GlobalAnalysis ofInvariantObjects

GAIO is an experimental software for the numerical investigation of dynamical systems. It provides methods for

• the setwise approximation of invariant sets like (relative) global attractors, invariant manifolds of (in principle) arbitrary dimension [1,2];
• the approximation of natural invariant measures [3,4].
• the approximation of cyclic behavior and identification of "almost invariant sets" [4,5].

GAIO implements data structures and corresponding algorithms that allow for a numerical analysis of dynamical systems on a global level. The methods build upon a multilevel approach to the computation and storage of coverings of the objects being approximated. Typical objects include invariant sets like (global) attractors, chain recurrent sets, the maximal invariant set and invariant manifolds. On top of the computation of these topological structures a global statistical analysis of the system may be performed. To this end the computed covering is used in order to discretize a suitable transfer operator of the system. Spectral properties of the resulting finite-state Markov chain reveal statistical information like invariant measures and almost invariant sets.

GAIO is coded in C,C++ and Fortran and features interfaces to Numerical Python and MATLAB.