An approach to designing dynamical systems with a three-dimensional state space is described that can be used to build a variety of non-periodic oscillators. The state space is taken to be a 3sphere, which is identified with the manifold of unit quaternions. Any such system can be described as a quaternion-valued ordinary differential equation, which is digitally realized using an approximation as a finite difference e quation. Two examples are shown. Compared to previous applications of dynamical systems used to generate audio samples, the approach described here offers a wide choice of specific flows which can neither diverge nor approach a stable limit point.