Distributed Delay Terms Research Articles

Euler-Maruyama and Milstein Approximations for Stochastic Functional Differential Equations with Distributed Memory Term

We consider the problem of strong approximations of the solution of stochastic functional differential equations of Ito form with a distributed delay term in the drift and diffusion coefficient. We provide necessary background material, and give convergence proofs for the Euler-Maruyama and the Milestein scheme. Numerical examples illustrate the theoretical results.

Norm continuity and stability for a functional differential equation in Hilbert space

A functional differential equation in Hilbert space with initial data on [−h,0] is considered. An unbounded operator A and a square integrable weight function are acting in the distributed delay term. For a not necessarily continuous weight function the norm continuity of the associated solution semigroup is established at every t>h. In the case that the spectrum of A is real and negative, the asymptotic stability of the solution is obtained.

Optimization of olfactory model in software to give 1/ f power spectra reveals numerical instabilities in solutions governed by aperiodic (chaotic) attractors

We present a general connectionist model for an olfactory system. The dynamical behavior of each node (neural ensemble) of the model is governed by a second-order ordinary differential equation (ODE) followed by an asymmetric sigmoidal function, relating the aggregate activity of neurons to system parameters and stimuli from an outside environment. A digital implementation of the general connectionist model simulates the characteristics of a mammalian olfactory system having modifiable synaptic connections and spatio-temporal interactions among neural ensembles. Each of four distributed delay terms is represented by a second-order ODE. A parameter optimization algorithm is an integral component of the model. The parameter optimization discussed in this paper results in aperiodic oscillations having a near 1/ f-type power spectrum with a peak in the gamma range, simulating the electro-encephalographic (EEG) potentials from the neural olfactory system. Random optimization is used for a rough search in a global parameter domain and the parameter self-adaptation rule serves for fine tuning within a local domain after the global search. By design the model is started from an unstable zero point by an external impulse input. It requires 650–850 ms to pass through an initializing transient before settling into a strange attractor. The attractor persists for at least 1500 ms, which is 7.5–20 times longer than the duration of the maximal stationary states observed in the EEGs and it is stable under perturbation by simulated sensory inputs giving response amplitudes less than 2 times the basal aperiodic activity. However, the optimization to 1/ f-type activity reveals an inherent limit on digital simulation of chaotic states, owing to attractor crowding such that the size of basins decreases with increasing size of the model, until it approaches the size of digitizing step in computation, here a 64-bit word (∼10 −16). Outputs that are not optimized to approach 1/ f-type power spectra transit earlier to limit cycle activity, usually well before 2000 ms. The duration of stationarity is increased by randomizing the terminal bit of the 64-bit words representing the state variables. The 1/ f-type solutions are also exquisitely sensitive to parameter truncation; parameter values must be saved in their full binary form for re-starting. The implications in terms of numerical instability, chaos, attractor crowding and the shadowing theorem are discussed.

Interaction of spatial diffusion and delays in models of genetic control by repression.

A class of models based on the Jacob and Monod theory of genetic repression for control of biosynthetic pathways in cells is considered. Both spatial diffusion and time delays are taken into account. A method is developed for representing the effects of spatial diffusion as distributed delay terms. This method is applied to two specific models and the interaction between the diffusion and the delays is treated in detail. The destabilization of the steady-state and the bifurcation of oscillatory solutions are studied as functions of the diffusivities and the delays. The limits of very small and very large diffusivities are analyzed and comparisons with well-mixed compartment models are made.

Observers for Systems with Discrete and Distributed Delays

An observer theory for linear systems containing discrete and distributed delays in its state- and output equations is presented. In particular, existence criteria for observers with finite-dimensional error dynamics are derived. The stabilizability and eigenvalue assignability of the observation error dynamics are expressed in terms of the invariant zeros of a matrix function P(s), depending on the discrete delay parameters and the parameters of a Gram-Schmidt expansion of the distributed delay terms.