  We propose a general strategy for reduced order modeling of systems that
display highly nonlinear oscillations. By considering a continuous family of
forced periodic orbits defined in relation to a stable fixed point and
subsequently leveraging phase-amplitude-based reduction strategies, we arrive
at a low order model capable of accurately capturing nonlinear oscillations
resulting from arbitrary external inputs. In the limit that oscillations are
small, the system dynamics relax to those obtained from local linearization,
i.e.,~that can be fully described using linear eigenmodes. For larger amplitude
oscillations, the behavior can be understood in terms of the dynamics of a
small number of nonlinear modes. We illustrate the proposed strategy in a
variety of examples yielding results that are substantially better than those
obtained using standard linearization-based techniques.
