Structural susceptibility and separation of time scales in the van der Pol oscillator.

We use an extension of the van der Pol oscillator as an example of a system with multiple time scales to study the susceptibility of its trajectory to polynomial perturbations in the dynamics. A striking feature of many nonlinear, multiparameter models is an apparently inherent insensitivity to large-magnitude variations in certain linear combinations of parameters. This phenomenon of "sloppiness" is quantified by calculating the eigenvalues of the Hessian matrix of the least-squares cost function. These typically span many orders of magnitude. The van der Pol system is no exception: Perturbations in its dynamics show that most directions in parameter space weakly affect the limit cycle, whereas only a few directions are stiff. With this study, we show that separating the time scales in the van der Pol system leads to a further separation of eigenvalues. Parameter combinations which perturb the slow manifold are stiffer and those which solely affect the jumps in the dynamics are sloppier.