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Type of Document	Dissertation
Author	Vela-Arevalo, Luz Vianey
Author's Email Address	luzvela AT cds.caltech.edu
URN	etd-03302004-115559
Persistent URL	http://resolver.caltech.edu/CaltechETD:etd-03302004-115559
Title	Time-frequency analysis based on wavelets for Hamiltonian systems
Degree	PhD
Option	Control and Dynamical Systems
Advisory Committee	Advisor Name Title Jerrold E. Marsden Committee Chair Emmanuel Candes Committee Member George Haller Committee Member John Doyle Committee Member Richard M. Murray Committee Member
Keywords	numerical methods celestial mechanics Morlet-Grossman wavelet Baggott Hamiltonian Sun-Jupiter system dissociation energy instantaneous frequency resonance channels triatomic molecule 3-dof
Date of Defense	2001-08-14
Availability	unrestricted
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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