Brain and Cognition 48, 107–116 (2002) doi:10.1006/brcg.2001.1307, available online at http://www.idealibrary.com on Synchronizing Movements with the Metronome: Nonlinear Error Correction and Unstable Periodic Orbits Ralf Engbert,* , ² Ralf Th. Krampe,² Ju¨rgen Kurths,* and Reinhold Kliegl² Departments of *Physics and ²Psychology, University of Potsdam, Germany Published online November 15, 2001 The control of human hand movements is investigated in a simple synchronization task. We propose and analyze a stochastic model based on nonlinear error correction; a mechanism which implies the existence of unstable periodic orbits. This prediction is tested in an experi- ment with human subjects. We find that our experimental data are in good agreement with numerical simulations of our theoretical model. These results suggest that feedback control of the human motor systems shows nonlinear behavior.  2001 Elsevier Science The application of nonlinear dynamics to the problem of human movement control relates to several central issues in extant biological and psychological models (Kelso, 1995, and references therein). Even very simple movements show strong random variability between successive realizations of the same target interval. Therefore, theoretical models for the production of rhythmic movements often focus on stochas- tic aspects (Vorberg & Wing, 1996). The analysis of these fluctuations in the frame- work of linear stochastic processes has provided important insights into the organiza- tion of the human movement control system. The investigation of the interaction between nonlinear determinism and stochastic variability remains a challenge to ex- perimental and modeling approaches. To this end, we use the concept of unstable periodic orbits as a novel approach to the analysis of movement control in this article. Unstable periodic orbits are fundamental for understanding chaotic dynamical sys- tems (Cvitanovic ´, 1988; Grebogi, Ott, & Yorke, 1988; Hunt & Ott, 1996; Lai, Na- gai, & Grebogi, 1997; Zaks, Park, Rosenblum, & Kurths, 1999). One important appli- cation of periodic orbit theory is the control of chaotic systems (Ott, Grebogi, & Yorke, 1990; Schiff et al., 1994; for a review, see Ott & Spano, 1995). Consequently, the detection of unstable periodic orbits in experimental data has become a central issue in nonlinear time series analysis (Pierson & Moss, 1995; Pei & Moss, 1996; So et al., 1997a, 1997b; So, Francis, Netoff, Gluckman, & Schiff, 1998). Systems with unstable periodic orbits are suggestive for the presence of deterministic chaos. At a more general level, the detection of these orbits provides evidence for determin- This work was supported by Deutsche Forschungsgemeinschaft (DFG), grant INK 12/A1 (project B1). This research is part of the interdisciplinary project Formal Models of Cognitive Complexity at the University of Potsdam. We thank G. Zo ¨ller and S. Hainzl (Potsdam) for valuable discussions on the detection of unstable periodic orbits and H.-H. Schulze (Marburg) and N. Stollenwerk (Cambridge/U.K.) for valuable comments on an early version of the manuscript. Address correspondence and reprint requests to Ralf Engbert, Department of Psychology, University of Potsdam, POB 601553, D-14415 Potsdam, Germany or E-mail: engbert@rz.uni-potsdam.de 107 0278-2626/01 $35.00  2001 Elsevier Science All rights reserved.