#### by William A. Sethares and Iven M. Mareels

Adaptive hybrids are one way of cancelling the echo path in telephone systems. This paper conducts a bifurcation analysis of a simplified model of an adaptive hybrid using two bifurcation parameters, the adaptive stepsize and the ratio of the two inputs. 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. For inputs with a spectral density, conditions are derived under which a single stable (averaged) equilibrium exists.

These results are interesting from two points of view. From the practical side, they provide an explanation of the intermittent bursting behavior of adaptive hybrids, demonstrating that bursting can be due to a slowly attractive periodic orbit (in which case the bursting eventually dies away), to a stable aperiodic orbit, or to a strange attractor (in which cases the bursting persists). From the theoretical side, these results are interesting because they provide a 'real world' example exhibiting a rich variety of nonlinear behaviors. Due to the particular form of the model, many of these behaviors can actually be proven.

This study of the dynamics of a misbehaving phone line first appeared in the *IEEE Trans. on Circuits and Systems *in January 1991.

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