Optimal synchronization to a limit cycle

Area of Science:

• Nonlinear dynamics
• Oscillatory systems
• Theoretical physics

Background:

• The van der Pol oscillator naturally approaches a limit cycle over infinite time.
• External forcing can accelerate the system's convergence to the limit cycle.

Purpose of the Study:

• To minimize non-conservative work required to drive the van der Pol oscillator to its limit cycle in finite time.
• To establish a speed-limit inequality relating time and work.
• To generalize findings to Liénard oscillators.

Main Methods:

• Phase plane analysis of the van der Pol oscillator.
• Calculus of variations to minimize work.
• Mathematical generalization to Liénard equation.

Main Results:

• A speed-limit inequality was derived, quantifying the trade-off between finite-time convergence and non-conservative work.
• The methodology was successfully generalized to the broader class of Liénard oscillators.
• Analysis of minimizing total external work was also performed.

Conclusions:

• Finite-time control of nonlinear oscillators is achievable with minimized work.
• The speed-limit inequality provides fundamental constraints for driving oscillatory systems.
• This work offers insights into optimal control strategies for nonlinear dynamics.