This thesis is concerned with changes in the behaviour of solutions to parameter-dependent stochastic delay differential equations.

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<p>The goal of this work is to develop a practical and comprehensive methodology to study the response and the stability of various delay differential equations (DDEs). The development of these new analysis techniques is motivated by the existence of delays in the governing equations of many physical systems such as turning and milling processes. </p><p>Delay differential equations appear in many models in science in engineering either as an intrinsic component (e.g. machining dynamics) or as a modeling decision (biology related dynamics). However, the infinite dimensionality of DDEs significantly complicates the resulting analysis from both an analytical and numerical perspective. Since the delay results in an infinite dimensional state-space, it is often necessary to use an approximate procedure to study DDEs and ascertain their stability.</p><p>Several approximate techniques appeared in literature to study the stability of DDEs. However, a large number of these techniques---such as D-subdivision, Cluster Treatment of Characteristic Roots and Continuous Time Approximation---are limited to autonomous DDEs. Moreover, the methods that are suitable for non-autonomous DDEs, e.g. the Semi-discretization approach, often result in a very large system of algebraic equations that can cause computational difficulties. Collocation-type methods, such as Chebyshev-collocation approach, have also been used to study DDEs. One major limitation of the conventional Chebyshev collocation approach is that it is strictly applicable to DDEs with continuous coefficients. An alternative approach that can handle DDEs with piecewise continuous coefficients is the Temporal Finite Element Analysis (TFEA). However, TFEA has only linear rates of convergence and is limited to h-convergence schemes. The limited rate of convergence in TFEA has prohibited its application to a wide class of DDEs such as DDEs with complicated coefficients or with distributed and multiple delays. </p><p>In this thesis, I develop a spectral element method for the stability analysis of DDEs. The spectral element method is a Galerkin-type approach that discretizes the infinite dimensional DDE into a finite set of algebraic equations (or a dynamic map). The stability of the system is then studied using the eigenvalues of the map. </p><p>In contrast to TFEA, the spectral element method was shown to have exponential rates of convergence and hp-refinement capabilities. Further, a comparison with the widely-used collocation methods showed that our approach can often yield higher rates of convergence. The higher rates of convergence of the developed approach enabled extending it to DDEs with multiple and distributed delays. I further extended this approach to calculating the periodic orbits of DDEs and their stability. </p><p>As an application of the methods developed in this thesis, I studied the stability of turning and milling models. For example, a distributed force model was proposed to characterize cutting forces in turning. The stability of the resulting delay integro-differential equation was studied using the methods developed in this study and they were shown to agree with practical observations. As another example, the stability of a milling process--- whose model contains piecewise coefficients---was investigated. The effect of multiple-flute engagement, which contributed to the complexity of the coefficients, was also investigated. The resulting stability charts revealed new stability observations in comparison to typical analysis methods. Specifically, I was able to show that unstable regions appear in what was deemed a stable region by prior analysis techniques.</p> / Dissertation

We consider a general stochastic differential delay equation (SDDE) with multiplicative colored noise. We study the limit as the time delays and the correlation times of the noises go to zero at the same rate. First, we derive the limiting equation for the equation obtained by Taylor expanding the SDDE to first order in the time delays. The limiting equation contains a noise-induced drift term that depends on the ratios of the time delays to the correlation times of the noises. We prove that, under appropriate assumptions, the solution of the equation obtained by the Taylor expansion converges to the solution of this limiting equation in probability with respect to the sup norm over compact time intervals. Next, we derive the limiting equation for the SDDE and prove a similar convergence result regarding convergence of the solution of the SDDE to the solution of this limiting equation. We see that the limiting equation corresponding to the equation obtained by the Taylor expansion is an approximation of the limiting equation corresponding to the SDDE. Finally, we study the effects of time delay on a particular model of active Brownian motion.

We study bifurcations of periodic solutions with spatio-temporal symmetries of functional differential equations (FDEs). The two main results are: (1) a centre manifold reduction around a periodic solution of FDEs with spatio-temporal symmetries, and (2) symmetry-breaking bifurcations for symmetric rings of delay-coupled lasers. For the case of ODEs, symmetry-breaking bifurcations from periodic solutions has already been studied. We extend this result to the case of symmetric FDEs using a Centre Manifold Theorem for symmetric FDEs which reduces FDEs into ODEs on an integral manifold around a periodic solution. We show that the integral manifold is invariant under the spatio-temporal symmetries which guarantees that the symmetry structure of the system of FDEs is preserved by this reduction. We also consider a problem in rings of delay-coupled lasers modeled using the Lang-Kobayashi rate equations. We classify the symmetry of bifurcating branches of solutions from steady-state and Hopf bifurcations that occur in 3-laser systems. This involves finding isotropy subgroups of the symmetry group of the system, and then using the Equivariant Branching Lemma and the Equivariant Hopf Theorem. We then utilize this result to find the bifurcating branches of solutions in DDE-Biftool. Symmetry often causes eigenvalues to have multiplicity, and in some cases, this could lead DDE-Biftool to incorrectly predict the bifurcation points. We address this issue by developing a method of finding bifurcation points which can be used for the general case of n-laser systems with unidirectional and bidirectional coupling. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2012-04-30 11:25:01.011

We compare an internal state method and a direct Runge-Kutta method for solving Volterra integro-differential equations and Volterra delay differential equations. The internal state method requires the kernel of the Volterra integral to be realizable as an impulse response function. We discover that when applicable, the internal state method is orders of magnitude more efficient than the direct numerical method. However, constructing state representation for realizable kernels can be challenging at times; therefore, we propose a rational approximation approach to avoid the problem. That is, we approximate the transfer function by a rational function, construct the corresponding linear system, and then approximate the Volterra integro-differential equation. We show that our method is convergent for the case where the kernel is nuclear. We focus our attention on time-invariant realizations but the case where the state representation of the kernel is a time-variant linear system is briefly discussed. / Ph. D.

Studying the zeros of a parameter dependent operator F defined on a Hilbert space H is a fundamental problem in mathematics. When the Hilbert space is finite dimensional, continuation provides, via predictor-corrector algorithms, efficient techniques to numerically follow the zeros of F as we move the parameter. In the case of infinite dimensional Hilbert spaces, this procedure must be applied to some finite dimensional approximation which of course raises the question of validity of the output. We introduce a new technique that combines the information obtained from the predictor-corrector steps with ideas from rigorous computations and verifies that the numerically produced zero for the finite dimensional system can be used to explicitly define a set which contains a unique zero for the infinite dimensional problem F: HxR->Im(F). We use this new validated continuation to study equilibrium solutions of partial differential equations, to prove the existence of chaos in ordinary differential equations and to follow branches of periodic solutions of delay differential equations. In the context of partial differential equations, we show that the cost of validated continuation is less than twice the cost of the standard continuation method alone.

L’objectif de cette thèse est d’étudier et de donner des outils pour la compréhension des problèmes de perturbations singulières pour des modèles épidémiques et des problèmes de dynamiques de populations. Les modèles considérés sont des équations structurées en âge qui peuvent dans certains cas se réécrire comme des équations à retard. L’étude de ces classes d’exemples s’est faite avec succès et a permis de comprendre et de mettre en évidence toute la complexité et l’étendue de ces problèmes. Comme on peut le remarquer dans la littérature, l’une des clés fondamentales à la compréhension de ces problèmes est l’étude des variétés normalement hyperboliques en dimension infinie que nous avons largement étudiées dans cette thèse. L’approche utilisée est la méthode de Lyapunov-Perron. Ce qui nous a amené à étudier les problèmes de persistance et d’existence de trichotomie (dichotomie) exponentielle qui sont des éléments fondamentaux dans l’utilisation de cette méthode. / In this thesis we aim to give tools to understand singular perturbations in epidemic model sand population dynamic models. We study some singularly perturbed delay differential equation which does not enter into the class frame work of geometric singular perturbation for delay differential equations. An example of singularly perturbed age structured model is also studied. The study of these examples allowed us to understand and highlight some complexities of these problems. One of the main tools in understanding such questions is the normally hyperbolic manifolds theory which is our central focus in this thesis. The approach used here is the Lyapunov-Perron method. Therefore the problems of persistence and existence of exponential trichotomy (dichotomy) are also stressed since there are one of the mainingredients of this method.

The thesis deals with so-called evolutionary equations, a class of abstract linear operator equations, which cover a huge class of partial differential equation with and without memory. We provide a unified Hilbert space framework for the well-posedness of such equations. Moreover, we inspect the exponential stability of those problems and construct spaces of admissible inital values and pre-histories, on which a strongly continuous semigroup could be associated with the given problem. The theoretical results are illustrated by several examples.

The thesis deals with so-called evolutionary equations, a class of abstract linear operator equations, which cover a huge class of partial differential equation with and without memory. We provide a unified Hilbert space framework for the well-posedness of such equations. Moreover, we inspect the exponential stability of those problems and construct spaces of admissible inital values and pre-histories, on which a strongly continuous semigroup could be associated with the given problem. The theoretical results are illustrated by several examples.