An explicit expression for the first return map in the center problem.(English)Zbl 1066.34026

After an appropriate change of variable, this paper is concerned with plane autonomous systems of the form $$dx/dt=-y+F(x,y)$$, $$dy/dt =x+G(x,y)$$, where $$F$$ and $$G$$ are real analytic functions whose Taylor expansions at the origin do not contain constant and linear terms. Generally, the origin can be either a center or a focus. The author investigates this classical Poincaré problem which characterizes $$F$$ and $$G$$ in order that the origin is a center. The author presents a method by which an explicit expression for the first return map is obtained for this center problem.

MSC:

 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
Full Text:

References:

 [1] Alwash, M.A.M., On the composition conjectures, Electron. J. differential equations, 69, 1-4, (2003) · Zbl 1287.34022 [2] V.I. Arnold, Yu. Il’yashenko, Ordinary differential equations, Encyclopedia of Mathematical Sciences, Vol. 1 (Dynamical Systems-I), Springer, Berlin, 1988. [3] Alwash, M.A.M.; Lloyd, N.G., Non-autonomous equations related to polynomial two-dimensional systems, Proc. royal soc. Edinburgh, 105A, 129-152, (1987) · Zbl 0618.34026 [4] Arnold, V.I., Problems on singularities and dynamical systems, (), 251-274 · Zbl 0883.58016 [5] Bendixson, I., Sur LES courbes définies par des équations différentielles, Acta math, 24, 1-88, (1901) · JFM 31.0328.03 [6] Christopher, C., An algebraic approach to the classification of centers in polynomial lienard systems, J. math. ann. appl, 229, 319-329, (1999) · Zbl 0921.34033 [7] Collins, C.B., Conditions for a center in a simple class of cubic systems, Differential integral equations, 10, 2, 333-356, (1997) · Zbl 0894.34022 [8] Frommer, M., Die integralkurven einer gewöhnlichen differential-gleichung erster ordnung in der umgebung rationaler unbestimmtheitsstellen, Math. ann, 99, 222-272, (1928) · JFM 54.0453.03 [9] Françoise, J.-P., The successive derivatives of the period function of a plane vector field, J. differential equations, 146, 2, 320-335, (1998) · Zbl 0943.34021 [10] Hur, Seok, Composition conditions and center problem, C. R. acad. sci. Paris ser. I, 333, 779-784, (2001) · Zbl 1009.34027 [11] Il’yashenko, Yu., Algebraic unsolvability and almost algebraic solvability of the problem for the center-focus, Funct. anal. priloz, 6, 3, 30-37, (1972) [12] Lyapunov, A.M., The general problem of the stability of motion, Internat. J. control, 55, 3, 521-790, (1992), (Translated by A. T. Fuller from Édouard Daraux’s French translation (1907) of the 1892 Russian original) [13] H. Poincaré, Sur les courbes définies par une équation différentielle, Oeuvres, t.1, Paris, 1892. · JFM 14.0666.01 [14] Sibirsky, K.S., Introduction to the algebraic theory of invariants of differential equations, nonlinear science: theory and applications, (1988), Manchester University Press Manchester [15] Schlomiuk, D., Algebraic particular integrals, integrability and the problem of the center, Trans. AMS, 338, 2, 799-841, (1993) · Zbl 0777.58028 [16] Zoladek, H., The problem of center for resonant singular points of polynomial vector fields, J. differential equations, 137, 1, 94-118, (1997) · Zbl 0885.34034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.