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- W4226254080 abstract "In this paper we study planar polynomial Kolmogorov's differential systems [ X_muquadsist{xf(x,y;mu),}{yg(x,y;mu),} ] with the parameter $mu$ varying in an open subset $LambdasubsetR^N$. Compactifying $X_mu$ to the Poincar'e disc, the boundary of the first quadrant is an invariant triangle $Gamma$, that we assume to be a hyperbolic polycycle with exactly three saddle points at its vertices for all $muinLambda.$ We are interested in the cyclicity of $Gamma$ inside the family ${X_mu}_{muinLambda},$ i.e., the number of limit cycles that bifurcate from $Gamma$ as we perturb $mu.$ In our main result we define three functions that play the same role for the cyclicity of the polycycle as the first three Lyapunov quantities for the cyclicity of a focus. As an application we study two cubic Kolmogorov families, with $N=3$ and $N=5$, and in both cases we are able to determine the cyclicity of the polycycle for all $muinLambda,$ including those parameters for which the return map along $Gamma$ is the identity." @default.
- W4226254080 created "2022-05-05" @default.
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- W4226254080 date "2022-03-24" @default.
- W4226254080 modified "2023-09-26" @default.
- W4226254080 title "On the cyclicity of Kolmogorov polycycles" @default.
- W4226254080 doi "https://doi.org/10.48550/arxiv.2203.12972" @default.
- W4226254080 hasPublicationYear "2022" @default.
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