Due to their stability and robustness properties, Stable Dynamical Systems (SDS) have received considerable attention as a means of representing motions in learning from demonstration tasks. Designing vector fields that fit complex trajectories while ensuring stability still remains a key challenge; although recent deep learning-based methods have shown substantial progress in this direction, their tendency to overfit to demonstration trajectories often leads to undesirable behaviors, particularly as tasks deviate from demonstrations. At a fundamental level, the only reliable way to address this lack of generalization is to provide supervision in out-of-demonstration regions. Focusing on two types of general behaviors, mimicking and contracting, we propose a Behavior-Controllable Stable Dynamics Model (BCSDM), a one-parameter family of SDS that allows users to adjust the system's overall behavior depending on user intent. We show how to extend BCSDM to accommodate demonstrations of multiple tasks, and also propose a Deep Operator Vector Field (DeepOVec) for memory-efficient encoding of multiple dynamical systems. Extensive experiments on tasks that involve mimicking or contracting behaviors demonstrate the advantages of BCSDMs over existing state-of-the-art SDS learning methods.