Small amplitude chaos and ergodicity in adaptive control

Abslraet--The theory for continuous iterated maps on the real line is used to study the dynamics of a continuous family of one parameter adaptive systems. We establish sufficient and necessary conditions for uniform asymptotic stability and ultimate boundedness, exact bifurcation boundaries and sufficient conditions for the existence of chaotic attractors and ergodicity of the estimated parameter. It is argued that the chaotic dynamics are structurally stable, that they persist in higher dimensional adaptive control systems, and that they do not necessarily lead to poor performance when the controller is modified to deal with the infinite drift and small divisor problems. A novel modification involving a sign change in the adaptive law is proposed.