Disturbance-generated bifurcation in a prototype adaptive system with e1-modification law

The e₁-modification law has been proposed by K.S. Narendra and A.M. Annaswamy (ibid., vol.32, no.2, pp.134-5, 1987) for robust model reference adaptive control in the presence of bounded disturbances. Using this law, it is shown by the present authors that, in the case of regulation, a prototype adaptive control system with a single unknown pole and an unknown constant disturbance at its input possesses multiple equilibria and undergoes various bifurcations. Simulations and local analysis give a complete characterization of the possible dynamics exhibited by the disturbance adaptive system. As the constant disturbance is varied, saddle-node bifurcation, (subcritical) Hopf bifurcation, and saddle-connection bifurcation occur in that order. Consequently, in this case, the e₁-modification law is qualitatively equivalent to the σ-modification law