Geodesic
Bifurcations of a polycycle formed by separatrices of a saddle with zero saddle value of a dynamical system with central symmetry
Matematičeskie zametki SVFU, Tome 30 (2023) no. 3, pp. 67-77 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider two-parameter families of planar vector fields with central symmetry. Assume that for zero values of the parameters, the field has a hyperbolic saddle at the origin $O$ and two symmetric loops of the separatrices of this saddle. The saddle value - the trace of the matrix of the linear part of the field at the point $O$ - is assumed to be zero. We describe the bifurcation diagram of a generic family - a partition of a neighborhood of the origin on the parameter plane into topological equivalence classes of dynamical systems defined by these vector fields in a fixed neighborhood $U$ of the polycycle formed by loops of separatrices. In particular, for each element of the partition, the number and type of the field belonging to $U$ are indicated.
Keywords: planar vector field, central symmetry, saddle, separatrix, limit cycle.
Mots-clés : bifurcation 
@article{SVFU_2023_30_3_a5,
     author = {V. Sh. Roitenberg},
     title = {Bifurcations of a polycycle formed by separatrices of a saddle with zero saddle value of a dynamical system with central symmetry},
     journal = {Matemati\v{c}eskie zametki SVFU},
     pages = {67--77},
     year = {2023},
     volume = {30},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVFU_2023_30_3_a5/}
}
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V. Sh. Roitenberg. Bifurcations of a polycycle formed by separatrices of a saddle with zero saddle value of a dynamical system with central symmetry. Matematičeskie zametki SVFU, Tome 30 (2023) no. 3, pp. 67-77. http://geodesic.mathdoc.fr/item/SVFU_2023_30_3_a5/