# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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:
 34C23 Bifurcation (ODE) 34C05 Location of integral curves, singular points, limit cycles (ODE) 37G05 Normal forms 37G10 Bifurcations of singular points 37G15 Bifurcations of limit cycles and periodic orbits 34C20 Transformation and reduction of ODE and systems, normal forms
##### Keywords:
limit cycle; nilpotent critical point; normal form
Full Text:
##### 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