Stability and chaos analysis for an ICA algorithmI

The stability and chaos of an ICA algorithm are investigated by analyzing the corresponding deterministic discrete time (DDT) system. The existence and stability of all the possible fixed points of the ICA algorithm are studied. While the nonlinear function contained in the algorithm is specified, an invariant set of the algorithm is obtained so that the non-divergence of the algorithm can be guaranteed. It is then derived in this invariant set that the behaviors of the algorithm are dominated by a one-dimensional map. The conditions for convergence and chaos are derived. In the outside of the invariant set, the corrected Marottoʹs theorem and computer-assisted method are applied to study the twodimensional case of the algorithm and the existence of chaos is proved. The attractors and bifurcation diagrams of the algorithm with different parameters are presented to further confirm the obtained results.