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)
Local analytic integrability for nilpotent centers. (English) Zbl 1037.34025

Authors’ abstract: Let X(x,y) and Y(x,y) be real analytic functions without constant and linear terms defined in a neighborhood of the origin. Assume that the analytic differential system x ˙=y+X(x,y),y ˙=Y(x,y), has a nilpotent center at the origin. The first integrals, formal or analytic, will be real except if we say explicitly the converse. We prove the following:

(1) If X=yf(x,y 2 ) and Y=g(x,y 2 ), then the system has a local analytic first integral of the form H=y 2 +F(x,y), where F starts with terms of order higher than two.

(2) If the system has a formal first integral, then it has a formal first integral of the form H=y 2 +F(x,y), where F starts with terms of order higher than two. In particular, if the system has a local analytic first integral defined at the origin, then it has a local analytic first integral of the form H=y 2 +F(x,y), where F starts with terms of order higher than two.

As an application, we characterize the nilpotent centers for the differential systems (3) x ˙=y+P 3 (x,y), y ˙=Q 3 (x,y), which have a local analytic first integral, where P 3 and Q 3 are homogeneous polynomials of degree three.

MSC:
34C05Location of integral curves, singular points, limit cycles (ODE)
34C20Transformation and reduction of ODE and systems, normal forms
34C25Periodic solutions of ODE