The stochastic exit problem for dynamical systems

The problem of "exit against a flow" for dynamical systems subject to small Gaussian white noise excitation is studied. Here the word "flow" refers to the behavior in phase space of the unperturbed system's state variables. "Exit against a flow" occurs if a perturbation causes the phase point to leave a phase space region within which it would normally be confined. In particular, there are two components of the problem of exit against a flow:

i) the mean exit time

ii) the phase-space distribution of exit locations.

When the noise perturbing the dynamical systems is small, the solution of each component of the problem of exit against a flow is, in general, the solution of a singularly perturbed, degenerate elliptic-parabolic boundary value problem.

Singular perturbation techniques are used to express the asymptotic solution in terms of an unknown parameter. The unknown parameter is determined using the solution of the adjoint boundary value problem.

The problem of exit against a flow for several dynamical systems of physical interest is considered, and the mean exit times and distributions of exit positions are calculated. The systems are then simulated numerically, using Monte Carlo techniques, in order to determine the validity of the asymptotic solutions.

Periodic solutions of integro-differential equations which arise in population dynamics

The problem of the existence and stability of periodic solutions of infinite-lag integra-differential equations is considered. Specifically, the integrals involved are of the convolution type with the dependent variable being integrated over the range (- ∞,t), as occur in models of population growth. It is shown that Hopf bifurcation of periodic solutions from a steady state can occur, when a pair of eigenvalues crosses the imaginary axis. Also considered is the existence of traveling wave solutions of a model population equation allowing spatial diffusion in addition to the usual temporal variation. Lastly, the stability of the periodic solutions resulting from Hopf bifurcation is determined with aid of a Floquet theory.

The first chapter is devoted to linear integro-differential equations with constant coefficients utilizing the method of semi-groups of operators. The second chapter analyzes the Hopf bifurcation providing an existence theorem. Also, the two-timing perturbation procedure is applied to construct the periodic solutions. The third chapter uses two-timing to obtain traveling wave solutions of the diffusive model, as well as providing an existence theorem. The fourth chapter develops a Floquet theory for linear integro-differential equations with periodic coefficients again using the semi-group approach. The fifth chapter gives sufficient conditions for the stability or instability of a periodic solution in terms of the linearization of the equations. These results are then applied to the Hopf bifurcation problem and to a certain population equation modeling periodically fluctuating environments to deduce the stability of the corresponding periodic solutions.

Topics in bifurcation theory

I. Existence and Structure of Bifurcation Branches

The problem of bifurcation is formulated as an operator equation in a Banach space, depending on relevant control parameters, say of the form G(u,λ) = 0. If dimN(G_u(u_O,λ_O)) = m the method of Lyapunov-Schmidt reduces the problem to the solution of m algebraic equations. The possible structure of these equations and the various types of solution behaviour are discussed. The equations are normally derived under the assumption that G^O_λεR(G^O_u). It is shown, however, that if G^O_λεR(G^O_u) then bifurcation still may occur and the local structure of such branches is determined. A new and compact proof of the existence of multiple bifurcation is derived. The linearized stability near simple bifurcation and "normal" limit points is then indicated.

II. Constructive Techniques for the Generation of Solution Branches

A method is described in which the dependence of the solution arc on a naturally occurring parameter is replaced by the dependence on a form of pseudo-arclength. This results in continuation procedures through regular and "normal" limit points. In the neighborhood of bifurcation points, however, the associated linear operator is nearly singular causing difficulty in the convergence of continuation methods. A study of the approach to singularity of this operator yields convergence proofs for an iterative method for determining the solution arc in the neighborhood of a simple bifurcation point. As a result of these considerations, a new constructive proof of bifurcation is determined.

Compressible flows at small Reynolds numbers

The problem of the slow viscous flow of a gas past a sphere is considered. The fluid cannot be treated incompressible in the limit when the Reynolds number Re, and the Mach number M, tend to zero in such a way that Re ~ o(M^2 ). In this case, the lowest order approximation to the steady Navier-Stokes equations of motion leads to a paradox discovered by Lagerstrom and Chester. This paradox is resolved within the framework of continuum mechanics using the classical slip condition and an iteration scheme that takes into account certain terms in the full Navier-Stokes equations that drop out in the approximation used by the above authors. It is found however that the drag predicted by the theory does not agree with R. A. Millikan's classic experiments on sphere drag.

The whole question of the applicability of the Navier-Stokes theory when the Knudsen number M/Re is not small is examined. A new slip condition is proposed. The idea that the Navier-Stokes equations coupled with this condition may adequately describe small Reynolds number flows when the Knudsen number is not too large is looked at in some detail. First, a general discussion of asymptotic solutions of the equations for all such flows is given. The theory is then applied to several concrete problems of fluid motion. The deductions from this theory appear to interpret and summarize the results of Millikan over a much wider range of Knudsen numbers (almost up to the free molecular or kinetic limit) than hitherto Believed possible by a purely continuum theory. Further experimental tests are suggested and certain interesting applications to the theory of dilute suspensions in gases are noted. Some of the questions raised in the main body of the work are explored further in the appendices.

Some approximate solutions of dynamic problems in the linear theory of thin elastic shells

Some aspects of wave propagation in thin elastic shells are considered. The governing equations are derived by a method which makes their relationship to the exact equations of linear elasticity quite clear. Finite wave propagation speeds are ensured by the inclusion of the appropriate physical effects.

The problem of a constant pressure front moving with constant velocity along a semi-infinite circular cylindrical shell is studied. The behavior of the solution immediately under the leading wave is found, as well as the short time solution behind the characteristic wavefronts. The main long time disturbance is found to travel with the velocity of very long longitudinal waves in a bar and an expression for this part of the solution is given.

When a constant moment is applied to the lip of an open spherical shell, there is an interesting effect due to the focusing of the waves. This phenomenon is studied and an expression is derived for the wavefront behavior for the first passage of the leading wave and its first reflection.

For the two problems mentioned, the method used involves reducing the governing partial differential equations to ordinary differential equations by means of a Laplace transform in time. The information sought is then extracted by doing the appropriate asymptotic expansion with the Laplace variable as parameter.

A mathematical study of finite-amplitude rock-folding

The problem of the finite-amplitude folding of an isolated, linearly viscous layer under compression and imbedded in a medium of lower viscosity is treated theoretically by using a variational method to derive finite difference equations which are solved on a digital computer. The problem depends on a single physical parameter, the ratio of the fold wavelength, L, to the "dominant wavelength" of the infinitesimal-amplitude treatment, L_d. Therefore, the natural range of physical parameters is covered by the computation of three folds, with L/L_d = 0, 1, and 4.6, up to a maximum dip of 90°.

Significant differences in fold shape are found among the three folds; folds with higher L/L_d have sharper crests. Folds with L/L_d = 0 and L/L_d = 1 become fan folds at high amplitude. A description of the shape in terms of a harmonic analysis of inclination as a function of arc length shows this systematic variation with L/L_d and is relatively insensitive to the initial shape of the layer. This method of shape description is proposed as a convenient way of measuring the shape of natural folds.

The infinitesimal-amplitude treatment does not predict fold-shape development satisfactorily beyond a limb-dip of 5°. A proposed extension of the treatment continues the wavelength-selection mechanism of the infinitesimal treatment up to a limb-dip of 15°; after this stage the wavelength-selection mechanism no longer operates and fold shape is mainly determined by L/L_d and limb-dip.

Strain-rates and finite strains in the medium are calculated f or all stages of the L/L_d = 1 and L/L_d = 4.6 folds. At limb-dips greater than 45° the planes of maximum flattening and maximum flattening rat e show the characteristic orientation and fanning of axial-plane cleavage.

Application of microwave diagnostics to copper chloride and carbon dioxide lasers

Part A

A problem restricting the development of the CuCl laser has been the decrease in output power with increases of tube temperature above 400°C. At that temperature the CuCl vapor pressure is about .1 torr. This is a small fraction of the buffer gas pressure (He at 10 torr).

The aim of the project was to measure the peak radiation temperature (assumed related to the mean energy of electrons) in the laser discharge as a function of the tube temperature. A 24 gHz gated microwave radiometer was used.

It was found that at the tube temperatures at which the output power began to deteriorate, the electron radiation temperature showed a sharp increase (compared with radiation temperature in pure buffer).

Using the above result, we have postulated that this sudden increase is a result of Penning ionization of the Cu atoms. As a consequence of this process the number of Cu atoms available for lasing decrease.

PART B

The aim of the project was to study the dissociation of CO₂ in the glow discharge of flowing CO₂ lasers.

A TM₀₁₁ microwave (3 gHz) cavity was used to measure the radially averaged electron density n_e and the electron-neutral collision frequency in the laser discharge. An estimate of the electric field is made from these two measurements. A gas chromatograph was used to measure the chemical composition of the gases after going through the discharge. This instrument was checked against a mass spectrometer for accuracy and sensitivity.

Several typical laser mixtures were .used: CO₂-N₂-He (1,3,16), (1,3,0), (1,0,16), (1,2,10), (1,2,0), (1,0,10), (2,3,15), (2,3,0), (2,0,15), (1,3,16)+ H₂O and pure CO₂. Results show that for the conditions studied the dissociation as a function of the electron density is uniquely determined by the STP partial flow rate of CO₂, regardless of the amount of N₂ and/or He present. The presence of water vapor in the discharge decreased the degree of dissociation.

A simple theoretical model was developed using thermodynamic equilibrium. The electrons were replaced in the calculations by a distributed heat source.

The results are analyzed with a simple kinetic model.

Theory of s-d exchange scattering in dilute magnetic alloys

The problem of s-d exchange scattering of conduction electrons off localized magnetic moments in dilute magnetic alloys is considered employing formal methods of quantum field theoretical scattering. It is shown that such a treatment not only allows for the first time, the inclusion of multiparticle intermediate states in single particle scattering equations but also results in extremely simple and straight forward mathematical analysis. These equations are proved to be exact in the thermodynamic limit. A self-consistent integral equation for electron self energy is derived and approximately solved. The ground state and physical parameters of dilute magnetic alloys are discussed in terms of the theoretical results. Within the approximation of single particle intermediate states our results reduce to earlier versions. The following additional features are found as a consequence of the inclusion of multiparticle intermediate states;

(i) A non analytic binding energy is pre sent for both, antiferromagnetic (J < o) and ferromagnetic (J > o) couplings of the electron plus impurity system.

(ii) The correct behavior of the energy difference of the conduction electron plus impurity system and the free electron system is found which is free of unphysical singularities present in earlier versions of the theories.

(iii) The ground state of the conduction electron plus impurity system is shown to be a many-body condensate state for J < o and J > o, both. However, a distinction is made between the usual terminology of "Singlet" and "Triplet" ground states and nature of our ground state.

(iv) It is shown that a long range ordering, leading to an ordering of the magnetic moments can result from a contact interaction such as the s-d exchange interaction.

(v) The explicit dependence of the excess specific heat of the Kondo systems is obtained and found to be linear in temperatures as T→ o and T ℓnT for 0.3 T_K ≤ T ≤ 0.6 T_K. A rise in (ΔC/T) for temperatures in the region 0 < T ≤ 0.1 T_K is predicted. These results are found to be in excellent agreement with experiments.

(vi) The existence of a critical temperature for Ferromagnetic coupling (J > o) is shown. On the basis of this the apparent contradiction of the simultaneous existence of giant moments and Kondo effect is resolved.

Radar methods for the exploration of glaciers

The problem of finding the depths of glaciers and the current methods are discussed briefly. Radar methods are suggested as a possible improvement for, or adjunct to, seismic and gravity survey methods. The feasibility of propagating electromagnetic waves in ice and the maximum range to be expected are then investigated theoretically with the aid of experimental data on the dielectric properties of ice. It is found that the maximum expected range is great enough to measure the depth of many glaciers at the lower radar frequencies if there is not too much liquid water present. Greater ranges can be attained by going to lower frequencies.

The results are given of two expeditions in two different years to the Seward Glacier in the Yukon Territory. Experiments were conducted on a small valley glacier whose depth was determined by seismic sounding. Many echoes were received but their identification was uncertain. Using the best echoes, a profile was obtained each year, but they were not in exact agreement with each other. It could not be definitely established that echoes had been received from bedrock. Agreement with seismic methods for a considerable number of glaciers would have to be obtained before radar methods could be relied upon. The presence of liquid water in the ice is believed to be one of the greatest obstacles. Besides increasing the attenuation and possibly reflecting energy, it makes it impossible to predict the velocity of propagation. The equipment used was far from adequate for such purposes, so many of the difficulties could be attributed to this. Partly because of this, and the fact that there are glaciers with very little liquid water present, radar methods are believed to be worthy of further research for the exploration of glaciers.

Theory of viscous and thermal attenuation of sound by small spheres

The problem is to calculate the attenuation of plane sound waves passing through a viscous, heat-conducting fluid containing small spherical inhomogeneities. The attenuation is calculated by evaluating the rate of increase of entropy caused by two irreversible processes: (1) the mechanical work done by the viscous stresses in the presence of velocity gradients, and (2) the flow of heat down the thermal gradients. The method is first applied to a homogeneous fluid with no spheres and shown to give the classical Stokes-Kirchhoff expressions. The method is then used to calculate the additional viscous and thermal attenuation when small spheres are present. The viscous attenuation agrees with Epstein's result obtained in 1941 for a non-heat-conducting fluid. The thermal attenuation is found to be similar in form to the viscous attenuation and, for gases, of comparable magnitude. The general results are applied to the case of water drops in air and air bubbles in water.

For water drops in air the viscous and thermal attenuations are camparable; the thermal losses occur almost entirely in the air, the thermal dissipation in the water being negligible. The theoretical values are compared with Knudsen's experimental data for fogs and found to agree in order of magnitude and dependence on frequency. For air bubbles in water the viscous losses are negligible and the calculated attenuation is almost completely due to thermal losses occurring in the air inside the bubbles, the thermal dissipation in the water being relatively small. (These results apply only to non-resonant bubbles whose radius changes but slightly during the acoustic cycle.)

Generalizations and extensions of the Fokker-Planck-Kolmogorov equations

The problem of determining probability density functions of general transformations of random processes is considered in this thesis. A method of solution is developed in which partial differential equations satisfied by the unknown density function are derived. These partial differential equations are interpreted as generalized forms of the classical Fokker-Planck-Kolmogorov equations and are shown to imply the classical equations for certain classes of Markov processes. Extensions of the generalized equations which overcome degeneracy occurring in the steady-state case are also obtained.

The equations of Darling and Siegert are derived as special cases of the generalized equations thereby providing unity to two previously existing theories. A technique for treating non-Markov processes by studying closely related Markov processes is proposed and is seen to yield the Darling and Siegert equations directly from the classical Fokker-Planck-Kolmogorov equations.

As illustrations of their applicability, the generalized Fokker-Planck-Kolmogorov equations are presented for certain joint probability density functions associated with the linear filter. These equations are solved for the density of the output of an arbitrary linear filter excited by Markov Gaussian noise and for the density of the output of an RC filter excited by the Poisson square wave. This latter density is also found by using the extensions of the generalized equations mentioned above. Finally, some new approaches for finding the output probability density function of an RC filter-limiter-RC filter system driven by white Gaussian noise are included. The results in this case exhibit the data required for complete solution and clearly illustrate some of the mathematical difficulties inherent to the use of the generalized equations.

Wave uplift pressures on horizontal platforms

The major objective of the study has been to investigate in detail the rapidly-varying peak uplift pressure and the slowly-varying positive and negative uplift pressures that are known to be exerted by waves against the underside of a horizontal pier or platform located above the still water level, but not higher than the crests of the incident waves.

In a "two-dimensional" laboratory study conducted in a 100-ft long by 15-in.-wide by 2-ft-deep wave tank with a horizontal smooth bottom, individually generated solitary waves struck a rigid, fixed, horizontal platform extending the width of the tank. Pressure transducers were mounted flush with the smooth soffit, or underside, of the platform. The location of the transducers could be varied.

The problem of a d equate dynamic and spatial response of the transducers was investigated in detail. It was found that unless the radius of the sensitive area of a pressure transducer is smaller than about one-third of the characteristic width of the pressure distribution, the peak pressure and the rise-time will not be recorded accurately. A procedure was devised to correct peak pressures and rise-times for this transducer defect.

The hydrodynamics of the flow beneath the platform are described qualitatively by a si1nple analysis, which relates peak pressure and positive slowly-varying pressure to the celerity of the wave front propagating beneath the platform, and relates negative slowly-varying pressure to the process by which fluid recedes from the platform after the wave has passed. As the wave front propagates beneath the platform, its celerity increases to a maximum, then decreases. The peak pressure similarly increases with distance from the seaward edge of the platform, then decreases.

Measured peak pressure head, always found to be less than five times the incident wave height above still water level, is an order of magnitude less than reported shock pressures due to waves breaking against vertical walls; the product of peak pressure and rise-time, considered as peak impulse, is of the order of 20% of reported shock impulse due to waves breaking against vertical walls. The maximum measured slowly-varying uplift pressure head is approximately equal to the incident wave height less the soffit clearance above still water level. The normalized magnitude and duration of negative pressure appears to depend principally on the ratio of soffit clearance to still water depth and on the ratio of platform length to still water depth.

The complex angular momentum theory of the production of three particles in collisions of two strongly interacting particles at high energy

The problem of the continuation to complex values of the angular momentum of the partial wave amplitude is examined for the simplest production process, that of two particles → three particles. The presence of so-called "anomalous singularities" complicates the procedure followed relative to that used for quasi two-body scattering amplitudes. The anomalous singularities are shown to lead to exchange degenerate amplitudes with possible poles in much the same way as "normal" singularities lead to the usual signatured amplitudes. The resulting exchange-degenerate trajectories would also be expected to occur in two-body amplitudes.

The representation of the production amplitude in terms of the singularities of the partial wave amplitude is then developed and applied to the high energy region, with attention being paid to the emergence of "double Regge" terms. Certain new results are obtained for the behavior of the amplitude at zero momentum transfer, and some predictions of polarization and minima in momentum transfer distributions are made. A calculation of the polarization of the ρ^o meson in the reaction π ^- p → π ^- ρ^op at high energy with small momentum transfer to the proton is compared with data taken at 25 Gev by W. D. Walker and collaborators. The result is favorable, although limited by the statistics of the available data.

The conjugate representation of signal envelope

The problem of the representation of signal envelope is treated, motivated by the classical Hilbert representation in which the envelope is represented in terms of the received signal and its Hilbert transform. It is shown that the Hilbert representation is the proper one if the received signal is strictly bandlimited but that some other filter is more appropriate in the bandunlimited case. A specific alternative filter, the conjugate filter, is proposed and the overall envelope estimation error is evaluated to show that for a specific received signal power spectral density the proposed filter yields a lower envelope error than the Hilbert filter.

An investigation of the stresses and deflections of swept plates

The problem in this investigation was to determine the stress and deflection patterns of a thick cantilever plate at various angles of sweepback.

The plate was tested at angles of sweepback of zero, twenty, forty, and sixty degrees under uniform shear load at the tip, uniformly distributed load and torsional loading.

For all angles of sweep and for all types of loading the area of critical stress is near the intersection of the root and trailing edge. Stresses near the leading edge at the root decreased rapidly with increase in angle of sweep for all types of loading. In the outer portion of the plate near the trailing edge the stresses due to the uniform shear and the uniformly distributed load did not vary for angles of sweep up to forty degrees. For the uniform shear and the uniformly distributed loads for all angles of sweep the area in which end effect is pronounced extends from the root to approximately three quarters of a chord length outboard of a line perpendicular to the axis of the plate through the trailing edge root. In case of uniform shear and uniformly distributed loads the deflections near the edge at seventy-five per cent semi-span decreased with increase in angle of sweep. Deflections near the trailing edge under the same loading conditions increased with increase in angle of sweep for small angles and then decreased at the higher angles of sweep. The maximum deflection due to torsional loading increased with increase in angle of sweep.