Вестник БГУ. Математика, информатика
Библиографическое описание:
СРАВНЕНИЕ ЧИСЛА ПРЕДЕЛЬНЫХ ЦИКЛОВ ОБОБЩЕННЫХ СИСТЕМ ЛЬЕНАРА, ПОЛУЧЕННЫХ РАЗЛИЧНЫМИ МЕТОДАМИ // Вестник БГУ. Математика, информатика. - 2017. №1. . - С. 10-17.
Заглавие:
СРАВНЕНИЕ ЧИСЛА ПРЕДЕЛЬНЫХ ЦИКЛОВ ОБОБЩЕННЫХ СИСТЕМ ЛЬЕНАРА, ПОЛУЧЕННЫХ РАЗЛИЧНЫМИ МЕТОДАМИ
Финансирование:
Коды:
Аннотация:
В статье приводятся данные вычислений наибольшего числа малоамплитудных предельных циклов для двух типов обобщенных полиномиальных систем дифференциальных уравнений, формально совпадающих на некотором множестве частных случаев. Первая методика основана на методе усреднения и авторском алгоритме вычисления ляпуновских величин, вторая – заключалась в исследовании влияния малых возмущений на гамильтонову систему. Вопреки ожиданиям, различные методики в некоторых случаях дают несовпадающие результаты.
Ключевые слова:
предельный цикл, обобщенное уравнение Льенара, 16-я проблема Гильберта, слабый фокус.
Список литературы:
1. J. Llibre, A.C. Merew and M.A. Teixeira. Limit cycles of the generalized polynomial Lienard differential equation. Math. Proc. Camb. Phil. Soc. (2009), 1-21.
2. S. Lynch and C.J. Christopher. Limit cycles in highly non-linear differential equation. J. Sound Vib. 224 (1999), 505-517.
3. Yu. Ilyashenko, A. Panov, Some upper estimates of the number of limit cycles of planar vector fields with application to Lienard equations, Moscow Math. J. vol. 1, no. 4, 583-599.
4. N.G. Lloyd. Limit cycles of polynomial systems-some recent developments. London. Math. Soc. Lecture Note Ser. 127, Cambridge University Press. (1988), 192-234.
5. A. Lienard. Etude des oscillations entrenues. Revue Generale de l’Elictricite.23 (1928), 946-954.
6. G.S. Rychkov. The maximum number of limit cycles of system 3 5 x& = y - a1 x
- a2 x , y& =- x is two. Differential’nye Uravneniya. 11 (1975), 380-391.
7. A. Lins, W. de Melo and C.C. Pugh. On Lienard’s equation. Lecture Notes in math. 597 (1977), Springer, 335-357.
8. W. A. Coppel. Some quadratic systems with at most one limit cycles. Dynam- ics Reportsd. vol. 2 Wiley (1998), 61-68.
9. F. Dumortier and C. Rousseau. Cubic Lienard equations with linear damping. Nonlinearity. 3 (1990), 1015-1039.
10. F. Dumortier and C. Li. On the uniqueness of limit cycles surrounding one or more singularities for Lienard equations. Nonlinearity. 9 (1996), 1489-1500.
11. F. Dumortier and Chengzhi. Quadratic Lienard equations with quadratic damping. J. Diff. Eqs. 139 (1997), 41-59.
12. N. G. Lloyd and S. Lynch. Small-amplitude limit cycles of certain Lienard systems. Pro. Royal. Soc. London Ser. A 418 (1988), 199-208.
13. C. J. Christopher and S. Lynch. Small-amplitude limit cycle bifurcates for Lienard systems with quadratic or cubic damping or restoring forces. Nonlinearity. 12 (1999), 1099-1112.
14. A. Gasuall and J. Torregrosa. Small-amplitude limit cycles in Lienard systems via multiplicity. J. Diff. Eqs. 159 (1998), 1015-1039/
15. P. Yu and M. Han. Limit cycles in generalized Lienard systems. Chaos soli- tons fractals. 30 (2006), 1048-1068.
16. J. Llibre, A.C. Merew and M.A. Teixeira. Limit cycles of the generalized
Lienard polynomial differential equations. Math. Proc. Camb. Phil. Soc. (2009), 1-21.
17. J. Llibre and A. Makhlouf. Limit cycles of a class of generalized Lienard polynomial equations. Math. Proc. Camb. Phil. Soc. (2014), 10-32.
18. A. Lins, W. Demelo and C. C. Pugh. On Lienard’s equation. Lecture Notes in math. 597 (1977), Springer, 335-357.
19. A. M. Urbina, G. L. de la Barra, G. Leon, M. L. de la Barra, M. Canas. Limit cycles of Lienard equations with nonlinear damping. Canad Math Bull. 36 (1993), 251-256.
20. M. Hamanda and A. Maklouf. Limit cycles of class of generalized Lienard polynomial equations. Global Journal of Pure and Applied Mathematics. vol. 12, num- ber 2 (2016), 1831-1843.
21. S. Lynch. Symbolic computation of Lyapunov quantities and the second part of Hilbert’s sixteenth problem. Trends in Mathematics: Differential Equations with Symbolic Computation, (2006), 1-22.
2. S. Lynch and C.J. Christopher. Limit cycles in highly non-linear differential equation. J. Sound Vib. 224 (1999), 505-517.
3. Yu. Ilyashenko, A. Panov, Some upper estimates of the number of limit cycles of planar vector fields with application to Lienard equations, Moscow Math. J. vol. 1, no. 4, 583-599.
4. N.G. Lloyd. Limit cycles of polynomial systems-some recent developments. London. Math. Soc. Lecture Note Ser. 127, Cambridge University Press. (1988), 192-234.
5. A. Lienard. Etude des oscillations entrenues. Revue Generale de l’Elictricite.23 (1928), 946-954.
6. G.S. Rychkov. The maximum number of limit cycles of system 3 5 x& = y - a1 x
- a2 x , y& =- x is two. Differential’nye Uravneniya. 11 (1975), 380-391.
7. A. Lins, W. de Melo and C.C. Pugh. On Lienard’s equation. Lecture Notes in math. 597 (1977), Springer, 335-357.
8. W. A. Coppel. Some quadratic systems with at most one limit cycles. Dynam- ics Reportsd. vol. 2 Wiley (1998), 61-68.
9. F. Dumortier and C. Rousseau. Cubic Lienard equations with linear damping. Nonlinearity. 3 (1990), 1015-1039.
10. F. Dumortier and C. Li. On the uniqueness of limit cycles surrounding one or more singularities for Lienard equations. Nonlinearity. 9 (1996), 1489-1500.
11. F. Dumortier and Chengzhi. Quadratic Lienard equations with quadratic damping. J. Diff. Eqs. 139 (1997), 41-59.
12. N. G. Lloyd and S. Lynch. Small-amplitude limit cycles of certain Lienard systems. Pro. Royal. Soc. London Ser. A 418 (1988), 199-208.
13. C. J. Christopher and S. Lynch. Small-amplitude limit cycle bifurcates for Lienard systems with quadratic or cubic damping or restoring forces. Nonlinearity. 12 (1999), 1099-1112.
14. A. Gasuall and J. Torregrosa. Small-amplitude limit cycles in Lienard systems via multiplicity. J. Diff. Eqs. 159 (1998), 1015-1039/
15. P. Yu and M. Han. Limit cycles in generalized Lienard systems. Chaos soli- tons fractals. 30 (2006), 1048-1068.
16. J. Llibre, A.C. Merew and M.A. Teixeira. Limit cycles of the generalized
Lienard polynomial differential equations. Math. Proc. Camb. Phil. Soc. (2009), 1-21.
17. J. Llibre and A. Makhlouf. Limit cycles of a class of generalized Lienard polynomial equations. Math. Proc. Camb. Phil. Soc. (2014), 10-32.
18. A. Lins, W. Demelo and C. C. Pugh. On Lienard’s equation. Lecture Notes in math. 597 (1977), Springer, 335-357.
19. A. M. Urbina, G. L. de la Barra, G. Leon, M. L. de la Barra, M. Canas. Limit cycles of Lienard equations with nonlinear damping. Canad Math Bull. 36 (1993), 251-256.
20. M. Hamanda and A. Maklouf. Limit cycles of class of generalized Lienard polynomial equations. Global Journal of Pure and Applied Mathematics. vol. 12, num- ber 2 (2016), 1831-1843.
21. S. Lynch. Symbolic computation of Lyapunov quantities and the second part of Hilbert’s sixteenth problem. Trends in Mathematics: Differential Equations with Symbolic Computation, (2006), 1-22.
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