Emergence of Strange Attractors from Singularities

Abstract

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 \(n\geqslant 4\).