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arXiv · 0912.3490

Some results on homoclinic and heteroclinic connections in planar systems

Armengol Gasull·Hector Giacomini·Joan Torregrosa
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Abstract

Consider a family of planar systems depending on two parameters $(n,b)$ and having at most one limit cycle. Assume that the limit cycle disappears at some homoclinic (or heteroclinic) connection when $Φ(n,b)=0.$ We present a method that allows to obtain a sequence of explicit algebraic lower and upper bounds for the bifurcation set ${Φ(n,b)=0}.$ The method is applied to two quadratic families, one of them is the well-known Bogdanov-Takens system. One of the results that we obtain for this system is the bifurcation curve for small values of $n$, given by $b=\frac5 7 n^{1/2}+{72/2401}n- {30024/45294865}n^{3/2}- {2352961656/11108339166925} n^2+O(n^{5/2})$. We obtain the new three terms from purely algebraic calculations, without evaluating Melnikov functions.

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Armengol Gasull, Hector Giacomini, Joan Torregrosa. 2009-12-17. Some results on homoclinic and heteroclinic connections in planar systems. https://doi.org/10.1088/0951-7715%2F23%2F12%2F001

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