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OBSERVING LYAPUNOV EXPONENTS OF INFINITE-DIMENSIONAL DYNAMICAL SYSTEMS
William Ott1, Mauricio A Rivas2, James West3
1Department of Mathematics, University of Houston, URL : http://www.math.uh.edu/~ott/.
Journal of Statistical Physics
|January 10, 2017
Summary
Lyapunov exponents of infinite-dimensional systems can be observed in projected dynamics. Embedding theorems confirm that typical nonlinear projections of compact invariant sets yield accurate Lyapunov exponent calculations from experimental data.
Area of Science:
- Dynamical Systems Theory
- Chaos Theory
- Nonlinear Dynamics
Background:
- Infinite-dimensional dynamical systems are crucial for modeling complex phenomena.
- Observing Lyapunov exponents directly in these systems is challenging.
- Projecting dynamics into finite dimensions is a common experimental approach.
Purpose of the Study:
- To determine if Lyapunov exponents of infinite-dimensional systems can be observed via nonlinear projections.
- To develop theoretical foundations for empirical observation of these exponents.
- To establish conditions for validating experimental Lyapunov exponent measurements.
Main Methods:
- Development of embedding theorems for C^1 maps on Hilbert spaces.
- Analysis of compact invariant sets in infinite-dimensional systems.
- Formulation of hypotheses on projected dynamics rather than the full system.
Main Results:
- Affirmative answer to the observability of Lyapunov exponents through projection.
- Demonstration of embedding theorems for relevant dynamical systems.
- Identification of checkable conditions for empirical validation of Lyapunov exponents.
Conclusions:
- Lyapunov exponents of infinite-dimensional systems are observable through typical nonlinear projections.
- The developed embedding theorems provide a theoretical basis for experimental validation.
- An empirical approach with checkable conditions is proposed for practical applications.
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