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)
Isochronicity into a family of time-reversible cubic vector fields. (English) Zbl 1025.37011

The authors consider planar cubic vector fields with a nondegenerate center at the origin of the (x,y)-plane. Without restriction, the linear part is (-y,x). The aim is to characterize, within a certain subfamily, those vector fields for which the center is isochronous. The subfamily chosen consists of all time-reversible vector fields possessing an integrating factor of the form (1+x) -k . Recall that a vector field is time-reversible, if it is invariant under some fixed rotation combined with time reversion. It is proved that in this subfamily, there exist exactly six genuinely cubic vector fields with an isochronous center at the origin. They are written down explicitly.

The proof consists of two parts: 1) The first five period constants, obtained by computer algebra methods, must vanish and thus yield necessary conditions. 2) Then sufficient conditions, such as the existence of a transversal commuting vector field, are used to select the final list of six vector fields.

MSC:
37C10Vector fields, flows, ordinary differential equations
34C14Symmetries, invariants (ODE)
34C25Periodic solutions of ODE
37C80Symmetries, equivariant dynamical systems