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.

A perturbation procedure for nonlinear oscillations (The dynamics of two oscillators with weak nonlinear coupling)

This thesis considers in detail the dynamics of two oscillators with weak nonlinear coupling. There are three classes of such problems: non-resonant, where the Poincaré procedure is valid to the order considered; weakly resonant, where the Poincaré procedure breaks down because small divisors appear (but do not affect the O(1) term) and strongly resonant, where small divisors appear and lead to O(1) corrections. A perturbation method based on Cole's two-timing procedure is introduced. It avoids the small divisor problem in a straightforward manner, gives accurate answers which are valid for long times, and appears capable of handling all three types of problems with no change in the basic approach.

One example of each type is studied with the aid of this procedure: for the nonresonant case the answer is equivalent to the Poincaré result; for the weakly resonant case the analytic form of the answer is found to depend (smoothly) on the difference between the initial energies of the two oscillators; for the strongly resonant case we find that the amplitudes of the two oscillators vary slowly with time as elliptic functions of ϵ t, where ϵ is the (small) coupling parameter.

Our results suggest that, as one might expect, the dynamical behavior of such systems varies smoothly with changes in the ratio of the fundamental frequencies of the two oscillators. Thus the pathological behavior of Whittaker's adelphic integrals as the frequency ratio is varied appears to be due to the fact that Whittaker ignored the small divisor problem. The energy sharing properties of these systems appear to depend strongly on the initial conditions, so that the systems not ergodic.

The perturbation procedure appears to be applicable to a wide variety of other problems in addition to those considered here.

Photodissociation and reaction dynamics of chlorine-containing species important in stratospheric ozone chemistry

Chlorine oxide species have received considerable attention in recent years due to their central role in the balance of stratospheric ozone. Many questions pertaining to the behavior of such species still remain unanswered and plague the ability of researchers to develop accurate chemical models of the stratosphere. Presented in this thesis are three experiments that study various properties of some specific chlorine oxide species.

In the first chapter, the reaction between ClONO_2 and protonated water clusters is investigated to elucidate a possible reaction mechanism for the heterogeneous reaction of chlorine nitrate on ice. The ionic products were various forms of protonated nitric acid, NO_2 +(H_20)_m, m = 0, 1, 2. These products are analogous to products previously reported in the literature for the neutral reaction occurring on ice surfaces. Our results support the hypothesis that the heterogeneous reaction is acid-catalyzed.

In the second chapter, the photochemistry of ClONO_2 was investigated at two wavelengths, 193 and 248 nm, using the technique of photofragmentation translational spectroscopy. At both wavelengths, the predominant dissociation pathways were Cl + NO_3 and ClO + NO_2. Channel assignments were confirmed by momentum matching the counterfragments from each channel. A one-dimensional stratospheric model using the new 248 nm branching ratio determined how our results would affect the predicted Cl_x and NO_x partitioning in the stratosphere.

Chapter three explores the photodissociation dynamics of Cl_2O at 193, 248 and 308 nm. At 193 nm, we found evidence for the concerted reaction channel, Cl_2 + O. The ClO + Cl channel was also accessed, however, the majority of the ClO fragments were formed with sufficient internal energies for spontaneous secondary dissociation to occur. At 248 and 308 nm, we only observed only the ClO + Cl channel. . Some of the ClO formed at 248 nm was formed internally hot and spontaneously dissociated. Bimodal translational energy distributions of the ClO and Cl products indicate two pathways leading to the same product exist.

Appendix A, B and C discuss the details of data analysis techniques used in Chapters 1 and 2. The development of a molecular beam source of ClO dimer is presented in Appendix D.

Dynamics and scaling of self-excited passive vortex generators for underwater propulsion

A series of experiments was conducted on the use of a device to passively generate vortex rings, henceforth a passive vortex generator (PVG). The device is intended as a means of propulsion for underwater vehicles, as the use of vortex rings has been shown to decrease the fuel consumption of a vehicle by up to 40% Ruiz (2010).

The PVG was constructed out of a collapsible tube encased in a rigid, airtight box. By adjusting the pressure within the airtight box while fluid was flowing through the tube, it was possible to create a pulsed jet with vortex rings via self-excited oscillations of the collapsible tube.

A study of PVG integration into an existing autonomous underwater vehicle (AUV) system was conducted. A small AUV was used to retrofit a PVG with limited alterations to the original vehicle. The PVG-integrated AUV was used for self-propelled testing to measure the hydrodynamic (Froude) efficiency of the system. The results show that the PVG-integrated AUV had a 22% increase in the Froude efficiency using a pulsed jet over a steady jet. The maximum increase in the Froude efficiency was realized when the formation time of the pulsed jet, a nondimensional time to characterize vortex ring formation, was coincident with vortex ring pinch-off. This is consistent with previous studies that indicate that the maximization of efficiency for a pulsed jet vehicle is realized when the formation of vortex rings maximizes the vortex ring energy and size.

The other study was a parameter study of the physical dimensions of a PVG. This study was conducted to determine the effect of the tube diameter and length on the oscillation characteristics such as the frequency. By changing the tube diameter and length by factors of 3, the frequency of self-excited oscillations was found to scale as f~D_0^{-1/2} L_0^0, where D_0 is the tube diameter and L_0 the tube length. The mechanism of operation is suggested to rely on traveling waves between the tube throat and the end of the tube. A model based on this mechanism yields oscillation frequencies that are within the range observed by the experiment.

Capturing protein dynamics with time-resolved luminescence spectroscopy

The presented doctoral research utilizes time-resolved spectroscopy to characterize protein dynamics and folding mechanisms. We resolve millisecond-timescale folding by coupling time-resolved fluorescence energy transfer (trFRET) to a continuous flow microfluidic mixer to obtain intramolecular distance distributions throughout the folding process. We have elucidated the folding mechanisms of two cytochromes---one that exhibits two-state folding (cytochrome cb₅₆₂) and one that has both a kinetic refolding intermediate ensemble and a distinct equilibrium unfolding intermediate (cytochrome c₅₅₂). Our data reveal that the distinct structural features of cytochrome c₅₅₂ contribute to its thermostability.

We have also investigated intrachain contact dynamics in unfolded cytochrome cb₅₆₂ by monitoring electron transfer, which occurs as the heme collides with a ruthenium photosensitizer, covalently bound to residues along the polypeptide. Intrachain diffusion for chemically denatured proteins proceeds on the microsecond timescale with an upper limit of 0.1 microseconds. The power-law dependence (slope = -1.5) of the rate constants on the number of peptide bonds between the heme and Ru complex indicate that cytochrome cb₅₆₂ is minimally frustrated.

In addition, we have explored the pathway dependence of electron tunneling rates between metal sites in proteins. Our research group has converted cytochrome b₅₆₂ to a c-type cytochrome with the porphyrin covalently bound to cysteine sidechains. We have investigated the effects of the changes to the protein structure (i.e., increased rigidity and potential new equatorial tunneling pathways) on the electron transfer rates, measured by transient absorption, in a series of ruthenium photosensitizer-modified proteins.

Dynamics of cell–matrix mechanical interactions in three dimensions

The forces cells apply to their surroundings control biological processes such as growth, adhesion, development, and migration. In the past 20 years, a number of experimental techniques have been developed to measure such cell tractions. These approaches have primarily measured the tractions applied by cells to synthetic two-dimensional substrates, which do not mimic in vivo conditions for most cell types. Many cell types live in a fibrous three-dimensional (3D) matrix environment. While studying cell behavior in such 3D matrices will provide valuable insights for the mechanobiology and tissue engineering communities, no experimental approaches have yet measured cell tractions in a fibrous 3D matrix.

This thesis describes the development and application of an experimental technique for quantifying cellular forces in a natural 3D matrix. Cells and their surrounding matrix are imaged in three dimensions with high speed confocal microscopy. The cell-induced matrix displacements are computed from the 3D image volumes using digital volume correlation. The strain tensor in the 3D matrix is computed by differentiating the displacements, and the stress tensor is computed by applying a constitutive law. Finally, tractions applied by the cells to the matrix are computed directly from the stress tensor.

The 3D traction measurement approach is used to investigate how cells mechanically interact with the matrix in biologically relevant processes such as division and invasion. During division, a single mother cell undergoes a drastic morphological change to split into two daughter cells. In a 3D matrix, dividing cells apply tensile force to the matrix through thin, persistent extensions that in turn direct the orientation and location of the daughter cells. Cell invasion into a 3D matrix is the first step required for cell migration in three dimensions. During invasion, cells initially apply minimal tractions to the matrix as they extend thin protrusions into the matrix fiber network. The invading cells anchor themselves to the matrix using these protrusions, and subsequently pull on the matrix to propel themselves forward.

Lastly, this thesis describes a constitutive model for the 3D fibrous matrix that uses a finite element (FE) approach. The FE model simulates the fibrous microstructure of the matrix and matches the cell-induced matrix displacements observed experimentally using digital volume correlation. The model is applied to predict how cells mechanically sense one another in a 3D matrix. It is found that cell-induced matrix displacements localize along linear paths. These linear paths propagate over a long range through the fibrous matrix, and provide a mechanism for cell-cell signaling and mechanosensing. The FE model developed here has the potential to reveal the effects of matrix density, inhomogeneity, and anisotropy in signaling cell behavior through mechanotransduction.

Role of feedback and dynamics in a gene regulatory network

Cells exhibit a diverse repertoire of dynamic behaviors. These dynamic functions are implemented by circuits of interacting biomolecules. Although these regulatory networks function deterministically by executing specific programs in response to extracellular signals, molecular interactions are inherently governed by stochastic fluctuations. This molecular noise can manifest as cell-to-cell phenotypic heterogeneity in a well-mixed environment. Single-cell variability may seem like a design flaw but the coexistence of diverse phenotypes in an isogenic population of cells can also serve a biological function by increasing the probability of survival of individual cells upon an abrupt change in environmental conditions. Decades of extensive molecular and biochemical characterization have revealed the connectivity and mechanisms that constitute regulatory networks. We are now confronted with the challenge of integrating this information to link the structure of these circuits to systems-level properties such as cellular decision making. To investigate cellular decision-making, we used the well studied galactose gene-regulatory network in \textit{Saccharomyces cerevisiae}. We analyzed the mechanism and dynamics of the coexistence of two stable on and off states for pathway activity. We demonstrate that this bimodality in the pathway activity originates from two positive feedback loops that trigger bistability in the network. By measuring the dynamics of single-cells in a mixed sugar environment, we observe that the bimodality in gene expression is a transient phenomenon. Our experiments indicate that early pathway activation in a cohort of cells prior to galactose metabolism can accelerate galactose consumption and provide a transient increase in growth rate. Together these results provide important insights into strategies implemented by cells that may have been evolutionary advantageous in competitive environments.

Flight dynamics in Drosophila through a dynamically-scaled robotic approach

Flies are particularly adept at balancing the competing demands of delay tolerance, performance, and robustness during flight, which invites thoughtful examination of their multimodal feedback architecture. This dissertation examines stabilization requirements for inner-loop feedback strategies in the flapping flight of Drosophila, the fruit fly, against the backdrop of sensorimotor transformations present in the animal. Flies have evolved multiple specializations to reduce sensorimotor latency, but sensory delay during flight is still significant on the timescale of body dynamics. I explored the effect of sensor delay on flight stability and performance for yaw turns using a dynamically-scaled robot equipped with a real-time feedback system that performed active turns in response to measured yaw torque. The results show a fundamental tradeoff between sensor delay and permissible feedback gain, and suggest that fast mechanosensory feedback provides a source of active damping that compliments that contributed by passive effects. Presented in the context of these findings, a control architecture whereby a haltere-mediated inner-loop proportional controller provides damping for slower visually-mediated feedback is consistent with tethered-flight measurements, free-flight observations, and engineering design principles. Additionally, I investigated how flies adjust stroke features to regulate and stabilize level forward flight. The results suggest that few changes to hovering kinematics are actually required to meet steady-state lift and thrust requirements at different flight speeds, and the primary driver of equilibrium velocity is the aerodynamic pitch moment. This finding is consistent with prior hypotheses and observations regarding the relationship between body pitch and flight speed in fruit flies. The results also show that the dynamics may be stabilized with additional pitch damping, but the magnitude of required damping increases with flight speed. I posit that differences in stroke deviation between the upstroke and downstroke might play a critical role in this stabilization. Fast mechanosensory feedback of the pitch rate could enable active damping, which would inherently exhibit gain scheduling with flight speed if pitch torque is regulated by adjusting stroke deviation. Such a control scheme would provide an elegant solution for flight stabilization across a wide range of flight speeds.

Dynamics of rotationally resolved multiphoton ionization processes in molecules

This dissertation presents the results of studies of several rotationally- resolved resonance enhanced multiphoton ionization (REMPI) processes in some simple molecular systems. The objective of these studies is to quantitatively identify the underlying dynamics of this highly state-specific process which utilizes the narrow bandwidth radiation of a laser to ionize a molecule by first preparing an excited state via multiphoton absorption and subsequently ionizing that state before it can decay. Coupled with high-resolution photoelectron spectroscopy, REMPI is clearly an important probe of molecular excited states and their photoioniza tion dynamics.

A key feature of our studies is that they are carried out using accurate Hartree-Fock orbitals to describe the photoelectron orbitals of the molecular ions. The use of such photoelectron orbitals is important in rotationally-resolved studies where the angular momentum coupling in the photoelectron orbital plays a significant role in the photoionization dynamics. In these studies the Hartree-Fock molecular molecular photoelectron orbitals are obtained by numerical solution of a Lippmann-Schwinger integral equation.

Studies reported here include investigations of (i) ionic rotational branching ratios and their energy dependence for REMPI via the A^2Σ^+(3sσ) and D^2Σ^+(3pσ)states of NO, (ii) the influence of angular momentum constraints on branching ratios at low photoelectron energies for REMPI via low-J levels of the resonant intermediate state, (iii) the strong dependence of photoelectron angular distributions on final ionic rotational state and on the alignment in REMPI of the A^2Σ^+ state of NO, (iv) vibrational state dependence of ionic rotational branching ratios arising from rapid orbital evolution in resonant states (E'^2Σ^+(3pσ) of CH), (v) the influence of rovibronic interactions on the rotational branching ratios seen in REMPI via the D^2Σ^+(3pσ) state of NO, and (vi) effects of laser intensity on the photoionization dynamics of REMPI.

Investigations of charge-carrier dynamics at semiconductor/liquid interfaces

Theoretical and experimental investigations of charge-carrier dynamics at semiconductor/liquid interfaces, specifically with respect to interfacial electron transfer and surface recombination, are presented.

Fermi's golden rule has been used to formulate rate expressions for charge transfer of delocalized carriers in a nondegenerately doped semiconducting electrode to localized, outer-sphere redox acceptors in an electrolyte phase. The treatment allows comparison between charge-transfer kinetic data at metallic, semimetallic, and semiconducting electrodes in terms of parameters such as the electronic coupling to the electrode, the attenuation of coupling with distance into the electrolyte, and the reorganization energy of the charge-transfer event. Within this framework, rate constant values expected at representative semiconducting electrodes have been determined from experimental data for charge transfer at metallic electrodes. The maximum rate constant (i.e., at optimal exoergicity) for outer-sphere processes at semiconducting electrodes is computed to be in the range 10^-17-10^-16 cm⁴ s^-1, which is in excellent agreement with prior theoretical models and experimental results for charge-transfer kinetics at semiconductor/liquid interfaces.

Double-layer corrections have been evaluated for semiconductor electrodes in both depletion and accumulation conditions. In conjuction with the Gouy-Chapman-Stern model, a finite difference approach has been used to calculate potential drops at a representative solid/liquid interface. Under all conditions that were simulated, the correction to the driving force used to evaluate the interfacial rate constant was determined to be less than 2% of the uncorrected interfacial rate constant.

Photoconductivity decay lifetimes have been obtained for Si(111) in contact with solutions of CH₃OH or tetrahydrofuran containing one-electron oxidants. Silicon surfaces in contact with electrolyte solutions having Nernstian redox potentials > 0 V vs. SCE exhibited low effective surface recombination velocities regardless of the different surface chemistries. The formation of an inversion layer, and not a reduced density of electrical trap sites on the surface, is shown to be responsible for the long charge-carrier lifetimes observed for these systems. In addition, a method for preparing an air-stable, low surface recombination velocity Si surface through a two-step, chlorination/alkylation reaction is described.

On the behavior of pliable plate dynamics in wind : application to vertical axis wind turbines

Numerous studies have shown that flexible materials improve resilience and durability of a structure. Several studies have investigated the behavior of elastic plates under the influence of a free stream, such as studies of the fluttering flag and others of shape reconfiguration, due to a free stream.

The principle engineering contribution of this thesis is the design and development of a vertical axis wind turbine that features pliable blades which undergo various modes of behavior, ultimately leading to rotational propulsion of the turbine. The wind turbine design was tested in a wind tunnel and at the Caltech Laboratory for Optimized Wind Energy. Ultimately, the flexible blade vertical axis wind turbine proved to be an effective way of harnessing the power of the wind.

In addition, this body of work builds on the current knowledge of elastic cantilever plates in a free stream flow by investigating the inverted flag. While previous studies have focused on the fluid structure interaction of a free stream on elastic cantilever plates, none had studied the plate configuration where the trailing edge was clamped, leaving the leading edge free to move. Furthermore, the studies presented in this thesis establish the geometric boundaries of where the large-amplitude flapping occurs.

Direct simulation of quantum dynamics in complex systems

Accurate simulation of quantum dynamics in complex systems poses a fundamental theoretical challenge with immediate application to problems in biological catalysis, charge transfer, and solar energy conversion. The varied length- and timescales that characterize these kinds of processes necessitate development of novel simulation methodology that can both accurately evolve the coupled quantum and classical degrees of freedom and also be easily applicable to large, complex systems. In the following dissertation, the problems of quantum dynamics in complex systems are explored through direct simulation using path-integral methods as well as application of state-of-the-art analytical rate theories.

Studies in nuclear reactor dynamics : I. The accuracy of point kinetics. II. The effect of delayed neutrons on the spectrum of the group-diffusion operator

This thesis is a theoretical work on the space-time dynamic behavior of a nuclear reactor without feedback. Diffusion theory with G-energy groups is used.

In the first part the accuracy of the point kinetics (lumped-parameter description) model is examined. The fundamental approximation of this model is the splitting of the neutron density into a product of a known function of space and an unknown function of time; then the properties of the system can be averaged in space through the use of appropriate weighting functions; as a result a set of ordinary differential equations is obtained for the description of time behavior. It is clear that changes of the shape of the neutron-density distribution due to space-dependent perturbations are neglected. This results to an error in the eigenvalues and it is to this error that bounds are derived. This is done by using the method of weighted residuals to reduce the original eigenvalue problem to that of a real asymmetric matrix. Then Gershgorin-type theorems .are used to find discs in the complex plane in which the eigenvalues are contained. The radii of the discs depend on the perturbation in a simple manner.

In the second part the effect of delayed neutrons on the eigenvalues of the group-diffusion operator is examined. The delayed neutrons cause a shifting of the prompt-neutron eigenvalue s and the appearance of the delayed eigenvalues. Using a simple perturbation method this shifting is calculated and the delayed eigenvalues are predicted with good accuracy.

Spin dynamics of pulsed nuclear magnetic double resonance in solids

In the first part of this thesis, experiments utilizing an NMR phase interferometric concept are presented. The spinor character of two-level systems is explicitly demonstrated by using this concept. Following this is the presentation of an experiment which uses this same idea to measure relaxation times of off-diagonal density matrix elements corresponding to magnetic-dipole-forbidden transitions in a ^(13)C-^1H, AX spin system. The theoretical background for these experiments and the spin dynamics of the interferometry are discussed also.

The second part of this thesis deals with NMR dipolar modulated chemical shift spectroscopy, with which internuclear bond lengths and bond angles with respect to the chemical shift principal axis frame are determined from polycrystalline samples. Experiments using benzene and calcium formate verify the validity of the technique in heteronuclear (^(13)C-^1H) systems. Similar experiments on powdered trichloroacetic acid confirm the validity in homonuclear (^1H- ^1H) systems. The theory and spin dynamics are explored in detail, and the effects of a number of multiple pulse sequences are discussed.

The last part deals with an experiment measuring the ^(13)C chemical shift tensor in K_2Pt(CN)_4Br_(0.3) • 3H_2O, a one-dimensional conductor. The ^(13)C spectra are strongly affected by ^(14)N quadrupolar interactions via the ^(13)C - ^(14)N dipolar interaction. Single crystal rotation spectra are shown.

An appendix discussing the design, construction, and performance of a single-coil double resonance NMR sample probe is included.

Macroscopically Dissipative Systems with Underlying Microscopic Dynamics : Properties and Limits of Measurement

While some of the deepest results in nature are those that give explicit bounds between important physical quantities, some of the most intriguing and celebrated of such bounds come from fields where there is still a great deal of disagreement and confusion regarding even the most fundamental aspects of the theories. For example, in quantum mechanics, there is still no complete consensus as to whether the limitations associated with Heisenberg's Uncertainty Principle derive from an inherent randomness in physics, or rather from limitations in the measurement process itself, resulting from phenomena like back action. Likewise, the second law of thermodynamics makes a statement regarding the increase in entropy of closed systems, yet the theory itself has neither a universally-accepted definition of equilibrium, nor an adequate explanation of how a system with underlying microscopically Hamiltonian dynamics (reversible) settles into a fixed distribution.

Motivated by these physical theories, and perhaps their inconsistencies, in this thesis we use dynamical systems theory to investigate how the very simplest of systems, even with no physical constraints, are characterized by bounds that give limits to the ability to make measurements on them. Using an existing interpretation, we start by examining how dissipative systems can be viewed as high-dimensional lossless systems, and how taking this view necessarily implies the existence of a noise process that results from the uncertainty in the initial system state. This fluctuation-dissipation result plays a central role in a measurement model that we examine, in particular describing how noise is inevitably injected into a system during a measurement, noise that can be viewed as originating either from the randomness of the many degrees of freedom of the measurement device, or of the environment. This noise constitutes one component of measurement back action, and ultimately imposes limits on measurement uncertainty. Depending on the assumptions we make about active devices, and their limitations, this back action can be offset to varying degrees via control. It turns out that using active devices to reduce measurement back action leads to estimation problems that have non-zero uncertainty lower bounds, the most interesting of which arise when the observed system is lossless. One such lower bound, a main contribution of this work, can be viewed as a classical version of a Heisenberg uncertainty relation between the system's position and momentum. We finally also revisit the murky question of how macroscopic dissipation appears from lossless dynamics, and propose alternative approaches for framing the question using existing systematic methods of model reduction.