A general family of morphed nonlinear phase oscillators with arbitrary limit cycle shape

We present a general family of nonlinear phase oscillators which can exhibit arbitrary limit cycle shapes and infinitely large basins of attraction. This general family is the superset of familiar control methods like PD-control over a periodic reference, and rhythmic Dynamical Movement Primitives. The general methodology is based on morphing the limit cycle of an existing phase oscillator with phase-based scaling functions to obtain a desired limit cycle behavior. The introduced methodology can be represented as first, second, or n-th order dynamical systems. The elegance of the formulation provides the possibility to define explicit arbitrary convergence behavior for simple cases. We analyze the stability properties of the methodology with the Poincaré-Bendixson theorem and the Contraction Theory, and use numerical simulations to show the properties of some oscillators that are a subset of this general family.