Emergence of Strange Attractors from Singularities

This paper is a summary of results that prove the abundance of one-dimensional strange attractors near a Shil'nikov configuration, as well as the presence of these configurations in generic unfoldings of singularities in $\mathbb{R}^{3}$ of minimal codimension. Finding these singularities in families of vector fields is analytically possible and thus provides a tractable criterion for the existence of chaotic dynamics. Alternative scenarios for the possible abundance of two-dimensional attractors in higher dimension are also presented. The role of Shil'nikov configuration is now played by a certain type of generalised tangency which should occur for families of vector fields $X_{\mu }$ unfolding generically some low codimension singularity in $\mathbb{R}^{n}$ with $n\geqslant 4$.