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arXiv · 2407.03435

Optimal synchronisation to a limit cycle

C. Ríos-Monje·C. A. Plata·D. Guéry-Odelin·A. Prados
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Abstract

In the absence of external forcing, all trajectories on the phase plane of the van der Pol oscillator tend to a closed, periodic, trajectory -- the limit cycle -- after infinite time. Here, we drive the van der Pol oscillator with an external time-dependent force to reach the limit cycle in a given finite time. Specifically, we are interested in minimising the non-conservative contribution to the work when driving the system from a given initial point on the phase plane to any final point belonging to the limit cycle. There appears a speed limit inequality, which expresses a trade-off between the connection time and cost -- in terms of the non-conservative work. We show how the above results can be { generalized to the broader family of non-linear oscillators given by} the Li\'enard equation. Finally, we also look into the problem of minimising the total work done by the external force.

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C. Ríos-Monje, C. A. Plata, D. Guéry-Odelin, A. Prados. 2024-07-03. Optimal synchronisation to a limit cycle. https://doi.org/10.1063/5.0227287

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