Clocking convergence to a stable limit cycle of a periodically driven nonlinear pendulum

Area of Science:

• Nonlinear Dynamics
• Chaos Theory
• Control Systems

Background:

• Periodically driven nonlinear pendulums exhibit complex transient behaviors.
• Understanding convergence to stable limit cycles is crucial for system analysis.
• Existing methods for assessing transient processes can be complex.

Purpose of the Study:

• To analyze the convergence to a stable limit cycle in a periodically driven nonlinear pendulum.
• To assess transient processes using the concept of H-rank.
• To propose a method for controlling transient processes and minimizing convergence time.

Main Methods:

• Application of the H-rank of a scalar sequence for transient process assessment.
• Utilizing the circle map to illustrate the manifold of non-asymptotic convergence.
• Stroboscopic representation of transient data for analysis.

Main Results:

• The H-rank effectively assesses transient processes in the driven pendulum system.
• A manifold of non-asymptotic convergence to a stable limit cycle is demonstrated in the stroboscopic representation.
• The existence of a complex structure of non-asymptotic convergence is illustrated.

Conclusions:

• Transient dynamics of driven nonlinear pendulums can be effectively analyzed using H-rank.
• A controllable manifold for non-asymptotic convergence to limit cycles exists.
• A simple external impulse method can minimize the transition time to stable limit cycles.