Biological Cybernetics

, 95:645 | Cite as

Engineering entrainment and adaptation in limit cycle systems

From biological inspiration to applications in robotics
  • Jonas BuchliEmail author
  • Ludovic Righetti
  • Auke Jan Ijspeert
Original Paper


Periodic behavior is key to life and is observed in multiple instances and at multiple time scales in our metabolism, our natural environment, and our engineered environment. A natural way of modeling or generating periodic behavior is done by using oscillators, i.e., dynamical systems that exhibit limit cycle behavior. While there is extensive literature on methods to analyze such dynamical systems, much less work has been done on methods to synthesize an oscillator to exhibit some specific desired characteristics. The goal of this article is twofold: (1) to provide a framework for characterizing and designing oscillators and (2) to review how classes of well-known oscillators can be understood and related to this framework. The basis of the framework is to characterize oscillators in terms of their fundamental temporal and spatial behavior and in terms of properties that these two behaviors can be designed to exhibit. This focus on fundamental properties is important because it allows us to systematically compare a large variety of oscillators that might at first sight appear very different from each other. We identify several specifications that are useful for design, such as frequency-locking behavior, phase-locking behavior, and specific output signal shape. We also identify two classes of design methods by which these specifications can be met, namely offline methods and online methods. By relating these specifications to our framework and by presenting several examples of how oscillators have been designed in the literature, this article provides a useful methodology and toolbox for designing oscillators for a wide range of purposes. In particular, the focus on synthesis of limit cycle dynamical systems should be useful both for engineering and for computational modeling of physical or biological phenomena.


Instantaneous Frequency Central Pattern Generator Phase Oscillator Phase Point Phase Response Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Jonas Buchli
    • 1
    Email author
  • Ludovic Righetti
    • 1
  • Auke Jan Ijspeert
    • 1
  1. 1.Biologically Inspired Robotics Group, School of Computer & Communication SciencesEcole Polytechnique Fédérale de LausanneLausanneSwitzerland

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