John Guckenheimer

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  Using Randomness in Locomotion
  John Guckenheimer

  Animal locomotion results from interactions of rhythmic movements of the body with the environment.

  
  

  A biomechanical model for the movements may yield an unstable periodic orbit of a dynamical system. In these circumstances, the animal must exercise active control to maintain stability. This lecture will discuss the problem of estimating the Floquet multipliers which characterize local stability properties of a periodic orbit. The information needed depends upon perturbations from the periodic orbit. Animals can either rely upon fluctuations in the environment (''noise'') or generate excitations that move the organism off the orbit to obtain the required information. We discuss these alternatives in terms of the distribution of residuals between the animal's trajectory and the target periodic orbit for its motion.

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  Predictive Models of Running
  John Guckenheimer
  This talk describes joint work with Horst-Moritz Maus, Shai Revzen, Christian Ludwig, Johann Reger and Andre Seyfarth. We utilize motion capture movies of individuals to construct low dimensional models of treadmill running. Our starting point was to assume that the data could be represented as a noisy limit cycle in a hybrid (piecewise smooth) dynamical system. Springmass (SLIP) models are concrete physical systems that have long been used as caricatures of running. Feedback that determines foot placement is an essential ingredient of the SLIP models. Our primary goal was to find still low dimensional extensions of SLIP that predict center of mass location for one and two steps ahead almost as well as is possible using all of the motion capture data.
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  Math to Bio and Bio to Math
  John Guckenheimer
  The interchange between dynamical systems theory with biology has had lasting impact upon both. As biology becomes increasingly quantitative, this relationship is likely to strengthen still further. This lecture will review my experience as a mathematician working at the interface with biology, emphasizing the role of multiple time scales in biological models. It will also look discuss why the solution of outstanding mathematical questions is essential to progress within biology.