| Abstract: |
| Dumortier, Roussarie and Sotomayor gave a complete investigation of the unfolding of Bogdanov-Takens bifurcations of codimension $3$ based on the hypothesis that there are at most two limit cycles. From then on, this hypothesis becomes a longstanding open problem. According to Dumortier, there are three unsolved classes: elliptic,
saddle, focus. For all these three cases, the problem are changed to estimate the number of zeros of some Abelian integrals. By giving new criteria, we prove that the associated Abelian integrals have at most two zeros for elliptic case and saddle case, that is, we give an affirmative answer of this hypothesis for these two cases. |
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