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.


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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