Parametric Resonance in Networked Oscillators

TL;DR

The paper studies parametric resonance in networks of linear oscillators with periodically modulated edge strengths, casting edge forcing as a perturbation of the graph Laplacian. By projecting onto the Laplacian eigenspace, it derives a scalar Mathieu-type description for full-network forcing and, for subnet forcing, a vector-valued, multiple-scale perturbation framework that identifies first-order instability tongues at frequencies given by sums of Laplacian eigenfrequencies. The key results connect tongue locations and widths to eigenvalues and eigenvectors of the Laplacian, with controllability considerations clarifying when forcing can excite specific modes. The analysis is specialized to ring and torus topologies to obtain explicit, topology-dependent stability diagrams and scaling laws, while generalizing to arbitrary subnet forcing and discussing extensions to damping and practical design implications.

Abstract

We investigate parametric resonance in oscillator networks subjected to periodically time-varying oscillations in the edge strengths. Such models are inspired by the well-known parametric resonance phenomena for single oscillators, as well as the potential rich phenomenology when such parametric excitations are present in a variety of applications like deep brain stimulation, AC power transmission networks, as well as vehicular flocking formations. We consider cases where a single edge, a subgraph, or the entire network is subjected to forcing, and in each case, we characterize an interesting interplay between the parametric resonance modes and the eigenvalues/vectors of the graph Laplacian. Our analysis is based on a novel treatment of multiple-scale perturbation analysis that we develop for the underlying high-dimensional dynamic equations.

Figures (11)

• Figure 1: The stability diagram for the undamped Mathieu equation ( left), and for the Mathieu equation with slight damping ( right), where the gray-shaded regions are the unstable regions in the $(\epsilon,\omega)$ plane of the parametric excitation amplitude $\epsilon$ and frequency $\omega$. In the undamped case, the system is unstable for arbitrarily small excitation amplitude $\epsilon$ at excitation frequencies that are $2,1,1/2,1/3,...$ times the natural frequency $\omega_{\rm n}$. The shaded areas near $\omega\approx \omega_{\rm n}/k, ~k=1/2,1,2,3,4,..$ are called the "Arnold tongues", which get progressively thinner with increasing $k$. Thus with slight damping, only the first tongue at $\omega\approx 2\omega_{\rm n}$ represents an instability phenomenon that can be triggered with small excitation amplitude $\epsilon$. Only the first three tongues are shown in the diagrams above.
• Figure 2: Stability diagrams for networked second order oscillators corresponding connected in a triangle (top) and square (bottom) layout with specified nominal edge weights. The stability diagrams are overlapping copies of scaled versions of Figure \ref{['fig:mathieu-instability-rescaled']}, one per mode of the system. The region of instability is the union of the unstable regions for each mode.
• Figure 3: Depiction of the first-order tongues of the single-edge excitation model (\ref{['eq:linearized-swing-dynamics-perturbed-one-edge']}). All possible locations of such tongues are given by sums $\omega_{lm}= { \Omega}_l + { \Omega}_m$ of natural frequencies ${ \Omega}_k \in \sqrt{{\rm eigs}(L)}$ of the unperturbed system, which in turn are determined by the eigenvalues of the network Laplacian $L$. The "widths" $a_{lm}$ of each tongue are determined by both eigenvalues and eigenvectors of $L$ by (\ref{['a_widths.eq']}).
• Figure 4: The numerically computed stability diagram for a triangular graph with unequal edge weights. The overlaid red lines are the critical parametric forcing frequencies predicted (to first order) by the regular perturbation analysis of Section \ref{['sec:regular-perturbation-analysis']}. The narrow tongues at $\omega \approx 2.9$ and $\omega \approx 4.7$ correspond to the predictions from higher order perturbations (\ref{['eq:basic-instability-criterion-filtered-Oeps2-plus']} and \ref{['eq:basic-instability-criterion-filtered-Oeps2-minus']}). We observe that while the Arnold tongue positions are predicted correctly by the perturbation analysis, the prediction of $2.85$ is a false positive. The perturbation analysis is refined using a multiple time-scale method in \ref{['sec:vector-valued-multiple-scale-perturbation-analysis']}.
• Figure 5: A comparison of the numerically computed stability diagram with the stability estimates from perturbation analysis for swing dynamics on triangular graph with one time-periodically perturbed edge. The wide Arnold tongue positions in $(\omega, \epsilon)$ space (black curves) correspond to first order perturbation analysis and are predicted correctly (dashed red lines). The multiple time-scale analysis from section \ref{['sec:vector-valued-multiple-scale-perturbation-analysis']} eliminates the false positives observed in figure \ref{['fig:basic_instability_criterion_unfiltered']}. Additionally, the estimate of the slope of the Arnold tongues as $\epsilon \to 0$ computed using \ref{['eq:k-to-omega-transformation-2']} (solid red lines) match the numerical results closely, and thus provide a simple way of predicting the degree of instability of each critical parametric forcing frequency $\omega$ from the properties of the graph Laplacian. The narrower, partially resolved Arnold tongues correspond to higher order terms in the perturbation expansion \ref{['eq:swing-dynamics-2nd-order-regular-perturbation']}.

Theorems & Definitions (3)

• Theorem 1: Instability criteria
• Lemma 1: Eigenvector test for harmonic oscillator networks
• proof