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Singular Lyapunov spectra and conservation laws
T. Bohr1, G. Grinstein, C. Jayaprakash
1The Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen O, DenmarkIBM T. J. Watson Research Center, Yorktown Heights, New York 10598Department of Physics, The Ohio State University, Columbus, Ohio 43210.
Chaos (Woodbury, N.Y.)
|June 1, 1995
Summary
Conservation laws can create singularities in Lyapunov exponents for dynamical systems. This discovery aids in identifying previously unknown conservation laws in physics.
Area of Science:
- Physics
- Dynamical Systems Theory
- Nonlinear Dynamics
Background:
- Lyapunov exponents characterize the sensitive dependence on initial conditions in dynamical systems.
- Extended dynamical systems possess spatial degrees of freedom, leading to complex behaviors.
- Conservation laws are fundamental principles that constrain system evolution.
Purpose of the Study:
- To investigate the impact of conservation laws on the spectrum of Lyapunov exponents.
- To demonstrate the emergence of spectral singularities due to conservation laws.
- To explore the application of this phenomenon for detecting hidden conservation laws.
Main Methods:
- Analytical arguments were employed to derive theoretical predictions.
- Numerical simulations provided evidence to support the analytical findings.
- The study focused on extended dynamical systems with low spatial dimensionality.
Main Results:
- The presence of conservation laws was shown to induce singularities in the spectrum of Lyapunov exponents.
- A direct correlation between conservation laws and spectral features was established.
- The findings hold for low-dimensional extended dynamical systems.
Conclusions:
- Conservation laws can fundamentally alter the spectral properties of dynamical systems.
- The observed spectral singularities offer a novel method for discovering hidden conservation laws.
- This research bridges the gap between conservation laws and spectral analysis in dynamical systems theory.