zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Monodromy, center-focus and integrability problems for quasi-homogeneous polynomial systems. (English) Zbl 1192.34035
Among all quasi-homogeneous polynomial vector fields on the plane, the authors characterize which ones are monodromic (that is, admit a Poincaré first return map in a neighborhood of the origin). Among monodromic systems they give conditions that distinguish foci from centers. Finally, they characterize integrability of such systems.

34C05Location of integral curves, singular points, limit cycles (ODE)
34C07Theory of limit cycles of polynomial and analytic vector fields
Full Text: DOI
[1] Mazzi, L.; Sabatini, M.: A characterization of centres via first integrals, Journal of differential equations 76, 222-237 (1988) · Zbl 0667.34036 · doi:10.1016/0022-0396(88)90072-1
[2] Li, W.; Llibre, J.; Nicolau, M.; Zhang, X.: On the differentiability of first integrals of two dimensional flows, Proceedings of the American mathematical society 130, 2079-2088 (2002) · Zbl 1010.34023 · doi:10.1090/S0002-9939-02-06310-4
[3] Moussu, R.: Symetrie et forme normale des centres et foyers degeneres, Ergodic theory and dynamical systems 2, 241-251 (1982) · Zbl 0509.34027 · doi:10.1017/S0143385700001553
[4] Strózyna, E.; Zoladek, H.: The analytic and formal normal form for the nilpotent singularity, Journal of differential equations 179, 479-537 (2002) · Zbl 1005.34034
[5] Bruno, A. D.: Local methods in nonlinear differential equations, (1989) · Zbl 0674.34002
[6] Brunella, M.; Miari, M.: Topological equivalence of a plane vector field with its principal part defined through Newton polyhedra, Journal of differential equations 85, 338-366 (1990) · Zbl 0704.58038 · doi:10.1016/0022-0396(90)90120-E
[7] Medvedeva, N. B.: A monodromic criterion for a singular point of a vector field on the plane, St. peterbourg mathematical journal 13, No. 2, 253-268 (2002) · Zbl 1003.37014
[8] Algaba, A.; Freire, E.; Gamero, E.; García, C.: Quasihomogeneous normal forms, Journal of computational and applied mathematics 150, 193-216 (2003) · Zbl 1022.34034 · doi:10.1016/S0377-0427(02)00660-X
[9] Mañosa, Victor: On the center problem for degenerate singular points of planar vector fields, International journal of bifurcation and chaos 12, No. 4, 687-707 (2002) · Zbl 1047.34022 · doi:10.1142/S0218127402004693
[10] Collins, C. B.: Algebraic conditions for a centre or a focus in some simple systems of arbitrary degree, Journal of mathematical analysis and applications 195, 719-735 (1995) · Zbl 0854.34033 · doi:10.1006/jmaa.1995.1385
[11] Dumortier, F.: Local study of planar vector fields: singularities and their unfolding, Structures in dynamics, studies in math. Physics 2, 161-242 (1991)
[12] Chavarriga, J.; Giacomini, H.; Giné, J.; Llibre, J.: Local analytic integrability for nilpotent centers, Ergodic theory and dynamical systems 23, 417-428 (2003) · Zbl 1037.34025 · doi:10.1017/S014338570200127X