# 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)
Linearizability of linear systems perturbed by fifth degree homogeneous polynomials. (English) Zbl 1127.34020

The authors study the linearizability problem of the planar complex system

$\stackrel{˙}{x}=x+P\left(x,y\right),\phantom{\rule{1.em}{0ex}}\stackrel{˙}{y}=-y+Q\left(x,y\right),\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $P\left(x,y\right)$ and $Q\left(x,y\right)$ are homogeneous polynomials of 5-th degree, in a neighborhood of the singular point $x=y=0$. They obtain necessary and sufficient conditions for linearizability of the germ of system (1) in form of explicit relations between the coefficients of the polynomials $P\left(x,y\right)$ and $Q\left(x,y\right)$. Using this result, the authors consider the isochronicity problem for the planar real system

$\stackrel{˙}{u}=-v+P\left(u,v\right),\phantom{\rule{1.em}{0ex}}\stackrel{˙}{v}=u+Q\left(u,v\right),\phantom{\rule{2.em}{0ex}}\left(2\right)$

in a neighborhood of the centre $u=v=0$, where all non-stationary trajectories are periodic. Here $P\left(u,v\right)$ and $Q\left(u,v\right)$ are also homogeneous polynomials of 5-th degree. If all non-stationary trajectories are periodic, the problem then arises to determine whether the period of oscillations is constant for all trajectories (a centre with such property is called isochronous centre).

From the results of Poincaré and Lyapunov it follows that the centre is isochronous if and only if the germ of system (2) is linearizable, i.e., if there exists an analytic transformation, which brings (2) into the linear system $\stackrel{˙}{u}=-v$, $\stackrel{˙}{v}=u$. Hence the authors obtain necessary and sufficient conditions of isochronicity for the germ of system (2). A new example of isochronous system of the given form is found.

##### MSC:
 34C20 Transformation and reduction of ODE and systems, normal forms 34C25 Periodic solutions of ODE 34C05 Location of integral curves, singular points, limit cycles (ODE)
SINGULAR