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 bifurcation of limit cycles and integrability of a class of nilpotent systems of differential equations. (English) Zbl 1201.34061

The authors consider analytic systems of differential equations in the plane whose origin is a nilpotent singularity

x ' =y+ i=1 P q-p+2is (x,y),y ' = i=1 Q q-p+2is (x,y),

where p,q and n are natural numbers, pq, s=(n+1)p-q>0 and (P j ,Q j ) are quasi-homogeneous vector fields of type (p,q) and degree j, with Q (2n+1)p-q (1,0)<0 (necessary condition of monodromy). For them, they give an algorithm which provides the Taylor expansion of the return map near the origin. This map is computed by using special polar coordinates, already introduced by Lyapunov, associated to the solution of dx/dθ=-y, dy/dθ=x 2n+1 .

By using their algorithm and other tools described below, they characterize the centers and study the cyclicity of some concrete families of systems of the type given above. These tools are called by the authors conservative-dissipative decomposition of the associated vector field and the following nice result is proved: The nilpotent systems

x ' =y+v y (x,y)K(v(x,y),y 2 )+yψ(v(x,y),y 2 ),y ' =-v x (x,y)K(v(x,y),y 2 ),

where v,k,ψ are analytic functions defined in a neighborhood of the origin and ψ(0,0)=0, are integrable analytically in a neighborhood of the origin. This last result extends particular cases considered in the literature, like v(x,y)x and others.

MSC:
34C23Bifurcation (ODE)
34C25Periodic solutions of ODE
34C07Theory of limit cycles of polynomial and analytic vector fields
37C10Vector fields, flows, ordinary differential equations
34C05Location of integral curves, singular points, limit cycles (ODE)
References:
[1]Algaba, A.; García, C.; Reyes, M.: The center problem for a family of systems of differential equations having a nilpotent singular point, J. math. Anal. appl. 340, 32-43 (2008) · Zbl 1156.34031 · doi:10.1016/j.jmaa.2007.07.043
[2]Álvarez, M. J.; Gasull, A.: Monodromy and stability for nilpotent critical points, Int. J. Bifur. chaos appl. Sci. eng. 15, No. 4, 1253-1265 (2005) · Zbl 1088.34021 · doi:10.1142/S0218127405012740
[3]Álvarez, M. J.; Gasull, A.: Generating limit cycles from a nilpotent critical point via normal forms, J. math. Anal. appl. 318, No. 1, 271-287 (2006) · Zbl 1100.34030 · doi:10.1016/j.jmaa.2005.05.064
[4]Andreev, A.: Investigation of the behaviour of the integral curves of a system of two differential equations in the neighborhood of a singular point, Trans. am. Math. soc. 8, 187-207 (1958) · Zbl 0079.11301
[5]Andreev, A.; Sadovskii, A. P.; Tskialyuk, V. A.: The center-focus problem for a system with homogeneous nonlinearities in the case zero eigenvalues of the linear part, Diff. equat. 39, No. 2, 155-164 (2003) · Zbl 1067.34030 · doi:10.1023/A:1025192613518
[6]Bruno, A. D.: Local methods in nonlinear differential equations, (1989)
[7]Chavarriga, J.; Giacomini, H.; Giné, J.; Llibre, J.: Local analytic integrability for nilpotent centers, Ergodic theory dynam. Syst. 23, 417-428 (2003) · Zbl 1037.34025 · doi:10.1017/S014338570200127X
[8]Gasull, A.; Torregrosa, J.: Center problem for several differential equations via cherka’s method, J. math. Anal. appl. 228, No. 2, 322-343 (1998) · Zbl 0926.34022 · doi:10.1006/jmaa.1998.6112
[9]Gasull, A.; Torregrosa, J.: A new algorithm for the computation of the Liapunov constants for some degenerated critical points, Nonlinear anal. 47, 4479-4490 (2001) · Zbl 1042.34528 · doi:10.1016/S0362-546X(01)00561-2
[10]Giacomini, H.; Giné, J.; Llibre, J.: The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems, J. diff. Equat. 227, No. 2, 406-426 (2006) · Zbl 1111.34026 · doi:10.1016/j.jde.2006.03.012
[11]H. Giacomini, J. Giné, J. Llibre, Corrigendum to: ”The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems [J. Diff. Equat. 227 (2006) 2, 406 – 426], J. Diff. Equat. 232 (2007) 2, 702. · Zbl 1111.34026 · doi:10.1016/j.jde.2006.03.012
[12]Liapunov, M. A.: Stability of motion, (1966)
[13]Moussu, R.: Symétrie et forme normale des centres et foyers dégénérés, Ergodic theory dynam. Syst. 2, 241-251 (1982) · Zbl 0509.34027 · doi:10.1017/S0143385700001553
[14]Nemytskii, V. V.; Stepanov, V. V.: Qualitative theory of differential equations, (1960) · Zbl 0089.29502
[15]Poincaré, H.: Mémoire sur LES courbes définies par LES équations différentielles, J. math. 37, 375-422 (1881) · Zbl 13.0591.01
[16]Sadovskii, A. P.: Problem of distinguishing a center and a focus for a system with a nonvanishing linear part, Translated diff. Urav. 12, No. 7, 1238-1246 (1976) · Zbl 0366.34019
[17]Strozyna, E.; Zoladek, H.: The analytic and formal normal form for the nilpotent singularity, J. diff. Equat. 179, 479-537 (2002) · Zbl 1005.34034 · doi:10.1006/jdeq.2001.4043