The characteristic equation for a linear delay differential equation (DDE) has countably infinite roots on the complex plane. We deal with linear DDEs that are on the verge of instability, i.e. a pair of roots of the characteristic equation (critical eigenvalues) lie on the imaginary axis of the complex plane, and all other roots (stable eigenvalues) have negative real parts. We show that, when the system is perturbed by small noise, under an appropriate change of time scale, the law of the amplitude of projection onto the critical eigenspace is close to the law of a certain one-dimensional stochastic differential equation (SDE) without delay. Further, we show that the projection onto the stable eigenspace is small. These results allow us to give an approximate description of the delay-system using an SDE (without delay) of just one dimension. The proof is based on the martingale problem technique.

Abstract. The characteristic equation for a linear delay differential equation (DDE) has countably infinite roots on the complex plane. This paper considers linear DDEs that are on the verge of instability, i.e. a pair of roots of the characteristic equation lie on the imaginary axis of the complex plane, and all other roots have negative real parts. It is shown that, when small noise perturbations are present, the probability distribution of the dynamics can be approximated by the probability distribution of certain one dimensional stochastic differential equation (SDE) without delay. This is advantageous because equations without delay are easier to simulate and one-dimensional SDE are analytically tractable. When the perturbations are also linear, it is shown that the stability depends on a specific complex number. The theory is applied to study oscillators with delayed feedback. Some errors in other articles that use multiscale approach are pointed out.

We present an efficient particle filtering algorithm for multiscale systems, which is adapted for simple atmospheric dynamics models that are inherently chaotic. Particle filters represent the posterior conditional distribution of the state variables by a collection of particles, which evolves and adapts recursively as new information becomes available. The difference between the estimated state and the true state of the system constitutes the error in specifying or forecasting the state, which is amplified in chaotic systems that have a number of positive Lyapunov exponents. In this paper, we propose a reduced-order particle filtering algorithm based on the homogenized multiscale filtering framework developed in Imkeller et al. "Dimensional reduction in nonlinear filtering: A homogenization approach," Ann. Appl. Probab. (to be published). In order to adapt the proposed algorithm to chaotic signals, importance sampling and control theoretic methods are employed for the construction of the proposal density for the particle filter. Finally, we apply the general homogenized particle filtering algorithm developed here to the Lorenz'96 [E. N. Lorenz, "Predictability: A problem partly solved," in Predictability of Weather and Climate, ECMWF, 2006 (ECMWF, 2006), pp. 40-58] atmospheric model that mimics mid-latitude atmospheric dynamics with microscopic convective processes.

Abstract. The phase space for the periodically driven nonlinear oscillator consists of many resonance zones. Let the strength of periodic excitation and the strength of the damping be indexed by a small parameter ε. It is well known that, as ε → 0, the measure of the set of initial conditions which lead to capture in a resonance zone goes to zero. In this paper we study the effect of weak noise on the escape from a resonance zone.

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