In this paper, the existence of 12 small limit cycles is proved for cubic order Z2-equivariant vector fields, which bifurcate from fine focus points. This is a new result in the study of the second part of the 16th Hilbert problem. The system under consideration has a saddle point, or a node, or a focus point (including center) at the origin, and two weak focus points which are symmetric about ...

if n is odd. Hence, (−1) nn n+1 is within of either 1 or −1. Since the distance from a to both 1 and −1 is at least 2 , we see that for all n ≥ N the distance from a to (−1) nn n+1 is at least . Therefore, at most finitely many elements of B can be within of a, so a cannot be a limit point of B. Hence, no number a such that |a| 6 = 1 can be a limit point of B, so we conclude that 1 and −1 are t...

Dispersing billiards introduced by Sinai are uniformly hyperbolic and have strong statistical properties (exponential decay of correlations and various limit theorems). However, if the billiard table has cusps (corner points with zero interior angles), then its hyperbolicity is nonuniform and statistical properties deteriorate. Until now only heuristic and experiments results existed predicting...

In this paper, we prove the existence of twelve small (local) limit cycles in a planar system with third-degree polynomial functions. The best result so far in literature for a cubic order planar system is eleven limit cycles. The system considered in this paper has a saddle point at the origin and two focus points which are symmetric about the origin. This system was studied by the authors and...

In this paper, we report the existence of twelve small limit cycles in a planar system with 3rd-degree polynomial functions. The system has Z2symmetry, with a saddle point, or a node, or a focus point at the origin, and two focus points which are symmetric about the origin. It is shown that such a Z2-equivariant vector field can have twelve small limit cycles. Fourteen or sixteen small limit cy...

In recent years, some generalized structures of topology were introduced. Supra was one the most important those generalizations. To contribute in this orientation, we devoted work to studying limit points and separation axioms on supra topological spaces by using $b$-open sets. We define them a similar way their counterparts spaces. general, demonstrate main properties investigate sufficient c...