Theory and applications of high codimension bifurcations

The study of bifurcation of high codimension singularities and cyclicity of related limit periodic sets has a long history and is essential in the theory and applications of differential equations and dynamical systems. It is also closely related to the second part of Hilbert's 16th problem. In 1994, Dumortier, Roussarie and Rousseau launched a program aiming at proving the finiteness part of Hilbert's 16th problem for the quadratic vector fields. For the program, 125 graphics need to be proved to have finite cyclicity. Since the launch of the program, most graphics have been proved to have finite cyclicity, and there are 40 challenging cases left. Among the rest of the graphics, there are 4 families of HH-graphics with a triple nilpotent singularity of saddle or elliptic type. Based on the work of Zhu and Rousseau, by using techniques including the normal form theory, global blow-up techniques, calculations and analytical properties of Dulac maps near the singular point of the blown-up sphere, properties of quadratic systems and the generalized derivation-division methods, we prove that these 4 families of HH-graphics (I1/12 ), (I1/13), (I1/9b) and (I1/11b) have finite cyclicity. Finishing the proof of the cyclicity of these 4 families of HH-graphics represents one important step towards the proof of the finiteness part of Hilbert's 16th problem for quadratic vector fields

Finite cyclicity of graphics through a nilpotent singularity of elliptic or saddle type

Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal

Limit Cycles Of The Generalized Polynomial Liénard Differential Equations

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.1482363383Blows, T.R., Lloyd, N.G., The number of small-amplitude limit cycles of Líenard equations (1984) Math. Proc. Camb. Phil. Soc., 95, pp. 359-366Buicǎ, A., Llibre, J., Averaging methods for finding periodic orbits via Brouwer degree (2004) Bull. Sci. Math., 128, pp. 7-22Christopher, C.J., Lynch, S., Small-amplitude limti cycle bifurcations for Líenard systems with quadratic or cubic damping or restoring forces (1999) Nonlinearity, 12, pp. 1099-1112Coppel, W.A., Some quadratic systems with at most one limit cycles (1998) Dynamics Reported, 2, pp. 61-68. , WileyDumortier, F., Panazzolo, D., Roussarie, R., More limit cycles than expected in Líenard systems (2007) Proc. Amer. Math. Soc., 135, pp. 1895-1904Dumortier, F., Li, C., On the uniqueness of limit cycles surrounding one or more singularities for Líenard equations (1996) Nonlinearity, 9, pp. 1489-1500Dumortier, F., Li, C., Quadratic Líenard equations with quadratic damping (1997) J. Diff. Eqs., 139, pp. 41-59Dumortier, F., Rousseau, C., Cubic Líenard equations with linear dampimg (1990) Nonlinearity, 3, pp. 1015-1039Gasull, A., Torregrosa, J., Small-amplitude limit cycles in Líenard systems via multiplicity (1998) J. Diff. Eqs., 159, pp. 1015-1039Ilyashenko, Y., Centennial history of Hilbert's 16th problem (2002) Bull. Amer. Math. Soc., 39, pp. 301-354Jibin, L.I., Hilbert's 16th problem and bifurcations of planar polynomial vector fields (2003) Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13, pp. 47-106Líenard, A., Étude des oscillations entrenues (1928) Revue Génerale de l' Électricité, 23, pp. 946-954Lins, A., De Melo, W., Pugh, C.C., On Líenard's equation (1977) Lecture Notes in Math, 597, pp. 335-357. , SpringerLloyd, N.G., Limit cycles of polynomial systems-some recent developments (1988) London Math. Soc. Lecture Note Ser., 127, pp. 192-234. , Cambridge University PressLloyd, N.G., Lynch, S., Small-amplitude limit cycles of certain Líenard systems (1988) Proc. Royal Soc. London Ser. A, 418, pp. 199-208Lloyd, N., Pearson, J., Symmetric in planar dynamical systems (2002) J. Symb. Comput., 33, pp. 357-366Lynch, S., Limit cycles if generalized Líenard equations (1995) Appl. Math. Lett., 8, pp. 15-17Lynch, S., Generalized quadratic Líenard equations (1998) Appl. Math. Lett., 11, pp. 7-10Lynch, S., Generalized cubic Líenard equations (1999) Appl. Math. Lett., 12, pp. 1-6Lynch, S., Christopher, C.J., Limit cycles in highly non-linear differential equations (1999) J. Sound Vib., 224, pp. 505-517Rychkov, 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. 380-391Smale, S., Mathematical problems for the next century (1998) Math. Intelligencer, 20, pp. 7-15Yu, P., Han, M., Limit cycles in generalized Líenard systems (2006) Chaos Solitons Fractals, 30, pp. 1048-106