Delayed feedback control of forced self-sustained oscillations

Area of Science:

• Nonlinear dynamics
• Control theory
• Chaos theory

Background:

• The van der Pol oscillator is a classic model for self-sustaining oscillations.
• Synchronization of nonlinear oscillators is crucial in various scientific and engineering fields.
• Unstable periodic orbits can limit the synchronization capabilities of oscillators under external forcing.

Purpose of the Study:

• To investigate the use of delayed feedback control to extend the synchronization domain of a van der Pol oscillator.
• To stabilize unstable periodic orbits outside the natural synchronization region.
• To derive analytical expressions for the extended synchronization domain and optimal control parameters.

Main Methods:

• Applying delayed feedback control to a weakly nonlinear van der Pol oscillator.
• Utilizing a Hopf bifurcation analysis to derive a simplified averaged equation.
• Analytical treatment of the averaged equation with delayed feedback.
• Numerical simulations of the original delay-differential equations to validate the analytical findings.

Main Results:

• Delayed feedback control successfully stabilizes unstable periodic orbits.
• The synchronization domain of the van der Pol oscillator is significantly extended.
• Analytical expressions for the extended synchronization domain and optimal control gain were derived.
• Control force required for stabilization is minimal and vanishes upon successful synchronization.

Conclusions:

• Delayed feedback control is an effective method for enhancing the synchronization of van der Pol oscillators.
• The analytical approach provides accurate predictions for the controlled system's behavior.
• This study offers a method to overcome synchronization limitations in nonlinear systems using minimal control effort.