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A cubic polynomial system with seven limit cycles at infinity. (English) Zbl 1096.65130

By the change of transformations ξ=x(x 2 +y 2 ) -2/3 , η=y(x 2 +y 2 ) -2/3 , t=(x 2 +y 2 ) -1 τ applied to the real planar cubic polynomial system without singular point at infinity the problem of limit cycles bifurcation at infinity is transferred into that at the origin. The computation of singular point values for transformated system allows to derive the conditions of the origin (respectively of the infinity for the original system) to be a center and the highest degree fine focus. In conclusion the system is constructed allowing the appearance of seven limit cycles in the neighbourhood of infinity.

Reviewer’s remarks: All computations have been done with the computer algebra system Mathematica.

MSC:
65P30Bifurcation problems (numerical analysis)
37M20Computational methods for bifurcation problems
37G15Bifurcations of limit cycles and periodic orbits
37G10Bifurcations of singular points