Dynamics of bow-tie shaped bursting: Forced pendulum with dynamic feedback

Area of Science:

• Nonlinear Dynamics
• Mechanical Systems Analysis
• Control Theory

Background:

• Driven pendulums are complex mechanical systems exhibiting diverse behaviors.
• Feedback control introduces rich dynamics, including oscillations.
• Previous studies have not detailed the specific bursting patterns observed here.

Purpose of the Study:

• To explore the parameter space of a driven pendulum with damping, torque, and feedback control.
• To identify and characterize novel oscillatory behaviors.
• To understand the underlying mechanisms of observed bursting phenomena.

Main Methods:

• Detailed analysis of the mechanical system's parameter space.
• Investigation of dynamic feedback control strategies.
• Identification of bifurcation points leading to distinct states.
• Analytical estimation of the resting period between bursts.

Main Results:

• Discovery of a unique bow-tie shaped bursting oscillatory behavior.
• Observation of this behavior in a specific parameter regime with small driving frequencies.
• Identification of saddle-focus bifurcations as the cause of transitions between quiescent and spiking states.
• Analytical estimation of the resting period (T_rest) between successive bursts.

Conclusions:

• The driven pendulum system exhibits complex bursting oscillations under specific feedback control conditions.
• Saddle-focus bifurcations play a crucial role in the generation and cessation of these bursting patterns.
• The analytical estimation of the resting period provides valuable insight into the system's dynamics.