On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields

1986 ◽  
Vol 17 (2) ◽  
pp. 67-101 ◽  
Author(s):  
R. Roussarie
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Planar Vector Fields ◽  
Planar Vector

Center cyclicity for some nilpotent singularities including the ℤ2-equivariant class

2020 ◽  
pp. 2050053
Author(s):  
Isaac A. García
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Weighted Degree ◽  
Integrating Factor ◽  
The Family ◽  
Planar Vector Fields ◽  
Planar Vector ◽  
Formal Inverse ◽  
Leading Term ◽  
Inverse Integrating Factor

This work concerns with polynomial families of real planar vector fields having a monodromic nilpotent singularity. The families considered are those for which the centers are characterized by the existence of a formal inverse integrating factor vanishing at the singularity with a leading term of minimum [Formula: see text]-quasihomogeneous weighted degree, being [Formula: see text] the Andreev number of the singularity. These families strictly include the case [Formula: see text] and also the [Formula: see text]-equivariant families. In some cases for such families we solve, under additional assumptions, the local Hilbert 16th problem giving global bounds on the maximum number of limit cycles that can bifurcate from the singularity under perturbations within the family. Several examples are given.

scholarly journals Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields

2020 ◽  
Vol 35 (3) ◽  
pp. 490-514
Author(s):  
João L. Cardoso ◽  
Jaume Llibre ◽  
Douglas D. Novaes ◽  
Durval J. Tonon
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Piecewise Linear ◽  
Simultaneous Occurrence ◽  
Planar Vector Fields ◽  
Planar Vector
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