Delayed feedback control of dynamical systems at a subcritical Hopf bifurcation

Area of Science:

• Nonlinear Dynamics and Control Systems
• Bifurcation Theory
• Chaos Theory

Background:

• Dynamical systems can exhibit unstable periodic orbits near bifurcation points.
• Subcritical Hopf bifurcations present challenges for stabilization due to their inherent instability.
• Controlling such systems often requires advanced feedback strategies.

Purpose of the Study:

• To develop and analyze a delayed feedback control strategy for unstable periodic orbits.
• To investigate the role of unstable degrees of freedom in the control loop.
• To explore the impact of nonlinear coupling between the system and controller.

Main Methods:

• Analytical treatment using the method of averaging near the bifurcation point.
• Incorporation of an unstable degree of freedom into the feedback control design.
• Modeling nonlinear coupling between the controlled dynamical system and the controller.

Main Results:

• Demonstration of effective stabilization of torsion-free unstable periodic orbits.
• Validation of the necessity for utilizing an unstable degree of freedom in feedback control.
• Confirmation of the influence of nonlinear coupling on control performance.

Conclusions:

• Delayed feedback control is a viable method for stabilizing unstable periodic orbits in systems with subcritical Hopf bifurcations.
• The proposed control strategy, incorporating unstable modes and nonlinear coupling, offers a robust approach.
• The findings are applicable to understanding and controlling complex nonlinear phenomena in various scientific and engineering fields.