Periodic orbits of the Lorenz system through perturbation theory

Different transformations are applied to the Lorenz system with the aim of reducing the initial three-dimensional system into others of dimension two. The symmetries of the linear part of the system are determined by calculating the matrices which commute with the matrix associated to the linear part. These symmetries are extended to the whole system up to an adequate order by using Lie transformations. After the reduction, we formulate the resulting systems using the invariants associated to each reduction. At this step, we calculate for each reduced system the equilibria and their stability. They are in correspondence with the periodic orbits and invariant sets of the initial system, the stability being the same.