Nonlinear dynamics in adaptive echo cancellation

Adaptive hybrids are one way of canceling the echo path in telephone systems. A simplified model of an adaptive hybrid using two bifurcation parameters, the adaptive stepsize and the ratio of the two inputs, is illustrated. As these parameters vary, the system exhibits a wide variety of behaviors, including stable and unstable equilibrium points, stable and unstable periodic orbits, and aperiodic orbits. The underlying bifurcations include Hopf, flip, period-doubling sequences, and a degenerate global bifurcation which gives rise to some very complex dynamics