Two-dimensionalhomogeneous cubic systems: classificationand normal forms—I
DOI:
https://doi.org/10.21638/11701/spbu01.2016.201Abstract
This work is the first in a series of papers devoted to the classification of two-dimensional homogeneous cubic systems based on the partition into classes of linear equivalence. Principles are developed to constructively distinguish the structure of the simplest system in each class and canonical set that defines the permissible values that can take its coefficients. The polynomial vector in the right part of such system identified with (2×4)-matrix is called the canonical form (CF) and the system itself— the normal cubic form. One of the main objectives of the series is to simplify the reduction of a system with a homogeneous cubic polynomial in the unperturbed part to the various structures of the generalized normal form (GNF). Under GNF we mean a system, perturbed part of which is in some sense the simplest form. Constructive implementation of the normalization process depends on the ability to explicitly specify the conditions of compatibility and possible solutions of so-called connective system which is understood as a countable set of linear algebraic equations systems that determine the normalizing transformation of the perturbed system. The principles mentioned above are based on the idea of the greatest possible simplification of the connective system. This will allow to reduce the initial perturbed system by an invertible linear substitution to the system with some CF in the unperturbed part, and then reduce the resulting system, optimal for normalization, by almost identical substitutions to various structures of the GNF. In this paper: 1) the general problem is set, the other problems close to it are formulated with the description of the existing results; 2) the connective system is derived, which determines the equivalence of any two perturbed systems with the same homogeneous cubic part, the possibilities of its simplification are discussed, also the GNF is defined and the method of resonant equations which allows to constructively obtain all of its structures is given; 3) special recording forms of homogeneous cubic systems in the presence of the common homogeneous factor in their right-hand parts having a power from one to three are introduced; the linear equivalence of such systems as well as systems without a common factor is studied; the key linear invariants are offered. Refs 20.Downloads
Download data is not yet available.
References
Литература
1. Басов В.В. Обобщенная нормальная форма и формальная эквивалентность систем дифференциальных уравнений с нулевым приближением (x3, -x3 ) // Дифференц. уравнения. 2004. Т. 40, 2 1 № 8. С. 1011-1022.
2. Белицкий Г.Р. Нормальные формы относительно фильтрующегося действия группы // Труды ММО. 1979. Т. 40. С. 2-46.
3. Kokubu H., Oka H., Wang D. Linear grading function and further redaction of normal forms // J. Diff. Eq. 1996. Vol. 132. P. 293-318.
4. Брюно А.Д., Петрович В.Ю. Нормальные формы системы ОДУ // Препринт ИПМ РАН. 2000. № 18. 24 с.
5. Baider A., Sanders J. Further reduction of the Takens-Bogdanov normal form // J. Diff. Eq. 1992. Vol. 99, N 2, P. 205-244.
6. Басов В.В., Михлин Л.С. Обобщенные нормальные формы систем ОДУ с невозмущенной частью (x2, ±x2n-1 ) // Вестник С-Петерб. ун-та. Сер. 1. 2015. Т. 2 (60), вып. 1. С. 14-22.
7. Vaganyan A.S. Generalized normal forms of infinitesimal symplectic and contact transformations in the neighbourhood of a singular point // arXiv:1212.4947. 2013. URL: http://arxiv.org/abs/1212.4947.
8. Брюно А.Д. Аналитическая форма дифференциальных уравнений // Труды ММО. 1971. Т. 25. С. 119-262; 1972, Т. 26. С. 199-238.
9. Брюно А.Д. Нормальная форма системы Гамильтона // УМН. 1988. Т. 43, вып. 1(259). С. 23-56.
10. Birkhoff G.D. Dynamical Systems // Amer. Math. Soc., Providence, Colloquium Publications. 1927. Vol. 9. P. xii 305.
11. Лычагин В.В. Локальная классификация нелинейных дифференциальных уравнений в частных производных первого порядка // УМН. 1975. Т. 30, вып. 1(181). С. 101-171.
12. Басов В.В., Федорова Е.В. Двумерные вещественные системы ОДУ с квадратичной невозмущенной частью: классификация и вырожденные обобщенные НФ // Дифференц. уравнения и процессы управления. 2010. № 4. С. 49-85. URL: http://www.math.spbu.ru/diffjournal.
13. Басов В.В., Скитович А.В. Обобщенная нормальная форма и формальная эквивалентность двумерных систем с нулевыми квадратичным приближением. I, II // Дифференц. уравнения. 2003. Т. 39, № 8. С. 1016-1029; 2005. Т. 41, № 8. С. 1011-1022.
14. Басов В.В. Обобщенная нормальная форма и формальная эквивалентность двумерных систем с нулевыми квадратичным приближением. III // Дифференц. уравнения. 2006. Т. 42, № 3. С. 308-319.
15. Басов В.В., Федорова Е.В. Обобщенная нормальная форма и формальная эквивалентность двумерных систем с нулевыми квадратичным приближением. IV // Дифференц. уравнения. 2009. Т. 45, № 3. С. 297-313.
16. Сибирский К.С. Введение в алгебраическую теорию инвариантов дифференциальных уравнений. Кишинев: «Штиинца», 1982. 168 с.
17. Басов В.В., Петрова С.Е. Обобщенные нормальные формы систем ОДУ с квадратичнокубической невозмущенной частью // Дифференц. уравнения и процессы управления. 2012. № 2. С. 154-217. URL: http://www.math.spbu.ru/diffjournal.
18. Takens F. Singularities of vector fields // IHES. 1974. Vol. 43, N 2. P. 47-100.
19. Белицкий Г.Р. Нормальные формы формальных рядов и ростков C ∞-отображений относительно действия группы // Изв. АН СССР. Сер. матем. 1976. Т. 40, № 4. С. 855-868.
20. Окунев Л.Я. Высшая алгебра. М.; Л.: Гос. изд-во тех.-теор. лит-ры, 1949. 434 c.
References
1. Basov V.V., “The generalized normal form and formal equivalence of systems of differential equations with zero approximation (x32 ,−x31 )”, Differential Equations 40(8), 1073—1085 (2004).
2. Belickii G.R., “Normal forms in relation to the filtering action of the group”, Trudy Mosk. Mat. Obshch. 40, 2–46 (1979) [in Russian].
3. Kokubu H., Oka H., Wang D., “Linear grading function and further redaction of normal forms”, J. of Differential Equations 132, 293–318 (1996).
4. Bruno A.D., Petrovich V.Yu., “Normal forms of system of ODE”, Preprint Inst. Prikl.Mat. RAN (18), (Moscow, 2000) [in Russian].
5. Baider A., Sanders J., “Further reduction of the Takens—Bogdanov normal form”, Journal of Differential Equations 99(2), 205–244 (1992).
6. Basov V.V., Mikhlin L. S., “Generalized normal forms of ODE systems with unperturbed part (x2,±x2n−1 1 )”, Vestnik of Saint-Petersburg University. Ser. 1 2(60), Issue 1, 14–22 (2015) [in Russian].
7. Vaganyan A. S., “Generalized normal forms of infinitesimal symplectic and contact transformations in the neighbourhood of a singular point”, arXiv:1212.4947 (2013), URL: http://arxiv.org/abs/1212.4947.
8. Bruno A.D., “An analytical form of differential equations”, Trudy Mosk. Mat. Obshch. 25, 119–262 (1971); 26, 199–238 (1972) [in Russian].
9. Bruno A.D., “The normal form of a Hamiltonian system”, Russian Mathematical Surveys 43, Issue 1, 25–66 (1988).
10. Birkhoff G.D., “Dynamical Systems”, Amer.Math. Soc., Providence, Colloquium Publications 9, xii+305 (1927).
11. Lychagin V.V., “Lokal classification of non-lineral first order partial differential equations”, Russian Mathematical Surveys 30, Issue 1, 105–175 (1975).
12. Basov V.V., Fedorova E.V., “Two-dimensional real systems of ODE with quadratic unperturbed parts: classification and degenerate generalized normal forms”, Differential equations and control processes (4), 49–85 (2010). URL: http://www.math.spbu.ru/diffjournal.
13. Basov V.V., Skitovich A.V., “A generalized normal form and formal equivalence of twodimensional systems with quadratic zero approximation: I, II”, Differential Equations 39(8), 1067—1081 (2003); 41(8), 1061—1074 (2005).
14. Basov V.V., “A generalized normal form and formal equivalence of two-dimensional systems with quadratic zero approximation: III”, Differential Equations 42(3), 327—339 (2006).
15. Basov V.V., Fedorova E.V., “A generalized normal form and formal equivalence of twodimensional systems with quadratic zero approximation: IV”, Differential Equations 45(3), 305—322 (2009).
16. Sibirskii K. S., An introduction to the algebraic theory of invariants of differential equations (Izd. Shtiintsa, Kishinev, 1982, 168 p.) [in Russian].
17. Basov V.V., Petrova S. E., “Generalized normal forms of systems of ODE with quadraticcubic unperturbed parts”, Differential equations and control processes (2), 154–217 (2012) URL: http://www.math.spbu.ru/diffjournal [in Russian].
18. Takens F., “Singularities of vector fields”, IHES 43(2), 47–100 (1974).
19. Belickii G.R., “Normal forms for formal series and germs of C∞-mappings with respect to the action of a group”, Mathematics of the USSR-Izvestiya 10(4), 809–821 (1976).
20. Okunev L.Ya., Higher algebra (Gos. izdat. teh.-teor. lit., Moscow, 1949, 434 p.) [in Russian].
Downloads
Published
2020-10-19
How to Cite
Basov, V. V. (2020). Two-dimensionalhomogeneous cubic systems: classificationand normal forms—I. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 3(2), 1. https://doi.org/10.21638/11701/spbu01.2016.201
Issue
Section
Mathematics
License
Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
