The authors conducted two experiments that served as a test bed for applying the recently developed uncontrolled manifold (UCM) approach to rhythmic motor coordination, which has been extensively investigated from a coordination dynamics perspective. The results of two experiments, one investigating within-person and one investigating between-persons rhythmic movement coordination, identified synergistic behaviors in both of those types of coordination. Stronger synergies were identified for in-phase than antiphase coordination, at the endpoints of the movement cycles compared with the midpoints, for movement frequencies closer to the intrinsic frequency of the coordinated limbs, and for within-person coordination. Frequency detuning did not weaken the strength of interlimb rhythmic coordination synergies. The results suggest the synergistic behavior captured by the UCM analysis may be identifiable with the strength of coupling between the coordinated limbs. The UCM analysis appears to distinguish coordination parameters that affect coupling strength from parameters that weaken coordination attractors.

A well-known lower bound for the number of fixed points of a self-map / : X -> X is the Nielsen number N(f). Unfortunately, the Nielsen number is difficult to calculate. The Lefschetz number £(/), on the other hand, is readily computable, but usually does not estimate the number of fixed points. It is known that N(f) = \L(f)\ for all maps on nilmanifolds (homogeneous spaces of nilpotent Lie groups) and that N(f) > \L(f)\ for all maps on solvmanifolds (homogeneous spaces of solvable Lie groups). Typically, though, the strict inequality holds, so the Nielsen number cannot be completely computed from the Lefschetz number. In the present work, we produce a large class of solvmanifolds for which N(f) = \L(f)\ for all self maps. This class includes exponential solvmanifolds: solvmanifolds for which the corresponding exponential map is surjective. Our methods provide Nielsen and Lefschetz number product theorems for the Mostow fibrations of these solvmanifolds, even though the maps on the fibers in general will belong to varying homotopy classes.

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Abstract. Given invariant sets A , B , and C , and connecting orbits A -» B and B -> C , we state very general conditions under which they bifurcate to produce an A -» C connecting orbit. In particular, our theorem is applicable in settings for which one has an admissible semiflow on an isolating neighborhood of the invariant sets and the connecting orbits, and for which the Conley index of the invariant sets is the same as that of a hyperbolic critical point. Our proof depends on the connected simple system associated with the Conley index for isolated invariant sets. Furthermore, we show how this change in connected simple systems can be associated with transition matrices, and hence, connection matrices. This leads to some simple examples in which the nonuniqueness of the connection matrix can be explained by changes in the connected simple system.