Random perturbations of delay differential equations at the verge of instability and periodically driven nonlinear oscillators

First part of this thesis (chapters 1-5) studies the effect of small noise perturbations on delay differential equations (DDE) whose fixed point is on the verge of instability. With appropriate scaling of coordinates, the dynamics close to the fixed point can be cast in the form of a linear DDE perturbed by small noise and small nonlinearities. The instability scenario considered is the following: a pair of roots of the characteristic equation (critical eigenvalues) lie on the imaginary axis of the complex plane, and all other roots have negative real parts (stable eigenvalues). The effect of deterministic perturbations on such DDE is analytically well-studied in the literature. For noise perturbations with noise as a Wiener process, the existing studies analyze the amplitude of the critical eigenmodes using the averaging approach, and in the spirit of multiscale analysis. This thesis is an attempt to make the averaging approach rigorous. Errors in other articles using multiscale approach are pointed out. Markovian noises of the exponentially ergodic type are also considered in this thesis. It is shown that as the strength of the perturbations decreases to zero, the probability law of the amplitude of critical eigenmodes, under an appropriate change of time scale, converges to the probability law of a one-dimensional stochastic differential equation (SDE) without delay. Further it is shown that the stable eigenmodes are small in an appropriate sense. For small perturbations, the SDE obtained gives a good approximate description of the original DDE. The advantage is that the SDE without delay is easier to anlayze and simulate numerically. The reduced dimensional description using the one-dimensional SDE can be used to understand what kind of noise advances the bifurcation and what postpones it. For the case of random perturbations with linear coefficients, a complex number is identified which alone dictates the stability of the system. Analogous results for a different instability scenario are also presented. In this scenario, the characteristic equation has zero as a simple root, and all other roots have negative real parts. When the noise perturbation is weaker than the deterministic perturbations, it is shown that the results of Freidlin-Wentzell can be applied to study large deviations from corresponding deterministic trajectories. The second part of this thesis (chapter 6) deals with random perturbations of periodically driven nonlinear oscillators. The recent surge of research articles in energy harvesting focuses on the cantilever beam devices which are used to convert small amplitude mechanical vibration into an electrical energy source. Prototypical beam type nonlinear energy harvesting models contain double well potentials, external or parametric periodic forcing terms, damping and ambient additive noise terms. An equation with all these features is considered. In the absence of noise, the phase space for such a periodically driven nonlinear oscillator consists of many resonance zones where the oscillator frequency and the driving frequency are commensurable. It is well known that, a small subset of initial conditions can lead to capture in one of the resonance zones. The effect of weak noise on the escape from a resonance zone is studied in the second part of this thesis. Using averaging techniques, a conjecture is made regarding the mean exit time from a resonance zone. Using this conjecture, the dependence of the exit rate on the parameters of the oscillator is illustrated. A qualitative picture of the dynamics of the oscillator under weak-driving-force, weak-damping, weak-noise limit is given