ON THE GLOBAL DYNAMICS OF ADAPTIVE SYSTEMS - A STUDY OF AN ELEMENTARYEXAMPLE

The inherent nonlinear character of adaptive systems poses serious the oretical problems for the analysis of their dynamics. On the other han d. the importance of their dynamic behavior is directly related to the practical interest in predicting such undesirable phenomena as nonlin ear oscillations, abrupt transients, intermittence or a high sensitivi ty with respect to initial conditions. A geometrical/qualitative descr iption of the phase portrait of a discrete-time adaptive system with u nmodeled disturbances is given. For this, the motions in the phase spa ce are referred to normally hyperbolic (structurally stable) locally i nvariant sets. The study is complemented with a local stability analys is of the equilibrium point and periodic solutions. The critical chara cter of adaptive systems under rather usual working conditions is disc ussed. Special emphasis is put on the causes leading to intermittence. A geometric interpretation of the effects of some commonly used palli atives to this problem is given. The ''dead-zone'' approach is studied in more detail. The predicted dynamics are compared with simulation r esults.