Optimal control in a noisy system

Area of Science:

• Nonlinear Dynamics
• Chaos Theory
• Control Theory

Background:

• Unstable periodic orbits (UPOs) are fundamental in understanding chaotic systems.
• Controlling UPOs is crucial for harnessing chaotic dynamics but challenging in noisy environments.

Purpose of the Study:

• To develop a robust method for controlling UPOs in the presence of noise.
• To adapt control strategies for varying noise intensities and system complexities.

Main Methods:

• Formulating control as an optimization problem to derive a control matrix (A).
• Applying the method to the Rossler, Lorenz, and a hyperchaotic system.
• Utilizing Lyapunov exponents and singular value decomposition (SVD) for analysis and noise reduction.

Main Results:

• The control strategy effectively stabilizes UPOs in tested chaotic systems, even with added noise.
• For low noise, a simple control matrix suffices; higher noise necessitates a full matrix.
• A noise-cleaning strategy using SVD improves UPO and exponent estimation.

Conclusions:

• The proposed optimization-based control is effective for stabilizing UPOs in noisy chaotic systems.
• The method's adaptability to noise levels and system types makes it broadly applicable.
• The SVD-based noise reduction technique offers a consistent approach for experimental applications.