Time-Frequency Analysis Based on Wavelets for Hamiltonian Systems

Citation

Vela-Arevalo, Luz Vianey (2002) Time-Frequency Analysis Based on Wavelets for Hamiltonian Systems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/8MBB-3Z60. https://resolver.caltech.edu/CaltechETD:etd-03302004-115559

Abstract

In this work, we present the method of time-frequency analysis based on wavelets for Hamiltonian systems and demonstrate its applications and consequences in the general dynamics of higher dimensional systems. By extracting instantaneous frequencies from the wavelet transform of numerical solutions, we can distinguish regular from chaotic motions, and characterize the global structure of the phase space. The method allows us to determine resonance areas that persists even for high energy levels. We can also show how the existence of resonant motion affects the dynamics of the chaotic motion: we detect when chaotic trajectories are temporarily trapped around resonance areas, or undergo transitions between different resonances. This process is a good indicator of intrinsic transport in the phase space. The method can be applied to a large class of systems, since it is not restricted to nearly integrable systems expressed in action-angle variables, which is the traditional framework for the definition of frequencies. The main contribution of this method is that we have included the time variable in the analysis. We can determine exactly when the trajectories exchange between different regions by keeping records of resonance trappings. This allows us to analyze chaotic trajectories and not only quasiperiodic trajectories. And, we do not require any assumption about the regularity of chaotic trajectories. We present three different applications of the method. The first application consists of the analysis of dynamics and global phase space structure of the classical version of a quantum Hamiltonian for the water molecule. In the second application, we study the planar circular restricted three body problem, and show how resonance transitions of chaotic orbits are related to transport between different regions of the Solar system. Finally, we applied our method to a vibrational three-degrees-of-freedom Hamiltonian of the planar OCS molecule. We study the global dynamics at an energy level close to dissociation, which corresponds to a highly excited state of the molecule.

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