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Lorenz attractors play an important role in the modern theory of dynamicalsystems. The reason is that they are robust, i.e. preserve their chaoticproperties under various kinds of perturbations. This means that suchattractors can exist in applied models and be observed in experiments. It isknown that discrete Lorenz attractors can appear in local and globalbifurcations of multidimensional diffeomorphisms. However, to date, onlypartial cases were investigated. In this paper bifurcations of homoclinic andheteroclinic cycles with quadratic tangencies of invariant manifolds arestudied. A full list of such bifurcations, leading to the appearance ofdiscrete Lorenz attractors is provided. In addition, with help of numericaltechniques, it was proved that if one reverses time in the diffeomorphismsdescribed above, the resulting systems also have such attractors. This resultis an important step in the systematic studies of chaos and hyperchaos.

Global and local bifurcations, three-dimensional Henon maps and discrete Lorenz attractors