Beyond Phase Reduction: Amplitude Collapse in Optimal Control of Coupled Oscillators
Abstract
We solve the four-dimensional Hamilton-Jacobi-Bellman (HJB) equation for two diffusively coupled Stuart-Landau-like oscillators to obtain full-state optimal feedback control. A sweep over coupling strength reveals a sharp change in the numerically optimal strategy: below a threshold coupling value, the controller steers the phase difference toward anti-phase while keeping both oscillators near the limit cycle, as reduced-order models would suggest. Above this threshold, the HJB solution changes qualitatively; the controller transiently collapses one oscillator's amplitude to near zero, thereby enabling large phase repositioning near the origin before rebuilding its amplitude. Direct gradient-based and stochastic optimization do not recover this lower-cost collapse trajectory from the initializations considered, suggesting that it occupies a region of the control landscape that is difficult to access by direct search. A joint sweep over nonisochronicity and coupling shows that collapse can occur even for an isochronous oscillator: phase repositioning near the origin can favor an off-cycle strategy. Nonisochronicity provides an additional energetic benefit through a phase-velocity surplus at small amplitude, quantitatively accounting for the observed reduction in control cost. Comparisons with uncoupled and coupled phase-reduced baselines show that phase models become increasingly inaccurate and cost significantly more energy for strong coupling. Results for coupled Van der Pol oscillators further demonstrate that exploitation of off-cycle dynamics is not specific to the Stuart-Landau-like oscillators.
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BibTeX
@misc{rajabi2026beyond,
title = {{Beyond Phase Reduction: Amplitude Collapse in Optimal Control of Coupled Oscillators}},
author = {Faranak Rajabi and Frédéric Gibou and Jeff Moehlis},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16449},
eprint = {2609.16449},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16449},
}
RIS
TY - UNPB
TI - Beyond Phase Reduction: Amplitude Collapse in Optimal Control of Coupled Oscillators
AU - Faranak Rajabi
AU - Frédéric Gibou
AU - Jeff Moehlis
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16449
UR - https://arxiv.org/abs/2609.16449
AB - We solve the four-dimensional Hamilton-Jacobi-Bellman (HJB) equation for two diffusively coupled Stuart-Landau-like oscillators to obtain full-state optimal feedback control. A sweep over coupling strength reveals a sharp change in the numerically optimal strategy: below a threshold coupling value, the controller steers the phase difference toward anti-phase while keeping both oscillators near the limit cycle, as reduced-order models would suggest. Above this threshold, the HJB solution changes qualitatively; the controller transiently collapses one oscillator's amplitude to near zero, thereby enabling large phase repositioning near the origin before rebuilding its amplitude. Direct gradient-based and stochastic optimization do not recover this lower-cost collapse trajectory from the initializations considered, suggesting that it occupies a region of the control landscape that is difficult to access by direct search. A joint sweep over nonisochronicity and coupling shows that collapse can occur even for an isochronous oscillator: phase repositioning near the origin can favor an off-cycle strategy. Nonisochronicity provides an additional energetic benefit through a phase-velocity surplus at small amplitude, quantitatively accounting for the observed reduction in control cost. Comparisons with uncoupled and coupled phase-reduced baselines show that phase models become increasingly inaccurate and cost significantly more energy for strong coupling. Results for coupled Van der Pol oscillators further demonstrate that exploitation of off-cycle dynamics is not specific to the Stuart-Landau-like oscillators.
ER -