Smart driving accelerates synchronization of nonlinear oscillators

What is it about?

A key feature of self-oscillatory systems is their tendency to a periodic behavior, a closed orbit known as a limit cycle, for long enough times (independently of their initial state). These systems approach their long-time periodic behavior over a certain characteristic "natural" timescale, but this timescale can be very long--for instance, when the nonlinear damping is small. In this paper, we investigate how this natural synchronization to the limit cycle can be accelerated by adding a smartly engineered driving force, in a wide family of oscillators with nonlinear damping--the Liénard equation. Also, we look into the energetic cost of such an accelerated synchronization to the limit cycle.

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Photo by Ed Wingate on Unsplash

Photo by Ed Wingate on Unsplash

Why is it important?

Self-oscillatory systems are ubiquitous in science, not only in physics but also in neuroscience, physiology, economics, etc. Our findings show how one can beat their "natural" timescale to approach the limit cycle by a very large margin. A first key result is the emergence of a speed limit inequality, i.e., a trade-off between the synchronization time and the energetic cost. Remarkably, the energetic cost can be kept controlled while obtaining a huge, formally diverging, acceleration of the synchronization. A second key result is our derivation of the optimal driving force, in the sense of minimizing energetic cost, for a given synchronization time.