×

On a class of polynomial differential systems of degree 4: phase portraits and limit cycles. (English) Zbl 1486.34072

The phase portraits in the Poincaré disc of a class of polynomial differential systems which are symmetric with respect to the \(x\)-axis are considered. Using the averaging theory of order five, the authors study the number of limit cycles which can bifurcate from the period annulus of the center when it is perturbed inside the class of all polynomial differential systems of degree four.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34C29 Averaging method for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] N.N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Mat. Sbornik 30 (1952), 181-196; Amer. Math. Soc. Transl. 100 (1954), 1-19. · Zbl 0059.08201
[2] R. Benterki and J. Llibre, Centers and limit cycles of polynomial differential systems of degree 4 via averaging theory, J. Comput. Appl. Math. 313 (2017), 273-283. · Zbl 1364.34035
[3] C.A. Buzzi, J. Llibre and J.C. Medrado, Phase portraits of reversible linear differential systems with cubic homogeneous polynomial nonlinearities having a non-degenerate center at the origin, Qual. Theory Dyn. Syst. 7 (2009), 369-403. · Zbl 1311.34061
[4] F. Dumortier, J. Llibre and J.C. Artes, Qualitative Theory of Planar Differential Systems, UniversiText, Springer-Verlag, New York, 2006. · Zbl 1110.34002
[5] J. Gine, Conditions for the existence of a center for the Kukles homogenenous systems, Comput. Math. Appl. 43 (2002), 1261-1269. · Zbl 1012.34025
[6] J. Gine, M. Grau and J. Llibre, Averaging theory at any order for computing periodic orbits, Comput. Phys. D. 250 (2013), 58-65. · Zbl 1267.34073
[7] J. Gine, J. Llibre and C. Valls, Centers for the Kukles homogeneous systems with odd degree, Bull. London Math. Soc. 47 (2015), 315-324. · Zbl 1323.34039
[8] J. Gine, J. Llibre and C. Valls, Centers for the Kukles homogeneous systems with even degree, J. Appl. Anal. an Comp. 7 (2017), 1534-1548. · Zbl 1456.34027
[9] J. Itikawa and J. Llibre, Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers, J. Comput. Appl. Math. 277 (2015), 171-191. · Zbl 1308.34041
[10] W. Kapteyn, On the midpoints of integral curves of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland (1911), 1446-1457. (in Dutch)
[11] W. Kapteyn, New investigations on the midpoints of integrals of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag Afd. Natuurk. 20 (1912), 1354-1365; 21, 27-33. (in Dutch)
[12] J. Llibre, B.D. Lopes and J.R. Moraes, Limit cycles of cubic polynomial differential systems with rational first integrals of degree 2, Appl. Math. and Comput. 250 (2015), 887-907. · Zbl 1328.34028
[13] J. Llibre, D. D. Novaes and M. A. Teixeira, Higher order averaging theory for finding periodic solution via Brouwer degree, Nonlinearity 27 (2014), 563-583. · Zbl 1291.34077
[14] K.E. Malkin, Criteria for the center for a certain differential equation, Volz. Mat. Sb. Vyp. 2 (1964), 87—91. (in Russian)
[15] L. Markus, Global structure of ordinary differential equations in the plane, Trans. Amer. Math Soc. 76 (1954), 127-148. · Zbl 0055.08102
[16] D. A. Neumann, Classification of continuous flows on 2-manifolds, Proc. Amer. Math. Soc. 48 (1975), 73-81. · Zbl 0307.34044
[17] M.M. Peixoto, Dynamical System, Proccedings of a Symposium held at the University of Bahia, 1973, Acad. Press, New York, 1973, pp. 389-420.
[18] N.I. Vulpe and K.S. Sibirskiı, Centro-affine invariant conditions for the existence of a center of a differential system with cubic nonlinearities, Dokl. Akad. Nauk SSSR 301 (1988), 1297-1301 (in Russian); English transl.: Soviet Math. Dokl. 38 (1989), 198-201. · Zbl 0674.34025
[19] D. Schlomiuk, Algebraic particular integrals, integrability and the problem of the center, Trans. Amer. Math. Soc. 338 (1993), 799-841. · Zbl 0777.58028
[20] E.P. Volokitin and V.V. Ivanov, Isochronicity and Commutation of polynomial vector fields, Sib. Math. J. 40 (1999), 22-37. · Zbl 0921.58053
[21] N.I. Vulpe, Affine-invariant conditions for the topological discrimination of quadratic systems with a center, Differential Equations 19 (1983), 273-280. · Zbl 0556.34019
[22] H. Żołądek, Quadratic systems with center and their perturbations, J. Differential Equations 109 (1994), 223-273. · Zbl 0797.34044
[23] H. Żołądek, The classification of reversible cubic systems with center, Topol. Methods Nonlinear Anal. 4 (1994), 79-136. · Zbl 0820.34016
[24] H. Żołądek, Remarks on: ”The classification of reversible cubic systems with center, Topol. Methods Nonlinear Anal. 4 (1994), 79-136, Topol. Methods Nonlinear Anal. 8 (1996), 335-342. · Zbl 0901.34036
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.