Universal energy scaling law for optimally excited nonlinear oscillators

Area of Science:

• Nonlinear dynamics
• Quantum mechanics
• Optimal control theory

Background:

• Nonlinear oscillators are fundamental in physics and engineering.
• Understanding energy absorption is crucial for system design and performance.
• Previous methods lacked a universally applicable approach for optimizing energy absorption.

Purpose of the Study:

• To compute the optimal temporal profile of an external driving force.
• To maximize energy absorption in driven nonlinear oscillators.
• To develop a universally applicable computational technique.

Main Methods:

• Utilizing optimal control theory with constrained force field amplitude.
• Deriving quasianalytical solutions for the optimal driving force.
• Applying the technique to the undamped Duffing oscillator and a quantum two-level system.

Main Results:

• The maximal energy absorption of any nonlinear oscillator grows quadratically with time (t^2) under an optimized force field.
• The asymptotic time-dependence of maximum amplitude growth follows X(t)∼t^{2/α}, determined by the nonlinearity degree α.
• For a two-level system, maximal inversion shows n-cycle thresholds and growth self-termination, not monotonic increase.

Conclusions:

• A universal power-law relationship governs maximal energy absorption in nonlinear oscillators.
• Optimal control provides a powerful framework for maximizing energy absorption.
• The findings have implications for designing systems that efficiently absorb energy, from mechanical to quantum regimes.