Optimal synchronization to a limit cycle

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 minimizing 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 Liénard equation. Finally, we also look into the problem of minimizing the total work done by the external force.