskii, Problem of distinguishing a center and a focus for a system with nonvanishing linear part, Differ. Uravnenija 12 (1976) 1238-1246 (Russian); Differ. Equations 12 (1976) 874-879 (English).
51. A.P. Sadovskii, Problem of distinguishing a center and a focus for a system with nonvanishing linear part, Differ. Uravnenija 12 (1976) 1238-1246 (Russian); Differ. Equations 12 (1976) 874-879 (English).
52. A.P. Sadovskii, Normal forms of systems of differential equations with a nonzero linear part, Differ. Uravnenija 18(5) (1982) 827-832 (Russian); Differ. Equations 18(5) (1982) (English).
53. A.P. Sadovskii, Normal forms of systems of differential equations with a nonzero linear part, Differ. Uravnenija 18(5) (1982) 827-832 (Russian); Differ. Equations 18(5) (1982) (English).
54. A. Soleev, A. Aranson, The first approximation of a reversible system of ordinary differential equations, Preprint 28, Inst. of Applied Mathematics, Moscow, 1995 (Russian).
55. F. Gustavson, On constructing formal integrals of a Hamiltonian system near an equillibrium point, The Astronomical Journal 71(8) (1966) 670-686.
56. M. Henon, C. Heiles, The applicability of the third integral of motion, some numerical experiments, The Astronomical Journal 69(1) (1964) 73-79.