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Global computation of phase-amplitude reduction for limit-cycle dynamics
1Department of Mathematics and Namur Institute for Complex Systems, University of Namur, B-5000 Namur, Belgium.
Chaos (Woodbury, N.Y.)
|August 3, 2018
Summary
This study introduces a novel method for computing global phase-amplitude coordinates, essential for understanding limit-cycle dynamics beyond classic phase reduction. The technique utilizes the Koopman operator to analyze system behavior, offering insights into responses to strong external inputs.
Area of Science:
- Dynamical Systems and Control Theory
- Nonlinear Dynamics
- Computational Mathematics
Background:
- Phase reduction is limited by strong convergence and weak inputs.
- Computing amplitude coordinates globally is a major challenge.
- Existing methods often focus on local quantities.
Purpose of the Study:
- To develop a method for computing global phase-amplitude coordinates.
- To overcome limitations of classic phase reduction.
- To analyze dynamics transversal to limit cycles.
Main Methods:
- Koopman (composition) operator theory.
- Laplace averages combined with harmonic balance.
- Forward integration applicable to N-dimensional systems.
Main Results:
- A method for computing the full set of phase-amplitude coordinates globally.
- Computation of isostables for limit cycles in 2D, 3D, and 4D state spaces.
- Demonstration of system responses to strong external inputs.
Conclusions:
- The proposed method effectively computes global phase-amplitude coordinates.
- This approach extends phase-amplitude reduction beyond local analyses.
- The method provides a powerful tool for analyzing complex dynamical systems.
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