A systematic delayed feedback control approach for unstable periodic orbits in chaotic systems with unknown...

Area of Science:

• Nonlinear Dynamics
• Control Theory
• Chaos Theory

Background:

• Delayed feedback control is common for stabilizing unstable periodic orbits in chaotic systems.
• Existing linear and nonlinear control methods often require system parameter knowledge and lack mechanisms to access natural system responses.
• Unstable periodic orbits are crucial for understanding chaotic system dynamics.

Purpose of the Study:

• To propose a novel adaptive control method for chaotic systems with entirely unknown parameters.
• To achieve stabilization of unstable periodic orbits and convergence to desired periodic responses.
• To develop a controller that steers system dynamics toward natural periodic responses, including unstable periodic orbits.

Main Methods:

• Development of an adaptive control strategy based on delayed system state information.
• Design of a mechanism to guide system dynamics towards natural periodic responses.
• Simulation-based demonstration and application to a Duffing oscillator.

Main Results:

• The proposed control method effectively stabilizes chaotic systems with unknown parameters.
• System states converge to a periodic response with a desired time period.
• The controller successfully steers the system towards natural periodic responses, including unstable periodic orbits, noninvasively.

Conclusions:

• The developed adaptive control strategy offers a noninvasive method for controlling chaotic systems with unknown parameters.
• This approach enables the stabilization of unstable periodic orbits and the exploration of natural system dynamics.
• The method is validated through simulations and its utility in revealing unstable periodic orbits within chaotic attractors is demonstrated.