Control of chaos via an unstable delayed feedback controller

Area of Science:

• Nonlinear dynamics
• Chaos theory
• Control theory

Background:

• Delayed feedback control is a common method for stabilizing unstable periodic orbits within chaotic attractors.
• This technique is limited to periodic orbits with finite torsion, excluding certain types of orbits.

Purpose of the Study:

• To propose a modification to delayed feedback control to overcome topological limitations.
• To enhance the applicability of delayed feedback control for stabilizing unstable states in chaotic systems.

Main Methods:

• Development of a modified delayed feedback control scheme using an unstable controller.
• Application and testing of the modified scheme on an unstable fixed point of a simple dynamic model.
• Demonstration of the scheme's efficacy on an unstable periodic orbit within the Lorenz system.

Main Results:

• The modified unstable delayed feedback controller successfully overcomes the topological limitations of the standard method.
• The proposed scheme demonstrates efficiency in stabilizing an unstable fixed point.
• The modified controller is also effective in stabilizing an unstable periodic orbit of the Lorenz system.

Conclusions:

• The modified unstable delayed feedback control offers a more robust method for controlling chaos.
• This approach expands the range of unstable states in chaotic systems that can be stabilized.
• The findings have implications for controlling complex dynamical systems in various scientific fields.