Ashwin, Peter; Buescu, Jorge; Stewart, Ian From attractor to chaotic saddle: A tale of transverse instability. (English) Zbl 0887.58034 Nonlinearity 9, No. 3, 703-737 (1996). Summary: Suppose that a dynamical system possesses an invariant submanifold, and the restriction of the system to this submanifold has a chaotic attractor \(A\). Under which conditions is \(A\) an attractor for the original system, and in what sense?We characterize the transverse dynamics near \(A\) in terms of the normal Lyapunov spectrum of \(A\). In particular, we emphasize the role of invariant measures on \(A\). Our results identify the points at which \(A\): (1) ceases to be asymptotically stable, possibly developing a locally riddled basin; (2) ceases to be an attractor; (3) becomes a transversely repelling chaotic saddle. We show, in the context of what we call ‘normal parameters’ how these transitions can be viewed as being robust. Finally, we discuss some numerical examples displaying these transitions. Cited in 107 Documents MSC: 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 34D35 Stability of manifolds of solutions to ordinary differential equations 37A99 Ergodic theory Keywords:invariant submanifold; chaotic attractor; invariant measures PDFBibTeX XMLCite \textit{P. Ashwin} et al., Nonlinearity 9, No. 3, 703--737 (1996; Zbl 0887.58034) Full Text: DOI Link