Project Details

Description

9972132

Nerurkar

The proposed research project deals with the study of stability and spectral

behavior of a family of linear differential equations with time-dependent

or random coefficients. Typically the underlying coefficients are

`recurrent' (in particular quasi-periodic) functions of time. This includes

examples of forced oscillations and (finite as well as infinite dimensional)

quantum oscillators with external (time dependent) quasi-periodic and

ergodic forcing fields. Many such examples have been numerically and

theoretically investigated recently. Motivated by these examples, the

project is geared towards finding precise analytical results. The standard

method of proving stability results is via well established techniques like

the KAM technique or the use of `hyperbolicity'. On the contrary, in this

project we shall mainly focus on the unstable behavior. We analyze the

stability properties of the linear system via the dynamical properties of

suitable `skew-product flows' associated with it. For example, instability

of forced oscillations is captured in terms of the absence of point spectrum

of the associated quasi-energy operator and this in turn is reflected in the

ergodic behavior of certain skew-product flows generated by the given

linear system. Consequently our technique is a blend of methods of

topological dynamics, ergodic theory, and ideas from control theory.

Viewing the desired perturbations as `controls with certain constraints'

allow us to apply methods of control theory to create chaotic behavior.

With this viewpoint and with the techniques developed thus far, the

project plans to examine questions of genericity and prevalence of chaotic

or unstable behavior, absence of zero Lyapunov exponents, ergodicity and

proximality for specific families of equations arising in physics and

engineering. Finally the non-linear implications of the theory developed

will be examined by developing results about time-dependent (or non-

stationary) linearization and normal forms.

This project investigates long term time evolution, particularly the

unstable and chaotic behavior of certain classes of dynamical systems.

Motivated by the recent theoretical and numerical study of forced quantum

oscillators with random or quasi-periodic forcing, this project plans to

provide the underlying analytical reasons and precisely predict the long

term behavior of such systems. The systems we consider model several

mechanisms of practical and theoretical interest such as (a) behavior of

electrons and other `spin 1/2 particles' subjected to external random

magnetic fields, (b) phonon conduction and polarization of light in optical

fibers, and (c) propagation through random media such as (quasi)-crystals

with irregularly distributed impurities. Typically most of the existing body

of knowledge is about two extreme cases, namely either when the external

forcing field has no randomness (e.g. is periodic) or is as random as can be

(technically this means the process is stochastically independent). Using

the analytical tools of ergodic theory, topological dynamics and control

theory, the project will investigate finer details relating the chaotic nature

of evolution of the system to the precise data about the nature of

`correlation function' and `rate of mixing' of these random forcing

processes. The project will attempt to find and identify the underlying

features of these systems that determine various types of unstable behavior

and obtain quantitative information about quantities such as the Lyapunov

exponents which are indicators of the degree of chaos. Some by-products

of our investigations will have important implications to engineering

disciplines such as filtering theory and to systems and control engineering.

StatusFinished
Effective start/end date9/1/998/31/02

Funding

  • National Science Foundation: $46,800.00

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