Thesis (Ph. D.)--Rutgers University, 2007. / "Graduate Program in Mathematics." Includes bibliographical references (p. 174-176).

Thesis (M.Sc.)--Memorial University of Newfoundland, 2003. / Bibliography: leaves 221-225. Also available online.

In this thesis we consider the traditional Van der Pol Oscillator with a forcing dependent on a delay in feedback. The delay is taken to be a nonlinear function of both position and velocity which gives rise to many different types of bifurcations. In particular, we study the Zero-Hopf bifurcation that takes place at certain parameter values using methods of centre manifold reduction of DDEs and normal form theory. We present numerical simulations that have been accurately predicted by the phase portraits in the Zero-Hopf bifurcation to confirm our numerical results and provide a physical understanding of the oscillator with the delay in feedback.

The Black-Scholes model and corresponding option pricing formula has led to a wide and extensive industry, used by financial institutions and investors to speculate on market trends or to control their level of risk from other investments. From the formation of the Chicago Board Options Exchange in 1973, the nature of options contracts available today has grown dramatically from the single-date contracts considered by Black and Scholes (1973) to a wider and more exotic range of derivatives. These include American options, which can be exercised at any time up to maturity, as well as options based on the weighted sums of assets, such as the Asian and basket options which we consider. Moreover, the underlying models considered have also grown in number and in this work we are primarily motivated by the increasing interest in past-dependent asset pricing models, shown in recent years by market practitioners and prominent authors. These models provide a natural framework that considers past history and behaviour, as well as present information, in the determination of the future evolution of an underlying process. In our studies, we explore option pricing techniques for arithmetic Asian and basket options under a Stochastic Delay Differential Equation (SDDE) approach. We obtain explicit closed-form expressions for a number of lower and upper bounds before giving a practical, numerical analysis of our result. In addition, we also consider the properties of the approximate numerical integration methods used and state the conditions for which numerical stability and convergence can be achieved.

by Hing-cheung So. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1995. / Includes bibliographical references (leaves 133-141). / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Time Delay Estimation (TDE) and its Applications --- p.1 / Chapter 1.2 --- Goal of the Work --- p.6 / Chapter 1.3 --- Thesis Outline --- p.9 / Chapter 2 --- Adaptive Methods for TDE --- p.10 / Chapter 2.1 --- Problem Description --- p.11 / Chapter 2.2 --- The Least Mean Square Time Delay Estimator (LMSTDE) --- p.11 / Chapter 2.2.1 --- Bias and Variance --- p.14 / Chapter 2.2.2 --- Probability of Occurrence of False Peak Weight --- p.16 / Chapter 2.2.3 --- Some Modifications of the LMSTDE --- p.17 / Chapter 2.3 --- The Adaptive Digital Delay-Lock Discriminator (ADDLD) --- p.18 / Chapter 2.4 --- Summary --- p.20 / Chapter 3 --- The Explicit Time Delay Estimator (ETDE) --- p.22 / Chapter 3.1 --- Derivation and Analysis of the ETDE --- p.23 / Chapter 3.1.1 --- The ETDE system --- p.23 / Chapter 3.1.2 --- Performance Surface --- p.26 / Chapter 3.1.3 --- Static Behaviour --- p.28 / Chapter 3.1.4 --- Dynamic Behaviour --- p.30 / Chapter 3.2 --- Performance Comparisons --- p.32 / Chapter 3.2.1 --- With the LMSTDE --- p.32 / Chapter 3.2.2 --- With the CATDE --- p.34 / Chapter 3.2.3 --- With the CRLB --- p.35 / Chapter 3.3 --- Simulation Results --- p.38 / Chapter 3.3.1 --- Corroboration of the ETDE Performance --- p.38 / Chapter 3.3.2 --- Comparative Studies --- p.44 / Chapter 3.4 --- Summary --- p.48 / Chapter 4 --- An Improvement to the ETDE --- p.49 / Chapter 4.1 --- Delay Modeling Error of the ETDE --- p.49 / Chapter 4.2 --- The Explicit Time Delay and Gain Estimator (ETDGE) --- p.52 / Chapter 4.3 --- Performance Analysis --- p.55 / Chapter 4.4 --- Simulation Results --- p.57 / Chapter 4.5 --- Summary --- p.61 / Chapter 5 --- TDE in the Presence of Multipath Propagation --- p.62 / Chapter 5.1 --- The Multipath TDE problem --- p.63 / Chapter 5.2 --- TDE with Multipath Cancellation (MCTDE) --- p.64 / Chapter 5.2.1 --- Structure and Algorithm --- p.64 / Chapter 5.2.2 --- Convergence Dynamics --- p.67 / Chapter 5.2.3 --- The Generalized Multipath Cancellator --- p.70 / Chapter 5.2.4 --- Effects of Additive Noises --- p.73 / Chapter 5.2.5 --- Simulation Results --- p.74 / Chapter 5.3 --- TDE with Multipath Equalization (METDE) --- p.86 / Chapter 5.3.1 --- The Two-Step Algorithm --- p.86 / Chapter 5.3.2 --- Performance of the METDE --- p.89 / Chapter 5.3.3 --- Simulation Results --- p.93 / Chapter 5.4 --- Summary --- p.101 / Chapter 6 --- Conclusions and Suggestions for Future Research --- p.102 / Chapter 6.1 --- Conclusions --- p.102 / Chapter 6.2 --- Suggestions for Future Research --- p.104 / Appendices --- p.106 / Chapter A --- Derivation of (3.20) --- p.106 / Chapter B --- Derivation of (3.29) --- p.110 / Chapter C --- Derivation of (4.14) --- p.111 / Chapter D --- Derivation of (4.15) --- p.113 / Chapter E --- Derivation of (5.21) --- p.115 / Chapter F --- Proof of unstablity of A°(z) --- p.116 / Chapter G --- Derivation of (5.34)-(5.35) --- p.118 / Chapter H --- Derivation of variance of αs11(k) and Δs11(k) --- p.120 / Chapter I --- Derivation of (5.40) --- p.123 / Chapter J --- Derivation of time constant of αΔ11(k) --- p.124 / Chapter K --- Derivation of (5.63)-(5.66) --- p.125 / Chapter L --- Derivation of (5.68)-(5.72) --- p.129 / References --- p.133

The research reported in this thesis builds on a novel model developed at the University of Minnesota to analyze the self-excited vibrations that occur when drilling with polycrystalline diamond cutter bits. The lumped parameter model of the drilling system takes into consideration the axial and the torsional vibrations of the bit. These vibrations are coupled through a bit-rock interaction law. At the bit-rock interface, the cutting process combined with the quasihelical motion of the bit leads to a regenerative effect that introduces a coupling between the axial and torsional modes of vibrations and a state-dependent delay in the governing equations, while the frictional contact process is associated with discontinuities in the boundary conditions when the bit sticks in its axial and angular motion. The response of this complex system is characterized by a fast axial dynamics superposed to the slow torsional dynamics. A two time scales analysis that uses a combination of averaging methods and a singular perturbation approach is proposed to study the dynamical response of the system. An approximate model of the decoupled axial dynamics permits to derive a pseudo analytical expression of the solution of the axial equation. Its averaged behavior influences the slow torsional dynamics by generating an apparent velocity weakening friction law that has been proposed empirically in earlier works. The analytical expression of the solution of the axial dynamics is used to derive an approximate analytical expression of the velocity weakening friction law related to the physical parameters of the system. This expression can be used to provide recommendations on the operating parameters and the drillstring or the bit design in order to reduce the amplitude of the torsional vibrations. Moreover, it is an appropriate candidate model to replace empirical friction laws encountered in torsional models used for control. In this thesis, we also analyze the axial and torsional vibrations by basing the model on a continuum representation of the drillstring rather than on the low dimensional lumped parameter model. The dynamic response of the drilling structure is computed using the finite element method. While the general tendencies of the system response predicted by the discrete model are confirmed by this computational model (for example that the occurrence of stick-slip vibrations as well as the risk of bit bouncing are enhanced with an increase of the weight-on-bit or a decrease of the rotational speed), new features in the self-excited response of the drillstring are detected. In particular, stick-slip vibrations are predicted to occur at natural frequencies of the drillstring different from the fundamental one (as sometimes observed in field operations), depending on the operating parameters. Finally, we describe the experimental strategy chosen for the validation of the model and discuss results of tests conducted with DIVA, an analog experimental set-up of the lumped parameter model. Some results of the experiments conducted in an artificial rock seem to validate the model studied here although the same experiments obtained with natural rocks were unsuccessful. Different problems with the design of the experimental setup were identified. By using the outcome of the analysis of the uncoupled dynamics, we could provide critical recommendation to elaborate and to design a simpler and stiffer analog experiment (TAZ) used to study the self excitation of the axial dynamics that ultimately lead to the excitation of the torsional dynamics.

The stability and boundedness theories are developed for impulsive differential equations with time delay. Definitions, notations and fundamental theory are presented for delay differential systems with both fixed and state-dependent impulses. It is usually more difficult to investigate the qualitative properties of systems with state-dependent impulses since different solutions have different moments of impulses. In this thesis, the stability problems of nontrivial solutions of systems with state-dependent impulses are ``transferred" to those of the trivial solution of systems with fixed impulses by constructing the so-called ``reduced system". Therefore, it is enough to investigate the stability problems of systems with fixed impulses. The exponential stability problem is then discussed for the system with fixed impulses. A variety of stability criteria are obtained and`numerical examples are worked out to illustrate the results, which shows that impulses do contribute to the stabilization of some delay differential equations. To unify various stability concepts and to offer a general framework for the investigation of stability theory, the concept of stability in terms of two measures is introduced and then several stability criteria are developed for impulsive delay differential equations by both the single and multiple Lyapunov functions method. Furthermore, boundedness and periodicity results are discussed for impulsive differential systems with time delay. The Lyapunov-Razumikhin technique, the Lyapunov functional method, differential inequalities, the method of variation of parameters, and the partitioned matrix method are the main tools to obtain these results. Finally, the application of the stability theory to neural networks is presented. In applications, the impulses are considered as either means of impulsive control or perturbations.Sufficient conditions for stability and stabilization of neural networks are obtained.

A Hamiltonian model for the radial and translational dynamics of clusters of coupled bubbles in an incompressible liquid developed by Ilinskii, Hamilton, and Zabolotskaya [J. Acoust. Soc. Am. 121, 786-795 (2007)] is extended to included the effects of compressibility in the host liquid. The bubbles are assumed to remain spherical and translation is allowed. The two principal effects of liquid compressibility are time delay in bubble interaction due to the finite sound speed and radiation damping due to energy lost to acoustic radiation. The incorporation of time delays produces a system of delay differential equations of motion instead of the system of ordinary differential equations in models of bubble interaction in an incompressible medium. The form of the Hamiltonian equations of motion is significantly different from the commonly used models based on Rayleigh-Plesset equations for coupled bubble dynamics, and it provides certain advantages in numerical integration of the time-delayed equations of motion. Corrections for radiation damping in clusters of interacting bubbles are developed in the form of a time-delayed expression for bubble self-action following the method of Ilinskii and Zabolotskaya [J. Acoust. Soc. Am. 92, 2837-2841 (1992)]. A set of approximate series expansions of this delayed expression is calculated to first order in the ratio of bubble radius to the characteristic wavelength of acoustic radiation from the bubble, and to varying orders in the ratio of bubble radius to characteristic bubble separation distance. Stability of the delay differential equations of motion is analyzed with four successive levels of approximation for the effects of radiation damping and time delay. The stability is analyzed with and without the effects of viscous and thermal damping. The effect of time delay and radiation damping on the pressure radiated by small systems of bubbles is considered. An approximate method to account for the delays in bubble interaction in a weakly compressible liquid is presented. This method converts the system of delay differential equations into an approximate system of ordinary differential equations, which may simplify numerical integration. Several sets of model equations incorporating propagation time delay in bubble interactions are solved numerically with existing algorithms specialized for delay differential equations. Numerical simulations of the dynamics of single bubbles, pairs of bubbles, and clusters of bubbles are used to compare the different levels of approximation for compressibility effects for low- and high-amplitude radial motion in systems of bubbles under free response and pulsed excitation by an external pressure source. / text

Thesis (M.S.)--University of Missouri-Columbia, 2007. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file (viewed on January 7, 2008) Includes bibliographical references.

Thesis (Ph. D.)--University of Texas at Austin, 1998. / Vita. Includes bibliographical references (leaves 133-140). Available also in a digital version from Dissertation Abstracts.