Collins and Stewart point out that the characteristic gaits of quadrupeds --- walk, trot, pace, bound, etc. --- can be described by spatio-temporal symmetries of periodic functions. For example, when a horse paces it moves both left legs in unison and then both right legs and so on. This form of motion is determined by two symmetries: (1) Interchange front and back legs, and (2) swap left and right legs with a half-period phase shift.

Biologists postulate the existence of central pattern generators (CPGs) in the neural system that send periodic signals to the legs. These CPGs can be thought of as electrical circuits that produce periodic signals and can be modeled by coupled systems of differential equations with symmetries based on permutation of the legs.

In this lecture we discuss animal gaits and describe how periodic solutions with prescribed spatio-temporal symmetry can be formed by Hopf bifurcation in systems with symmetry. We then a way to construct coupled ODE systems that will naturally produce periodic solutions with the symmetries of quadrupedal gaits. This network scales to many legged animals.

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1996-1997 Mathematics in High Performance Computing