The idea of storing memory states or patterns to be recognized or recalled as static attractors of the neural network dynamics implies that initial configurations of neurons in some neighborhood of a memory stated will be attracted to it. In applications, these attractors could represent memories, pattern classes, or stable control actions. A novel property of autonomous spatio-temporal pattern recognition and production with the Adaptive Time-delay Neural Network (ATNN) has been explored. The ATNN, a paradigm for training a nonlinear neural network with adaptive time-delays and weights, has a rich repertoire of capabilities that are used to perform signal production, and to learn repetitive spatial motions. ATNN has the property that initial segments on signals that contain large amounts of noise can be "cleaned up" to result in trained trajectory motions. The results on noise removal suggest that the trajectories trained into the ATNN networks are in fact attractors. We conducted experiments on the basins of attraction for the circle and figure eight attractors in these networks. The circular and figure eight trajectories can be considered limit cycle attractors or multistate oscillations because of the repetition of points along the closed figures. In spite of the fact that different initial arcs were used, including very noisy arcs, when starting the network trajectory, the nets always arrived at the trajectory for which they were trained. Thus the initial arcs were within the basin of attraction for the trained figures (attractors) both for the circle and figure eight. The impact of this result is that an autonomous or controlled system, such as a robotic arm, or other moving object, could be trained to generate repetitive desired motions, and could attain this repetitive motion from arbitrary starting trajectories.