Structural Susceptibility and Separation of Time Scales in the van der Pol Oscillator

TL;DR

This paper investigates how the van der Pol oscillator's multiple time scales influence its sensitivity to parameter changes, revealing a separation in eigenvalues that distinguishes stiff from sloppy directions in parameter space.

Contribution

It demonstrates that separating time scales in the van der Pol oscillator enhances the eigenvalue separation, clarifying the roles of different parameter directions in system sensitivity.

Findings

Eigenvalues span many orders of magnitude, indicating sloppiness.

Time scale separation increases eigenvalue separation.

Slow manifold perturbations are stiffer than transient effects.

Abstract

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, multi-parameter 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 which 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 transients in the dynamics are sloppier.