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Control of chaos by capture and release.
1Department of Mathematics, University of Colorado at Denver, Denver, CO 80211, USA.
Summary
We introduce a novel capture and release (CR) method for controlling flip saddles in chaotic systems. This technique utilizes the rotation of Möbius bands to guide system states, unlike prior methods relying on unstable dynamics.
Area of Science:
- Nonlinear Dynamics
- Chaos Theory
- Control Theory
Background:
- Flip saddles are key unstable periodic orbits in dissipative chaotic systems.
- Existing control methods, like Ott, Grebogi, and Yorke (OGY), often rely on the unstable manifold.
- Controlling chaotic systems with flip saddles presents unique challenges due to their specific dynamics.
Purpose of the Study:
- To propose and demonstrate a novel control method for flip saddles in dissipative chaotic systems.
- To offer an alternative to existing methods that utilize unstable system components.
- To leverage the geometric properties of flip saddle dynamics for state control.
Main Methods:
- Modeling the stable manifolds of a flip saddle's target orbit and its perturbation as rotating Möbius bands.
- Developing a capture and release (CR) method that exploits the relative rotation of these Möbius bands.
- Capturing a system state in the perturbed stable manifold and releasing it upon alignment of the bands.
Main Results:
- Demonstration of the CR method's efficacy in controlling flip saddles.
- The CR method successfully captures and guides system states using only stable manifold dynamics.
- The method operates within a stable subspace, avoiding reliance on the unstable flow component.
Conclusions:
- The capture and release (CR) method provides a new paradigm for controlling flip saddles.
- CR offers a robust alternative to existing methods by operating within the stable subspace.
- This approach enhances the understanding and manipulation of chaotic system dynamics.