To understand how individuals adapt to and anticipate each other in joint tasks, we employ a bidirectional delay–coupled dynamical system that allows for mutual adaptation and anticipation. In delay–coupled systems, anticipation is achieved when one system compares its own time‐delayed behavior, which implicitly includes past information about the other system’s behavior, with the other system’s instantaneous behavior. Applied to joint music performance, the model allows each system to adapt its behavior to the dynamics of the other. Model predictions of asynchrony between two simultaneously produced musical voices were compared with duet pianists’ behavior; each partner performed one voice while auditory feedback perturbations occurred at unpredictable times during live performance. As the model predicted, when auditory feedback from one musical voice was removed, the asynchrony changed: The pianist’s voice that was removed anticipated (preceded) the actions of their partner. When the auditory feedback returned and both musicians could hear each other, they rapidly returned to baseline levels of asynchrony. To understand how the pianists anticipated each other, their performances were fitted by the model to examine change in model parameters (coupling strength, time‐delay). When auditory feedback for one or both voices was removed, the fits showed the expected decrease in coupling strength and time‐delay between the systems. When feedback about the voice(s) returned, the coupling strength and time‐delay returned to baseline. These findings support the idea that when people perform actions together, they do so as a coupled bidirectional anticipatory system.

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A simple and systematic approach is developed for modeling and adaptive control of an unknown (or uncertain) chaotic system of the formx(n) = f(X) + g(X)u, using only input-output data obtained from the underlying dynamic system. Two different fuzzy identification methods, i.e. least-squares and gradient descent, are used for identifying the unknown functions f(X) and g(X). Based on the fuzzy modeling, an adaptive controller is devised, which works through sliding mode method. The presented procedure is illustrated by using the chaotic system-modified Duffing's equation as an example, on which simulation results demonstrate the effectiveness of the proposed adaptive algorithm.