Time-optimal chaos control by center manifold targeting.

Area of Science:

• Nonlinear dynamics
• Control theory
• Chaos theory

Background:

• Ott-Grebogi-Yorke (OGY) control targets the stable subspace of a dynamical system's orbit.
• Existing map-based control variants also focus on stable subspaces.
• This limits the speed and directness of achieving the target orbit.

Purpose of the Study:

• To propose an n-step control variation for dynamical systems.
• To enable direct targeting of the orbit, bypassing the stable subspace.
• To achieve time-optimal control in n steps, where n is the system dimension.

Main Methods:

• An n-step control procedure is developed.
• The method targets the controllable region of the system.
• Demonstrated using piecewise linear and nonlinear 2D maps.

Main Results:

• The proposed method sends any initial condition directly to the target orbit.
• Control is achieved in exactly n iterations, where n is the system dimension.
• The technique is shown to be time-optimal, accounting for potential errors.

Conclusions:

• The n-step control method offers a direct and time-optimal approach to targeting orbits in dynamical systems.
• The technique is extendable to higher-dimensional maps and flows.
• This provides a significant advancement over methods relying on stable subspaces.