Rubel, Michael Thomas (2007) A theory of stationarity and asymptotic approach in dissipative systems. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd01122007114557
Abstract
The approximate dynamics of many physical phenomena, including turbulence, can be represented by dissipative systems of ordinary differential equations. One often turns to numerical integration to solve them. There is an incompatibility, however, between the answers it can produce (i.e., specific solution trajectories) and the questions one might wish to ask (e.g., what behavior would be typical in the laboratory?) To determine its outcome, numerical integration requires more detailed initial conditions than a laboratory could normally provide. In place of initial conditions, experiments stipulate how tests should be carried out: only under statistically stationary conditions, for example, or only during asymptotic approach to a final state. Stipulations such as these, rather than initial conditions, are what determine outcomes in the laboratory.
This theoretical study examines whether the points of view can be reconciled: What is the relationship between one's statistical stipulations for how an experiment should be carried outstationarity or asymptotic approachand the expected results? How might those results be determined without invoking initial conditions explicitly?
To answer these questions, stationarity and asymptotic approach conditions are analyzed in detail. Each condition is treated as a statistical constraint on the systema restriction on the probability density of states that might be occupied when measurements take place. For stationarity, this reasoning leads to a singular, invariant probability density which is already familiar from dynamical systems theory. For asymptotic approach, it leads to a new, more regular probability density field. A conjecture regarding what appears to be a limit relationship between the two densities is presented.
By making use of the new probability densities, one can derive output statistics directly, avoiding the need to create or manipulate initial data, and thereby avoiding the conceptual incompatibility mentioned above. This approach also provides a clean way to derive reducedorder models, complete with local and global error estimates, as well as a way to compare existing reducedorder models objectively.
The new approach is explored in the context of five separate test problems: a trivial onedimensional linear system, a damped unforced linear oscillator in two dimensions, the isothermal RayleighPlesset equation, Lorenz's equations, and the Stokes limit of Burgers' equation in one space dimension. In each case, various output statistics are deduced without recourse to initial conditions. Further, reducedorder models are constructed for asymptotic approach of the damped unforced linear oscillator, the isothermal RayleighPlesset system, and Lorenz's equations, and for stationarity of Lorenz's equations.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  dissipative; dynamical; initial condition; statistical; turbulence 
Degree Grantor:  California Institute of Technology 
Major Option:  Aeronautics 
Thesis Committee: 

Defense Date:  6 October 2006 
Author Email:  mrubel (AT) galcit.caltech.edu 
Record Number:  etd01122007114557 
Persistent URL:  http://resolver.caltech.edu/CaltechETD:etd01122007114557 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  141 
Collection:  CaltechTHESES 
Deposited By:  Imported from ETDdb 
Deposited On:  12 Feb 2007 
Last Modified:  25 Dec 2012 14:57 
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