I will survey on flows presenting equilibria attached to regular orbits. The classical example for this is the Lorenz-attractor flow as well the geometric Lorenz flow.
More than three decades passed before the existence of the Lorenz attractor was rigorously established by Warwick Tucker with a computer-assisted proof in the year 2000. The difficulty in treating this kind of systems is both conceptual and numerical. On the one hand, the presence of the singularity accumulated by regular orbits prevents this invariant set to be uniformly hyperbolic. On the other hand, solutions slow down as they pass near the saddle equilibria and so numerical integration errors accumulate without bound. Trying to address this problem, a successful approach was developed by Afraimovich-Bykov-Shil'nikov and Guckenheimer-Williams independently, leading to the construction of a geometrical model displaying the main features of the behavior of the solutions of the Lorenz system of equations. In the 1990‚s a breakthrough was obtained by Carlos Morales, Enrique Pujals and Maria José Pacifico following very original ideas developed by Ricardo Mañé during the proof of the C1-stability conjecture, providing a characterization of robustly transitive attractors for three-dimensional flows, of which the Lorenz attractor is an example.
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