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)
Isochronous centers of a linear center perturbed by fifth degree homogeneous polynomials. (English) Zbl 0978.34028
The authors study plane time-reversible differential systems of the type $\dot x= -y+ X_5(x,y)$, $\dot y= x+ Y_5(x,y)$, where $X_5$, $Y_5$ are homogeneous polynomials of degree 5. They look for necessary and sufficient conditions for the origin to be an isochronous center. They find six classes of systems for which the origin is an isochronous center. For a seventh class it is proved that some necessary conditions for isochronicity hold. The final section is devoted to prove the existence of nontrivial, nonreversible isochronous centers of systems of the type $\dot x= -y+ X_5(x,y)$, $\dot y= x+ Y_5(x,y)$.

34C05Location of integral curves, singular points, limit cycles (ODE)
34D10Stability perturbations of ODE
Full Text: DOI
[1] N.N. Bautin, On the number of limit cycles which appears with the variation of coefficients from an equilibrium position of focus or center type (R), Mat. Sb. (30) (72) (1952) 181--196; Amer. Math. Soc. Transl. 100 (1954) 397--413. · Zbl 0059.08201
[2] Chavarriga, J.: Integrable systems in the plan with a center type linear part. Appl. math. 22, 285-309 (1994) · Zbl 0809.34002
[3] Chavarriga, J.; Giné, J.: Integrability of a linear center perturbed by fourth degree homogeneous polynomial. Publ. mat. 40, No. 1, 21-39 (1996) · Zbl 0851.34001
[4] Chavarriga, J.; Giné, J.: Integrability of a linear center perturbed by fifth degree homogeneous polynomial. Publ. mat. 41, No. 2, 335-356 (1996) · Zbl 0897.34030
[5] Chavarriga, J.; Giné, J.; Garcı\acute{}a, I. A.: Isochronous centers of cubic systems with degenerate infinity. Differential equations dyn. Systems 7, No. 2, 221-238 (1999) · Zbl 0982.34025
[6] Chavarriga, J.; Giné, J.; Garcı\acute{}a, I. A.: Isochronous centers of a linear center perturbed by fourth degree homogeneous polynomials. Bull. sci. Math. 123, 77-96 (1999) · Zbl 0921.34032
[7] Chicone, C.; Jacobs, M.: Bifurcation of critical periods for plane vector fields. Trans. amer. Math. soc. 312, 433-486 (1989) · Zbl 0678.58027
[8] Christopher, C. J.; Devlin, J.: Isochronous centres in planar polynomial systems. SIAM J. Math. anal. 28, 162-177 (1997) · Zbl 0881.34057
[9] Coppel, W. A.: A survey of quadratic systems. J. differential equations 2, 293-304 (1996) · Zbl 0143.11903
[10] N.G. Lloyd, Small amplitude limit cycles of polynomial Differential equations, in: W.N. Everitt, R.T. Lewis (Eds.), Ordinary Differential Equations and Operators, Lecture Notes in Maths., vol. 1032, Springer, Berlin, 1983, pp. 346--356. · Zbl 0527.34035
[11] Lloyd, N. G.; Christopher, J.; Devlin, J.; Pearson, J. M.; Yasmin, N.: Quadratic like cubic systems. Differential equations dynamical systems 5, No. 3--4, 329-345 (1997) · Zbl 0898.34026
[12] Loud, W. S.: Behavior of the period of solutions of certain plane autonomous systems near centers. Contrib. differential equations 3, 21-36 (1964) · Zbl 0139.04301
[13] Lunkevich, V. A.; Sibirskii, K. S.: Integrals of a general quadratic differential system in cases of a center. Differential equations 18, 563-568 (1982) · Zbl 0499.34017
[14] Mardesic, P.; Moser-Jauslin, L.; Rousseau, C.: Darboux linearization and isochronous centers with a rational first integral. J. differential equations 134, 216-268 (1997) · Zbl 0881.34041
[15] Mardesic, P.; Rousseau, C.; Toni, B.: Linearization of isochronous centers. J. differential equations 121, 67-108 (1995) · Zbl 0830.34023
[16] Pleshkan, I.: A new method of investigating the isochronicity of a system of two differential equations. Differential equations 5, 796-802 (1969)
[17] Rousseau, C.; Toni, B.: Local bifurcation in vector fields with homogeneous nonlinearities of the third degree. Canad. math. Bull. 36, 473-484 (1993) · Zbl 0792.58030
[18] Rousseau, C.; Toni, B.: Local bifurcation of critical periods in the reduced kukles system. Canad. J. Math. 49, 338-358 (1997) · Zbl 0885.34033
[19] Sabatini, M.: Characterizing isochronous centers by Lie brackets. Differential equations dynamical systems 5, No. 1, 91-99 (1997) · Zbl 0894.34021
[20] D. Schlomiuk, Algebraic and geometric aspects of the theory of polynomials vector fields, Bifurcations and Periodics Orbits of Vectors Fields, Kluwer Academic Publishers, Dordrecht, 1993, pp. 429--467. · Zbl 0790.34031
[21] Villarini, M.: Regularity properties of the period function near a centre of planar vector fields. Nonlinear analysis TMA 19, No. 8, 787-803 (1992) · Zbl 0769.34033
[22] Zoladek, H.: On certain generalization of the bautin’s theorem. Nonlinearity 7, 233-279 (1994)