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In this paper we study the existence of heteroclinic cycles in genericunfoldings of nilpotent singularities. Namely we prove that any nilpotentsingularity of codimension four in $\mathbb{R}^4$ unfolds generically abifurcation hypersurface of bifocal homoclinic orbits, that is, homoclinicorbits to equilibrium points with two pairs of complex eigenvalues. We alsoprove that any nilpotent singularity of codimension three in $\mathbb{R}^3$unfolds generically a bifurcation curve of heteroclinic cycles between twosaddle-focus equilibrium points with different stability indexes. Under genericassumptions these cycles imply the existence of homoclinic bifurcations.Homoclinic orbits to equilibrium points with complex eigenvalues are thesimplest configurations which can explain the existence of complex dynamics as,for instance, strange attractors. The proof of the arising of these dynamicsfrom a singularity is a very useful tool, particularly for applications.

Heteroclinic cycles arising in generic unfoldings of nilpotent singularities