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New configurations of 24 limit cycles in a quintic system. (English) Zbl 1153.37027

The second Hilbert’s 16th problem studies the questions of maximal numbers and relative position of limit cycles of planar polynomial vector fields. The weakened Hilbert’s 16th problem posed by Arnold studies the same questions from perturbations of Hamiltonian systems. An efficient method to obtain a large number of limit cycles is to perturb symmetric Hamiltonian systems.
In this paper the authors study the number and distributions of limit cycles in a Z3-equivariant quintic planar polynomial system. 24 limit cycles are found in this system and two different configurations of them are shown by combining the methods of double homoclinic loops bifurcation, Poincaré bifurcation and qualitative analysis. The two configurations of 24 limit cycles obtained in this paper are new respective previous works where other configurations of 24 limit cycles were found.

MSC:

37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
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
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