Asymptotic growth in nonlinear stochastic and deterministic functional differential equations

This thesis concerns the asymptotic growth of solutions to nonlinear functional differential equations, both random and deterministic. How quickly do solutions grow? How do growth rates of solutions depend on the memory and the nonlinearity of the system? What is the effect of randomness on the growth rates of solutions? We address these questions for classes of nonlinear functional differential equations, principally convolution Volterra equations of the second kind. We first study deterministic equations with sublinear nonlinearity and integrable kernels. For such systems, we prove that the growth rates of solutions are independent of the distribution of the memory. Hence we conjecture that stronger memory dependence is needed to generate growth rates which depend meaningfully on the delay structure. Using the theory of regular variation, we then demonstrate that solutions to a class of sublinear Volterra equations with non–integrable kernels grow at a memory dependent rate. We complete our treatment of sublinear equations by examining the impact of stochastic perturbations on our previous results; we consider the illustrative and important cases of Brownian and alpha–stable Lévy noise. In summary, if an appropriate functional of the forcing term has a limit L at infinity, solutions behave asymptotically like the underlying unforced equation when L = 0 and like the forcing term when L is infinite. Solutions inherit properties of both the forcing term and underlying unforced equation for finite and positive L. Similarly, we prove linear discrete Volterra equations with summable kernels inherit the behaviour of unbounded perturbations, random or deterministic. Finally, we consider Volterra integro–differential equations with superlinear nonlinearity and nonsingular kernels. We provide sharp estimates on the rate of blow–up if solutions are explosive, or unbounded growth if solutions are global. We also recover well–known necessary and sufficient conditions for finite–time blow–up via new methods

On the admissibility of unboundedness properties of forced deterministic and stochastic sublinear Volterra summation equations

In this paper we consider unbounded solutions of perturbed convolution Volterra summation equations. The equations studied are asymptotically sublinear, in the sense that the state--dependence in the summation is of smaller than linear order for large absolute values of the state. When the perturbation term is unbounded, it is elementary to show that solutions are also. The main results of the paper are mostly of the following form: the solution has an additional unboundedness property $U$ if and only if the perturbation has property $U$. Examples of property $U$ include monotone growth, monotone growth with fluctuation, fluctuation on $\mathbb{R}$ without growth, existence of time averages. We also study the connection between the times at which the perturbation and solution reach their running maximum, and the connection between the size of signed and unsigned running maxima of the solution and forcing term.Comment: 45 page

Subexponential Growth Rates in Functional Differential Equations

This paper determines the rate of growth to infinity of a scalar autonomous nonlinear functional differential equation with finite delay, where the right hand side is a positive continuous linear functional of $f(x)$. We assume $f$ grows sublinearly, and is such that solutions should exhibit growth faster than polynomial, but slower than exponential. Under some technical conditions on $f$, it is shown that the solution of the functional differential equation is asymptotic to that of an auxiliary autonomous ordinary differential equation with righthand side proportional to $f$ (with the constant of proportionality equal to the mass of the finite measure associated with the linear functional), provided $f$ grows more slowly than $l(x)=x/\log x$. This linear--logarithmic growth rate is also shown to be critical: if $f$ grows more rapidly than $l$, the ODE dominates the FDE; if $f$ is asymptotic to a constant multiple of $l$, the FDE and ODE grow at the same rate, modulo a constant non--unit factor.Comment: 10 page

Large scale ab-initio simulations of dislocations

We present a novel methodology to compute relaxed dislocations core configurations, and their energies in crystalline metallic materials using large-scale ab-intio simulations. The approach is based on MacroDFT, a coarse-grained density functional theory method that accurately computes the electronic structure with sub-linear scaling resulting in a tremendous reduction in cost. Due to its implementation in real-space, MacroDFT has the ability to harness petascale resources to study materials and alloys through accurate ab-initio calculations. Thus, the proposed methodology can be used to investigate dislocation cores and other defects where long range elastic effects play an important role, such as in dislocation cores, grain boundaries and near precipitates in crystalline materials. We demonstrate the method by computing the relaxed dislocation cores in prismatic dislocation loops and dislocation segments in magnesium (Mg). We also study the interaction energy with a line of Aluminum (Al) solutes. Our simulations elucidate the essential coupling between the quantum mechanical aspects of the dislocation core and the long range elastic fields that they generate. In particular, our quantum mechanical simulations are able to describe the logarithmic divergence of the energy in the far field as is known from classical elastic theory. In order to reach such scaling, the number of atoms in the simulation cell has to be exceedingly large, and cannot be achieved with the state-of-the-art density functional theory implementations