In this paper, we consider the problem of parameter sensitivity in models of complex dynamical systems through the lens of information geometry. We calculate the sensitivity of model behavior to variations in parameters. In most cases, models are sloppy, that is, exhibit an exponential hierarchy of parameter sensitivities. We propose a parameter classification scheme based on how the sensitivities scale at long observation times. We show that for oscillatory models, either with a limit cycle or a strange attractor, sensitivities can become arbitrarily large, which implies a high effective dimensionality on the model manifold. Sloppy models with a single fixed point have model manifolds with low effective dimensionality, previously described as a "hyper-ribbon." In contrast, models with high effective dimensionality translate into multimodal fitting problems. We define a measure of curvature on the model manifold which we call the winding frequency that estimates the density of local minima in the model's parameter space. We then show how alternative choices of fitting metrics can "unwind" the model manifold and give low winding frequencies. This prescription translates the model manifold from one of high effective dimensionality into the hyper-ribbon structures observed elsewhere. This translation opens the door for applications of sloppy model analysis and model reduction methods developed for models with low effective dimensionality.