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Spectral element method and the delayed feedback control of chaos
Dennis J Tweten1, Brian P Mann
1Mechanical Engineering and Materials Science Department, Duke University, Durham, North Carolina 27708, USA. dennis.tweten@duke.edu
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 11, 2012
Summary
A new spectral element method efficiently calculates Floquet exponents for unstable periodic orbits stabilized by extended delayed feedback control. This approach simplifies stability analysis for complex dynamical systems.
Area of Science:
- Dynamical Systems and Chaos Theory
- Nonlinear Dynamics
- Control Theory
Background:
- Unstable periodic orbits (UPOs) are crucial for understanding chaotic dynamics.
- Extended delayed feedback control (EDFC) is used to stabilize UPOs.
- Calculating Floquet exponents (FEs) is essential for determining UPO stability.
Purpose of the Study:
- To introduce a novel spectral element approach for determining Floquet exponents (FEs).
- To analyze the stability of unstable periodic orbits (UPOs) stabilized by extended delayed feedback control (EDFC).
- To provide a method that avoids solving time-dependent eigenproblems.
Main Methods:
- A spectral element approach is employed for numerical approximation of system stability.
- The method analyzes delay differential equations without requiring time-dependent eigenproblems.
- Applicable to UPOs arising from various bifurcations, including non-period-doubling.
Main Results:
- The spectral method accurately calculates FEs for UPOs in Duffing systems, showing good agreement with published results.
- Successfully analyzed a high-dimensional, asymmetrical system with a UPO arising from tori doubling after a Hopf bifurcation.
- Demonstrated the spectral approach's capability for complex dynamical systems.
Conclusions:
- The spectral element approach offers an efficient and accurate alternative for FE calculation.
- This method simplifies the stability analysis of EDFC-stabilized UPOs in various systems.
- The approach is robust and applicable to complex, high-dimensional, and asymmetrical dynamical systems.
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