Controlling transitions in a Duffing oscillator by sweeping parameters in time

Area of Science:

• Nonlinear Dynamics
• Oscillator Physics

Background:

• Duffing oscillators exhibit complex behaviors under external driving frequencies.
• Understanding state transitions is crucial for controlling nonlinear systems.

Purpose of the Study:

• To investigate controlled state transitions in a high-Q Duffing oscillator using a time-varying driving frequency.
• To analyze the role of unstable fixed points in mediating these transitions.

Main Methods:

• Simulating a weakly nonlinear Duffing oscillator with a frequency sweep from sigma i to sigma f at rate r.
• Analyzing the system's dynamics, focusing on transient behavior near saddle-type fixed points.

Main Results:

• Controlled transitions between two stable states were achieved via frequency sweeping.
• Transitions proceed through a long-lived transient state near a saddle-type unstable fixed point.
• The transient lifetime was found to scale as -(ln|r-rc|)lambda r, where rc is the critical sweep rate and lambda r is the repulsive eigenvalue.

Conclusions:

• Frequency sweeping offers a method for controlled state transitions in Duffing oscillators.
• The dynamics near unstable saddle points significantly influence the transition process.
• The derived scaling law provides insights into the transient behavior and its dependence on system parameters.