×

The center and cyclicity problems for quartic linear-like reversible systems. (English) Zbl 1446.34046

The paper is devoted to the study of a family of quartic reversible polynomial systems which are linear in one of the variables and have a non-degenerate center at the origin. A system of this class having two additional non-degenerate centers located outside the direct symmetry is constructed. Two configurations of these centers are possible: aligned and triangular. The problem of the center is solved in both situations. The manifolds of centers are represented in the form of irreducible decompositions of the manifold of ideal of Lyapunov quantities. In the triangular configuration of the centers, limit cycles obtained from the simultaneous degenerate Hopf bifurcation in the class of quartic polynomials are studied. It is proved that there exist fourth-degree polynomial perturbations such that at least \(13\) small limit cycles are generated from the centers of the system. Two configurations of such limit cycles are presented: \((4, 5, 4)\) and \((3, 7, 3)\).
It should be noted that in Section 4, difficulties in calculating the Gröbner basis of the ideal \(B\) of the five Lyapunov quantities can be easily overcome by replacing \(c=(a^2+b^2)s\), \(a=\sqrt{(A-b)b}\) . Then, instead of the ideal \(B\), we need to consider an ideal composed of the remainder of dividing \(L_i\), \(i=\overline{2,5}\), by \(L_1\) in the lexicographic order \(Z=\{s, A, b, d\}\). The Gröbner basis is found by adding the term \(1+t(A-b)b\) to the resulting ideal and in the lexicographic order \(\{t, s, A, b, d\}\). It has \(25\) components, where the first one is the following \[ (A-b-d)d(2+bd)(-2b+2d+Abd-2b^2d)(2+Ad-d^2)(-8-2bd+2d^2+Abd^2)(4+4bd+2d^2+Abd^2). \] The authors note that before the publication of Christopher’s theorem on the cyclicity of the center, similar results were derived by Chicone, Jacobs, Han. We also note that the main idea of the theorem has been used in the paper [A. E. Rudenok, Differ. Equations 23, No. 5, 561–569 (1987; Zbl 0632.34023); translation from Differ. Uravn. 23, No. 5, 825–834 (1987)].

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of 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
34C23 Bifurcation theory for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34C14 Symmetries, invariants of ordinary differential equations

Citations:

Zbl 0632.34023

Software:

SINGULAR; primdec
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Bondar, Y. L.; Sadovskii, A. P., Solution of the center and focus problem for a cubic system that reduces to the Liénard system, Differ. Uravn., 42, 1, 11-22 (2006), 141
[2] Chavarriga, J.; Giné, J., Integrability of a linear center perturbed by homogeneous polynomial, (Proceedings of the 2nd Catalan Days on Applied Mathematics (Odeillo, 1995). Proceedings of the 2nd Catalan Days on Applied Mathematics (Odeillo, 1995), Collect. Études (1995), Presses Univ. Perpignan: Presses Univ. Perpignan Perpignan), 61-75 · Zbl 0904.34022
[3] Chavarriga, J.; Giné, J., Integrability of cubic systems with degenerate infinity, (XIV CEDYA/IV Congress of Applied Mathematics (Spanish)(Vic, 1995) (1996), Univ. Barcelona: Univ. Barcelona Barcelona), 12
[4] Chicone, C.; Jacobs, M., Bifurcation of critical periods for plane vector fields, Trans. Amer. Math. Soc., 312, 2, 433-486 (1989) · Zbl 0678.58027
[5] Chicone, C.; Jacobs, M., Bifurcation of limit cycles from quadratic isochrones, J. Differential Equations, 91, 2, 268-326 (1991) · Zbl 0733.34045
[6] Christopher, C., Estimating limit cycle bifurcations from centers, (Differential Equations with Symbolic Computation. Differential Equations with Symbolic Computation, Trends Math. (2005), Birkhäuser: Birkhäuser Basel), 23-35 · Zbl 1108.34025
[7] Cozma, D., The problem of the center for cubic systems with two parallel invariant straight lines and one invariant cubic, ROMAI J., 11, 2, 63-75 (2015) · Zbl 1424.34105
[8] Cozma, D.; Şubă, A., The solution of the problem of center for cubic differential systems with four invariant straight lines, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.), 44, suppl, 517-530 (2000), 1998 Mathematical analysis and applications (Iaşi, 1997 · Zbl 1009.34026
[9] W. Decker, G.-M. Greuel, G. Pfister, H. Schönemann, Singular 4-1-1 — A computer algebra system for polynomial computations. http://www.singular.uni-kl.de, 2018.
[10] Decker, W.; Pfister, G.; Schönemann, H.; Laplagne, S., Primdec.Lib a Singular 4-1-1 Library for Computing the Primary Decomposition and Radical Ideals (2018)
[11] Dulac, H., Détermination et intégration d’une certaine class d’équations différentialles ayant pour point singulier un centre, Bull. Sci. Math., 32, 2, 230-252 (1908) · JFM 39.0374.01
[12] Dumortier, F.; Llibre, J.; Artés, J. C., (Qualitative Theory of Planar Differential Systems. Qualitative Theory of Planar Differential Systems, Universitext (2006), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1110.34002
[13] Ferčec, B.; Giné, J.; Romanovski, V. G.; Edneral, V. F., Integrability of complex planar systems with homogeneous nonlinearities, J. Math. Anal. Appl., 434, 1, 894-914 (2016) · Zbl 1337.34003
[14] Frommer, M., über Das auftreten von wirbeln und strudeln (geschlossener und spiraliger integralkurven) in der umgebung rationaler unbestimmtheitsstellen, Math. Ann., 109, 1, 395-424 (1934) · JFM 60.1094.01
[15] Gianni, P.; Trager, B.; Zacharias, G., Gröbner Bases and primary decomposition of polynomial ideals, J. Symb. Comput., 6, 2-3, 149-167 (1988), Computational aspects of commutative algebra · Zbl 0667.13008
[16] Han, M., Liapunov constants and hopf cyclicity of Liénard systems, Ann. Differential Equations, 15, 2, 113-126 (1999) · Zbl 0968.34029
[17] Ilyashenko, Y.; Yakovenko, S., (Lectures on Analytic Differential Equations. Lectures on Analytic Differential Equations, Graduate Studies in Mathematics, vol. 86 (2008), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 1186.34001
[18] Kaptyen, W., On the midpoints of integral curves of differntial equations of the first degree, Nederl. Adak. Wetecnsch. Versl. Afd. Natuurk. Konikl., 1446-1457 (1911)
[19] Kaptyen, W., New investigations on the midpoints of integral of differential equations of the first degree, Nederl. Adak. Wetecnsch. Versl. Afd. Natuurk. Konikl., 20, 1354-1365 (1912)
[20] Liapunov, A. M., Investigation of one of the special cases of the stability of motion, Mat. Sb., 17, 253-333 (1893)
[21] Liapunov, A. M., Problème Général de la Stabilité du Mouvement, (Annals of Mathematics Studies, Vol. 17 (1947), Princeton University Press, Oxford University Press: Princeton University Press, Oxford University Press Princeton, N. J. London)
[22] Lloyd, N. G.; Christopher, C. J.; Devlin, J.; Pearson, J. M.; Yasmin, N., Quadratic-like cubic systems, Differential Equations Dynam. Syst.. Differential Equations Dynam. Syst., Planar nonlinear dynamical systems, 5, 3-4, 329-345 (1995) · Zbl 0898.34026
[23] Poincaré, H., Mémoire sur les courbes définies par une équation différentielle, Sér. 3. Sér. 3, Planar nonlinear dynamical systems. Sér. 3. Sér. 3, Planar nonlinear dynamical systems, Sér.(3). Sér. 3. Sér. 3, Planar nonlinear dynamical systems. Sér. 3. Sér. 3, Planar nonlinear dynamical systems, Sér.(3), Sér.(4). Sér. 3. Sér. 3, Planar nonlinear dynamical systems. Sér. 3. Sér. 3, Planar nonlinear dynamical systems, Sér.(3). Sér. 3. Sér. 3, Planar nonlinear dynamical systems. Sér. 3. Sér. 3, Planar nonlinear dynamical systems, Sér.(3), Sér.(4), Sér.(4), 2, 151-217 (1886) · JFM 18.0314.01
[24] Prohens, R.; Torregrosa, J., New lower bounds for the Hilbert numbers using reversible centers, Nonlinearity, 32, 331 (2019) · Zbl 1414.34024
[25] Romanovski, V. G.; Shafer, D. S., The Center and Cyclicity Problems: A Computational Algebra Approach (2009), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA · Zbl 1192.34003
[26] Romanovski, V. G.; Shafer, D. S., Centers and limit cycles in polynomial systems of ordinary differential equations, (School on Real and Complex Singularities in São Carlos, 2012. School on Real and Complex Singularities in São Carlos, 2012, Adv. Stud. Pure Math., vol. 68 (2012), Math. Soc. Japan: Math. Soc. Japan Tokyo), 267-373 · Zbl 1383.34053
[27] Roussarie, R., (Bifurcation of Planar Vector Fields and Hilbert’s Sixteenth Problem. Bifurcation of Planar Vector Fields and Hilbert’s Sixteenth Problem, Progress in Mathematics, vol. 164 (1998), Birkhäuser Verlag: Birkhäuser Verlag Basel) · Zbl 0898.58039
[28] Saharnikov, N. A., On frommer’s conditions for the existence of a center, Akad. Nauk SSSR. Prikl. Mat. Meh., 12, 669-670 (1948)
[29] Sang, B., Center-focus problems for two classes of cubic differential systems, J. Nanjing Norm. Univ. Nat. Sci. Ed., 35, 2, 16-21 (2012) · Zbl 1274.34093
[30] Sang, B.; Niu, C., Solution of center-focus problem for a class of cubic systems, Chinese Ann. Math. Ser. B, 37, 1, 149-160 (2016) · Zbl 1343.34078
[31] Sibirskiĭ, K. S., On conditions for the presence of a center and a focus, Kišinev. Gos. Univ. Uč. Zap., 11, 115-117 (1954)
[32] Sibirskiĭ, K. S., The principle of symmetry and the problem of the center, Kišinev. Gos. Univ. Uč. Zap., 17, 27-34 (1955)
[33] Sibirskiĭ, K. S., On the number of limit cycles in the neighborhood of a singular point, Differencialnye Uravn., 1, 53-66 (1965) · Zbl 0196.35702
[34] Wang, P. S.; Guy, M. J.T.; Davenport, J. H., P-adic reconstruction of rational numbers, SIGSAM Bull., 16, 2, 2-3 (1982) · Zbl 0489.68032
[35] Zołądek, H., Quadratic systems with center and their perturbations, J. Differential Equations, 109, 2, 223-273 (1994) · Zbl 0797.34044
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.