Geometric criterium in the center problem. (English) Zbl 1356.34035
Among other results the authors prove the following theorem and use it to solve the center-focus problem in various cases, particularly degenerate ones.
Theorem. A smooth system of differential equations \(\dot x = F(x)\) on \({\mathbb R}^2\) with \(F(0) = 0\) admits a smooth Lyapunov function if and only if there exists a smooth system \(\dot x = X(x)\) that has a center at the origin and is such that \(X \wedge F\) does not change sign in a neighborhood of the origin.
Theorem. A smooth system of differential equations \(\dot x = F(x)\) on \({\mathbb R}^2\) with \(F(0) = 0\) admits a smooth Lyapunov function if and only if there exists a smooth system \(\dot x = X(x)\) that has a center at the origin and is such that \(X \wedge F\) does not change sign in a neighborhood of the origin.
Reviewer: Douglas S. Shafer (Charlotte)
MSC:
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
References:
| [1] | Algaba A., Gamero E., García C.: The integrability problem for a class of planar systems. Nonlinearity 22(2), 395-420 (2009) · Zbl 1165.34023 · doi:10.1088/0951-7715/22/2/009 |
| [2] | Algaba A., García C., Reyes M.: Characterization of a monodromic singular point of a planar vector field. Nonlinear Anal. 74, 5402-5414 (2011) · Zbl 1226.34031 · doi:10.1016/j.na.2011.05.023 |
| [3] | Algaba A., García C., Reyes M.: Existence of an inverse integrating factor, center problem and integrability of a class of nilpotent systems. Chaos Solitons Fractals 45, 869-878 (2012) · Zbl 1266.37023 · doi:10.1016/j.chaos.2012.02.016 |
| [4] | Algaba A., Fuentes N., García C.: Centers of quasi-homogeneous polynomial planar systems. Nonlinear Anal. Real World Appl. 13, 419-431 (2012) · Zbl 1238.34052 · doi:10.1016/j.nonrwa.2011.07.056 |
| [5] | Álvarez M.J., Gasull A.: Monodromy and stability for nilpotent critical points. Int. J. Bifurc. Chaos Appl. Sci. Eng. 15(4), 1253-1265 (2005) · Zbl 1088.34021 · doi:10.1142/S0218127405012740 |
| [6] | Bautin N.N.: On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type. Mat. Sb. N.S. 30(72), 181-196 (1952) · Zbl 0059.08201 |
| [7] | Bautin N.N.: On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type. Am. Math. Soc. Transl. 100(1), 397-413 (1954) |
| [8] | Bruno A.D.: Local Methods in Nonlinear Differential Equations. Springer, New York (1989) · Zbl 0674.34002 · doi:10.1007/978-3-642-61314-2 |
| [9] | Écalle, J.: Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac. In: Actualités Mathématiques (Current Mathematical Topics). Hermann, Paris (1992) · Zbl 1241.34003 |
| [10] | Gasull A., Llibre J., Mañosa V., Mañosas F.: The focus-centre problem for a type of degenerate system. Nonlinearity 13(3), 699- (2000) · Zbl 0962.34067 · doi:10.1088/0951-7715/13/3/311 |
| [11] | Giné J.: Sufficient conditions for a center at a completely degenerate critical point. Int. J. Bifurc. Chaos Appl. Sci. Eng. 12(7), 1659-1666 (2002) · Zbl 1047.34020 · doi:10.1142/S0218127402005315 |
| [12] | Giné J.: On the centers of planar analytic differential systems. Int. J. Bifurc. Chaos Appl. Sci. Eng. 17(9), 3061-3070 (2007) · Zbl 1158.34018 · doi:10.1142/S0218127407018865 |
| [13] | Giné J., Llibre J.: On the center conditions for analytic monodromic degenerate singularities. J. Bifurc. Chaos Appl. Sci. Eng. 22(12), 1250303 (2012) · Zbl 1258.34053 · doi:10.1142/S0218127412503038 |
| [14] | Giné J., Llibre J.: A method for characterizing nilpotent centers. J. Math. Anal. Appl. 413, 537-545 (2014) · Zbl 1323.34038 · doi:10.1016/j.jmaa.2013.12.013 |
| [15] | Giné J., Peralta-Salas D.: Existence of inverse integrating factors and Lie symmetries for degenerate planar centers. J. Differ. Equ. 252(1), 344-357 (2012) · Zbl 1298.34026 · doi:10.1016/j.jde.2011.08.044 |
| [16] | Hartman, P.: Ordinary differential equations. Wiley, New York (1964) (reprinted by SIAM, 2002) · Zbl 0125.32102 |
| [17] | Il’yashenko Y.S.: Algebraic unsolvability and almost algebraic solvability of the problem for the center-focus. Funct. Anal. Appl. 6(3), 30-37 (1972) |
| [18] | Il’yashenko, Y.S.: Finiteness Theorems for Limit Cycles. Translations of Mathematical Monographs, vol. 94. American Mathematical Society, Providence (1991). (Traslated from the Russian by H.H. McFaden) · Zbl 0753.34015 |
| [19] | Li J.: Hilbert’s 16th problem and bifurcations of planar polynomial vector fields. Int. J. Bifurc. Chaos Appl. Sci. Eng. 13(1), 47-106 (2003) · Zbl 1063.34026 · doi:10.1142/S0218127403006352 |
| [20] | Liapunov, A.M.: Stability of Motion. With a Contribution by V. A. Pliss and An Introduction by V.P. Basov. Mathematics in Science and Engineering, vol. 30. Academic Press, New York (1966). (Translated from the Russian by Flavian Abramovici and Michael Shimshoni) · Zbl 1165.34023 |
| [21] | Mañosa V.: On the center problem for degenerate singular points of planar vector fields. Int. J. Bifurc. Chaos Appl. Sci. Eng. 12(4), 687-707 (2002) · Zbl 1047.34022 · doi:10.1142/S0218127402004693 |
| [22] | Mattei J.F., Moussu R.: Holonomie et intégrales premières. Ann. Sci. École Norm. Sup. 13(4), 469-523 (1980) · Zbl 0458.32005 |
| [23] | Mazzi L., Sabatini M.: A characterization of centres via first integrals. J. Differ. Equ. 76(2), 222-237 (1988) · Zbl 0667.34036 · doi:10.1016/0022-0396(88)90072-1 |
| [24] | Medvedeva, N.B.: The first focus quantity of a complex monodromic singularity, (Russian). Trudy Sem. Petrovsk. 13, 106-122, 257 (1988) · Zbl 0664.34041 |
| [25] | Medvedeva NB: Translation in J. Soviet Math. 50(1), 1421-1436 (1990) · Zbl 0699.34031 · doi:10.1007/BF01097031 |
| [26] | Moussu R.: Symetrie et forme normale des centres et foyers degeneres. Ergod. Theory Dyn. Syst. 2, 241-251 (1982) · Zbl 0509.34027 · doi:10.1017/S0143385700001553 |
| [27] | Pearson J.M., Lloyd N.G., Christopher C.J.: Algorithmic derivation of centre conditions. SIAM Rev. 38(4), 619-636 (1996) · Zbl 0876.34033 · doi:10.1137/S0036144595283575 |
| [28] | Poincaré H.: Mémoire sur les courbes définies par les équations différentielles. J. Math. 37, 375-422 (1881) · JFM 13.0591.01 |
| [29] | Poincaré H.: Mémoire sur les courbes définies par les équations différentielles. J. Math. 8, 251-296 (1882) · JFM 14.0666.01 |
| [30] | Poincaré, H.: Oeuvres de Henri Poincaré, vol. I, pp. 3-84. Gauthier-Villars, Paris (1951) · Zbl 1323.34038 |
| [31] | Romanovski, V.G., Shafer, D.S.: The Center and Cyclicity Problems: A Computational Algebra Approach. Birkhäuser, Boston (2009) · Zbl 1192.34003 |
| [32] | Teixeira M.A., Yang J.: The center-focus problem and reversibility. J. Differ. Equ. 174, 237-251 (2001) · Zbl 0994.34020 · doi:10.1006/jdeq.2000.3931 |
| [33] | Varin V.P.: Poincaré map for some polynomial systems of differential equations. Math. Sb. 195(7), 3-20 (2004) · Zbl 1077.34032 · doi:10.4213/sm832 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
