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A quartic system with twenty-six limit cycles. (English) Zbl 1267.34059

The author begins with a quartic integrable non-Hamiltonian vector field on \({\mathbb R}^2\) that has a \({\mathbb Z}_2\)-symmetry and whose phase portrait contains four centers in two pairs, with each pair of period annuli enclosed in a larger period annulus. He proves that, under a suitable quartic perturbation that also has a \({\mathbb Z}_2\)-symmetry, a total of at least twenty-six limit cycles bifurcate from the six period annuli, six from each of the four small ones and one from each of the two large ones. The proof is a computer-aided counting of the number of zeros of pseudo-abelian integrals. To ensure that the results are correct the mathematical operations were performed in interval arithmetic with directed rounding. This result improves the lower bound on the Hilbert number for quartic polynomial systems on the plane from twenty-two to twenty-six.

MSC:

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
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
37M20 Computational methods for bifurcation problems in dynamical systems
65G20 Algorithms with automatic result verification
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References:

[1] Arnol’d [Arnol’d 90] V. I., Theory of Singularities and Its Applications pp 1– (1990)
[2] Blomquist, [Blomquist 05] F, Hofschuster, W. and Krämer, W. 2005. ”Real and Complex Taylor Arithmetic in C-XSC”. Preprint 2005/4, Universität Wuppertal. Available online (http://www.math.uni-wuppertal.de/xsc)
[3] DOI: 10.1016/j.jde.2005.01.009 · Zbl 1098.34024
[4] DOI: 10.1007/3-7643-7429-2_2
[5] Christopher [Christopher and Li 07] C., Limit Cycles of Differential Equations (2007)
[6] Griewank [Griewank 00] A, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation (2000) · Zbl 0958.65028
[7] Guckenheimer [Guckenheimer 95] J, Experiment. Math. 4 pp 153– (1995)
[8] Guckenheimer [Guckenheimer and Holmes 83] J., Applied Mathematical Sciences 42 (1983)
[9] DOI: 10.1142/S0218127496000497 · Zbl 0882.68133
[10] Hammer [Hammer 95] R, C++ Toolbox for Verified Computing (1995)
[11] DOI: 10.1142/S0218127407019895 · Zbl 1159.34029
[12] DOI: 10.1090/S0273-0979-02-00946-1 · Zbl 1004.34017
[13] Johnson, [Johnson 09] T. 2009. ”Computer-Aided Computation of Abelian Integrals and Robust Normal Forms”. PhD thesis, Uppsala University.
[14] DOI: 10.1080/14689360802641206 · Zbl 1175.37055
[15] DOI: 10.1142/S0218127410025405 · Zbl 1183.34042
[16] DOI: 10.1142/S0218127410026599 · Zbl 1193.34061
[17] Johnson [Johnson and Tucker 11] T., To appear in BIT-Numerical Mathematics (2011)
[18] DOI: 10.1016/j.jde.2009.01.038 · Zbl 1176.34037
[19] Moore [Moore 66] R. E., Interval Analysis (1966)
[20] Neumaier [Neumaier 90] A., Encyclopedia of Mathematics and Its Applications 37 (1990)
[21] Romanovski [Romanovski and Shafer 09] V. G., The Center and Cyclicity Problems: A Computational Algebra Approach (2009) · Zbl 1192.34003
[22] DOI: 10.1007/978-3-0348-8798-4
[23] DOI: 10.1007/BF01077820 · Zbl 0578.58035
[24] DOI: 10.1016/j.jmaa.2007.04.010 · Zbl 1134.34020
[25] DOI: 10.1016/S0022-247X(02)00018-5 · Zbl 1019.34042
[26] DOI: 10.1016/j.jmaa.2005.08.068 · Zbl 1106.34017
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