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.

Figure 1

(a) Eigenvalues of the Hessian matrix of the cost of fitting at $\mu =1$. (b) and (c) Top row: one period of time series $x\left(\tau \right)$ (dotted line) and $y\left(\tau \right)$ (solid line) for $0<\tau <1$ are shown for $\mu =1$ and 100 as a function of time along with schematic error bars for the data-fitting of the trajectory of variable $y$. (d) Eigenvalues for $\mu =100$ are shown on the right. As $\mu \to \infty$, the orbit collapses onto the critical manifold with the trajectory spending most of its time on the slow manifold and vanishingly short on the jumps. Also shown in (b) and (c), bottom row, is the orbit in the $xy$ plane (solid line) and the critical manifold (dashed line).

Figure 2

Eigenvalues of the Hessian matrix are shown here as a function of $\mu$. The range $1\le \mu \le 100$ corresponds to a ratio of time scales $1\le \epsilon \le 10000$. The five largest eigenvalues (solid lines) correspond to stiff directions in the parameter space: these directions perturb the slow manifold. The remainder (dashed lines) affect the transient part of the trajectory, which becomes smaller with an increasing separation of time scales, and hence these directions are decreasingly relevant.

Figure 3

Hessian eigenvectors are shown for $\mu =1,10,$ and 100. Each colored small square shows the magnitude of an eigenvector component (the scale bar is shown on the right). Eigenvectors for each $\mu$ are sorted so that the stiffer ones appear on the left; individual components are sorted so that “slow parameters” appear on the top. Note that with increasing $\mu$, the stiff and sloppy eigenvectors are separated by parameters: The stiff eigenvectors only have projections along the slow parameters, which perturb the slow manifold, whereas the sloppy directions have projections along the fast parameters, which mainly perturb the jumps.

Figure 4

Top three rows: Eigenpredictions $\delta y_k$ for $k=0,3,6,9,12$ at $\mu =1,$ 10, and 100 are shown as solid red lines for stiff modes and dashed green lines for sloppy modes. These curves show the response of perturbations if the parameters are changed infinitesimally along the Hessian eigenvectors: A parameter change of norm $\epsilon$ along eigendirection $n$ will change the trajectories by ${\lambda }_n\epsilon$ times the eigenpredicton. Dotted gray lines show the unperturbed van der Pol solution for comparison ($y$ scale on the right-hand side). As the time scales separate, the amplitudes of the sloppiest eigenpredictions increase (roughly in proportion to $\mu$), becoming increasingly concentrated at the jumps. The bottom row shows the eigencycles for $\mu =100$ as solid red lines and green dashed lines corresponding to the perturbations in row 3 (i.e., the new limit cycle for a perturbation of strength $\epsilon \sim 1/{\lambda }_n$). These curves show how the van der Pol orbit changes with perturbations along the Hessian eigenvectors. Both the stiff and the sloppy modes change the orbit at the jumps (occurring at the extrema in the dashed lines); the stiff modes also change behavior at the slow manifold, whereas the sloppy modes only affect the jumps.