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Generating limit cycles from a nilpotent critical point via normal forms. (English) Zbl 1100.34030
The authors develop a normal form theory which is applied to solve the center-focus problem for monodromic planar nilpotent singularities to generate limit cycles from this type of singularities.

MSC:
34C23Bifurcation (ODE)
34C05Location of integral curves, singular points, limit cycles (ODE)
37G05Normal forms
37G10Bifurcations of singular points
37G15Bifurcations of limit cycles and periodic orbits
34C20Transformation and reduction of ODE and systems, normal forms
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Full Text: DOI
References:
[1] M.J. Álvarez, A. Gasull, Monodromy and stability for nilpotent critical points, Internat. J. Bifur. Chaos, 2005, in press · Zbl 1088.34021
[2] Andreev, A. F.: Investigation of the behavior of the integral curves of a system of two differential equations in the neighbourhood of a singular point. Transl. amer. Math. soc. 8, 183-207 (1958) · Zbl 0079.11301
[3] Andreev, A. F.; Sadovskii, A. P.; Tsikalyuk, V. A.: The center -- focus problem for a system with homogeneous nonlinearities in the case of zero eigenvalues of the linear part. Differential equations 39, 155-164 (2003) · Zbl 1067.34030
[4] Andronov, A. A.; Leontovich, E. A.; Gordon, I. I.; Maier, A. G.: Theory of bifurcations of dynamical systems on a plane. (1973) · Zbl 0282.34022
[5] Chavarriga, J.; Giacomini, H.; Gine, J.; Llibre, J.: Local analytic integrability for nilpotent centers. Ergodic theory dynam. Systems 23, 417-428 (2003)
[6] Cima, A.; Gasull, A.; Mañosas, F.: Cyclicity of a family of vector fields. J. math. Anal. appl. 196, 921-937 (1995) · Zbl 0851.34027
[7] Farr, W. W.; Li, C.; Labouriau, I. S.; Langford, W. F.: Degenerate Hopf bifurcation formulas and Hilbert’s 16th problem. SIAM J. Math. anal. 20, 13-30 (1989) · Zbl 0682.58035
[8] Gasull, A.; Torregrosa, J.: Center problem for several differential equations via cherkas’ method. J. math. Anal. appl. 228, 322-343 (1998) · Zbl 0926.34022
[9] Jin, X. F.; Wang, D. M.: On the conditions of kukles for the existence of a centre. Bull. London math. Soc. 22, 1-4 (1990) · Zbl 0692.34027
[10] Lyapunov, A. M.: Stability of motion. Math. sci. Engrg. 30 (1966) · Zbl 0161.06303
[11] Moussu, R.: Symétrie et forme normale des centres et foyers dégénérés. Ergodic theory dynam. Systems 2, 241-251 (1982) · Zbl 0509.34027
[12] Roussarie, R.: Bifurcation of planar vector fields and Hilbert’s sixteenth problem. Progr. math. 164 (1998) · Zbl 0898.58039
[13] Shi, S. L.: On the structure of Poincaré -- Lyapunov constants for the weak focus of polynomial vector fields. J. differential equations 52, 52-57 (1984) · Zbl 0534.34059
[14] Stróẓyna, E.; &zdot, H.; Oładek: The analytic and formal normal form for the nilpotent singularity. J. differential equations 179, 479-537 (2002)
[15] Takens, F.: Singularities of vector fields. Inst. hautes études sci. Publ. math. 43, 47-100 (1974)
[16] Ye, Y. Q.: Theory of limit cycles. Transl. math. Monogr. 66 (1986)
[17] Zuppa, C.: Order of cyclicity of the singular point of Liénard’s polynomial vector fields. Bol. soc. Brasil math. 12, 105-111 (1981) · Zbl 0577.34023