Sensitivity of Attractor of the Lorenz model
Approach by Unstable Periodic Orbits.

A description of a deterministic chaotic system in terms of unstable periodic orbits (UPO) is used to develop a method of an a priori estimates of the sensitivity of statistical averages of the solution to small external influences. This method allows us to determine the forcing perturbation which maximizes the norm of the perturbation of a statistical moment of the solution on the attractor. The method is applied to the Lorenz model. The estimates of perturbations of two statistical moments were compared with directly calculated values. The comparison shows that some 100 UPOs is sufficient to realize this approach and to get a good accuracy. The linear approach remains valid up to rather high norms of the forcing perturbation.

The comparison of linear estimates with UPOs and direct calculation is presented below. The forcing is taken to be A*(0.2, 0.98, 0)

Some pictures
Click images to see what will happen when the norm of the forcing begins to grow. (animated gif files, about 500K)
Perturbation of the Lorenz attractor approximated by UPOs Perturbation of the Lorenz attractor calculated directly

For more information,
see
  • E. Kazantsev. Unstable periodic orbits and Attractor of the Lorenz Model. Rapport de recherche INRIA, 1998, No.3344 PS version ( .ps.gz file).
  • E. Kazantsev. Sensitivity of Attractor of the Lorenz Model to external influences: Approach by Unstable Periodic Orbits. Slides ( .ps.gz file).
  • E. Kazantsev. Sensitivity of Attractor of the Barotropic Ocean Model to external influences: Approach by Unstable Periodic Orbits. Slides ( .ps.gz file).

Eugene.Kazantsev@imag.fr
Christine.Kazantsev@imag.fr


Last modified the 10 of july 2000