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Controlling chaos in higher dimensional maps with constant feedback: an analytical approach
1cwieland@oec.uni-osnabrueck.de
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 21, 2002
Summary
We present two novel methods for controlling chaos in discrete dynamical systems using constant feedback. These techniques convert chaotic attractors into stable fixed points, applicable even without knowing system equations.
Area of Science:
- Dynamical Systems and Chaos Theory
- Nonlinear Dynamics
- Control Theory
Background:
- Chaos theory describes complex, unpredictable behavior in deterministic systems.
- Controlling chaos is crucial for stabilizing systems in engineering and science.
- Higher-dimensional discrete maps present unique challenges for chaos control.
Purpose of the Study:
- To introduce and validate two methods for controlling chaos in higher-dimensional discrete maps.
- To demonstrate the conversion of chaotic attractors into fixed point attractors using constant feedback.
- To provide a method for selecting appropriate constant feedback parameters.
Main Methods:
- Analytical derivation for a general class of function vectors.
- Utilizing time series information, eliminating the need for a priori system equations.
- Application of constant feedback control strategies.
Main Results:
- Chaotic attractors are analytically shown to be convertible into fixed point attractors.
- A systematic method for choosing constant feedback is presented.
- Demonstrated successful application to the Hénon map, a standard benchmark for chaotic systems.
Conclusions:
- The proposed methods offer effective and practical approaches to chaos control in discrete maps.
- The feedback control strategy is robust and does not require complete system knowledge.
- Varying the constant feedback allows access to desired periodic orbits, enhancing system predictability.