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Detecting order and chaos in three-dimensional Hamiltonian systems by geometrical methods
Yossi Ben Zion1, Lawrence Horwitz
1Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 13, 2007
Summary
A new geometrical method effectively distinguishes ordered from chaotic motion in 3D Hamiltonian systems, aligning with Lyapunov exponent calculations. This approach is demonstrated on unstable systems, including one derived from Yang-Mills theory.
Area of Science:
- Physics
- Dynamical Systems
- Chaos Theory
Background:
- Distinguishing between ordered and chaotic dynamics is crucial in Hamiltonian systems.
- Lyapunov characteristic exponents are a standard but computationally intensive method.
Purpose of the Study:
- To introduce and validate a novel geometrical method for classifying motion in 3D Hamiltonian systems.
- To compare the efficacy of the geometrical method against Lyapunov exponents.
Main Methods:
- Development of a geometrical approach to analyze phase space trajectories.
- Application of the method to various unstable 3D Hamiltonian systems.
- Comparative analysis with Lyapunov characteristic exponent computations.
Main Results:
- The geometrical method successfully differentiates ordered and chaotic motion.
- Results from the geometrical method show strong agreement with Lyapunov exponent calculations.
- Detailed analysis of a Yang-Mills derived Hamiltonian system is presented.
Conclusions:
- The proposed geometrical method offers a viable alternative for characterizing dynamics in Hamiltonian systems.
- This method provides an efficient way to identify chaotic behavior.
- The study validates the geometrical approach on complex systems relevant to theoretical physics.
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