Artículos de revistas
Limit Cycles Of The Generalized Polynomial Liénard Differential Equations
Registro en:
Mathematical Proceedings Of The Cambridge Philosophical Society. , v. 148, n. 2, p. 363 - 383, 2010.
3050041
10.1017/S0305004109990193
2-s2.0-77952547108
Autor
Llibre J.
Mereu A.C.
Teixeira M.A.
Institución
Resumen
We apply the averaging theory of first, second and third order to the class of generalized polynomial Liénard differential equations. Our main result shows that for any n, m ≥ 1 there are differential equations of the form ẍ + f(x)ẋ + g(x) = 0, with f and g polynomials of degree n and m respectively, having at least [(n + m 1)/2] limit cycles, where [·] denotes the integer part function. © 2009 Cambridge Philosophical Society. 148 2 363 383 Blows, T.R., Lloyd, N.G., The number of small-amplitude limit cycles of Líenard equations (1984) Math. Proc. Camb. Phil. Soc., 95, pp. 359-366 Buicǎ, A., Llibre, J., Averaging methods for finding periodic orbits via Brouwer degree (2004) Bull. Sci. Math., 128, pp. 7-22 Christopher, C.J., Lynch, S., Small-amplitude limti cycle bifurcations for Líenard systems with quadratic or cubic damping or restoring forces (1999) Nonlinearity, 12, pp. 1099-1112 Coppel, W.A., Some quadratic systems with at most one limit cycles (1998) Dynamics Reported, 2, pp. 61-68. , Wiley Dumortier, F., Panazzolo, D., Roussarie, R., More limit cycles than expected in Líenard systems (2007) Proc. Amer. Math. Soc., 135, pp. 1895-1904 Dumortier, F., Li, C., On the uniqueness of limit cycles surrounding one or more singularities for Líenard equations (1996) Nonlinearity, 9, pp. 1489-1500 Dumortier, F., Li, C., Quadratic Líenard equations with quadratic damping (1997) J. Diff. Eqs., 139, pp. 41-59 Dumortier, F., Rousseau, C., Cubic Líenard equations with linear dampimg (1990) Nonlinearity, 3, pp. 1015-1039 Gasull, A., Torregrosa, J., Small-amplitude limit cycles in Líenard systems via multiplicity (1998) J. Diff. Eqs., 159, pp. 1015-1039 Ilyashenko, Y., Centennial history of Hilbert's 16th problem (2002) Bull. Amer. Math. Soc., 39, pp. 301-354 Jibin, L.I., Hilbert's 16th problem and bifurcations of planar polynomial vector fields (2003) Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13, pp. 47-106 Líenard, A., Étude des oscillations entrenues (1928) Revue Génerale de l' Électricité, 23, pp. 946-954 Lins, A., De Melo, W., Pugh, C.C., On Líenard's equation (1977) Lecture Notes in Math, 597, pp. 335-357. , Springer Lloyd, N.G., Limit cycles of polynomial systems-some recent developments (1988) London Math. Soc. Lecture Note Ser., 127, pp. 192-234. , Cambridge University Press Lloyd, N.G., Lynch, S., Small-amplitude limit cycles of certain Líenard systems (1988) Proc. Royal Soc. London Ser. A, 418, pp. 199-208 Lloyd, N., Pearson, J., Symmetric in planar dynamical systems (2002) J. Symb. Comput., 33, pp. 357-366 Lynch, S., Limit cycles if generalized Líenard equations (1995) Appl. Math. Lett., 8, pp. 15-17 Lynch, S., Generalized quadratic Líenard equations (1998) Appl. Math. Lett., 11, pp. 7-10 Lynch, S., Generalized cubic Líenard equations (1999) Appl. Math. Lett., 12, pp. 1-6 Lynch, S., Christopher, C.J., Limit cycles in highly non-linear differential equations (1999) J. Sound Vib., 224, pp. 505-517 Rychkov, G.S., The maximum number of limit cycle of the system ẋ = y - A 1x3 - a2x5 , ẏ = -x is two (1975) Differential'Nye Uravneniya, 11, pp. 380-391 Smale, S., Mathematical problems for the next century (1998) Math. Intelligencer, 20, pp. 7-15 Yu, P., Han, M., Limit cycles in generalized Líenard systems (2006) Chaos Solitons Fractals, 30, pp. 1048-1068