Dense Short Solution Segments from Monotonic Delayed Arguments

We construct a delay functional d on an open subset of the space $$C^1_r=C^1([-r,0],\mathbb {R})$$ C r 1 = C 1 ( [ - r , 0 ] , R ) and find $$h\in (0,r)$$ h ∈ ( 0 , r ) so that the equation $$\begin{aligned} x'(t)=-x(t-d(x_t)) \end{aligned}$$ x ′ ( t ) = - x ( t - d ( x t ) ) defines a continuous semiflow of continuously differentiable solution operators on the solution manifold $$\begin{aligned} X=\{\phi \in C^1_r:\phi '(0)=-\phi (-d(\phi ))\}, \end{aligned}$$ X = { ϕ ∈ C r 1 : ϕ ′ ( 0 ) = - ϕ ( - d ( ϕ ) ) } , and along each solution the delayed argument $$t-d(x_t)$$ t - d ( x t ) is strictly increasing, and there exists a solution whose short segments$$\begin{aligned} x_{t,short}=x(t+\cdot )\in C^2_h,\quad t\ge 0, \end{aligned}$$ x t , s h o r t = x ( t + · ) ∈ C h 2 , t ≥ 0 , are dense in an infinite-dimensional subset of the space $$C^2_h$$ C h 2 . The result supplements earlier work on complicated motion caused by state-dependent delay with oscillatory delayed arguments.

Necessary conditions for quasi-singular controls in the stochastic optimal control problem with delayed argument

Рассмотрена задача оптимального управления, математические модели которых задаются нелинейными стохастическими дифференциальными уравнениями Ито с запаздывающим аргументом и диффузными компонентами, позволяющими учитывать действующие на систему случайные возмущения непрерывной природы. В предположении выпуклости области допустимого управления получено линеаризованное необходимое условие оптимальности. Исследован квазиособый случай. Описаны общие необходимые условия оптимальности квазиособых управлений. Рассмотрены частные случаи.

Optimal Perturbations of Systems with Delayed Argument for Control of Dynamics of Infectious Diseases Based on Multicomponent Actions

In this paper, we apply optimal perturbations to control mathematical models of infectious diseases expressed as systems of nonlinear differential equations with delayed argument. We develop the method for calculation of perturbations of the initial state of a dynamical system with delayed argument producing maximal amplification in the given local norm taking into account weights of perturbation components. For the model of experimental virus infection, we construct optimal perturbation for two types of stationary states, with low or high virus load, corresponding to different variants of chronic virus infection flow.

On The Correlation between Properties of One-Dimensional Mappings of Control Functions and Chaos in a Special Type Delay Differential Equation

A differential equation of a special form, which contains two control functions f and g and one delayed argument, is analyzed. This equation has a wide application in biology for the description of dynamic processes in population, physiological, metabolic, molecular-genetic, and other applications. Specific numerical examples show the correlation between the properties of the one-dimensional mapping, which is generated by the ratio f /g, and the presence of chaotic dynamics for such equation. An empirical criterion is formulated that allows one to predict the presence of a chaotic potential for a given equation by the properties of the one-dimensional mapping f /g.

Order Management System Proposal Using Inventory Balance Equation with Non-continuous Replenishment

The purpose of this paper is to present an inventory balance model including an order-up-to replenishment policy with partial backlogging. Pictured in the model is a situation, where goods are not replenished continuously, but only at predetermined intervals. The model is described by ordinary differential equations with delayed argument because of the assumption of a time lag between ordering and delivery.A computer simulation which helps to demonstrate and verify model behaviour is utilized for the numerical solution of the model. Sales data of a real company are used as the input data.Due to the comparison of the designed model outputs against the real state in the company, it was verified that it is possible to achieve a substantial reduction in warehousing costs without a disproportionate increase in the risk of inventory shortage. The authors note that modern methods of functional analysis can be successfully applied in solving an inventory balance model.

Chaos and hyperchaos in simple gene network with negative feedback and time delays

Today there are examples that prove the existence of chaotic dynamics at all levels of organization of living systems, except intracellular, although such a possibility has been theoretically predicted. The lack of experimental evidence of chaos generation at the intracellular level in vivo may indicate that during evolution the cell got rid of chaos. This work allows the hypothesis that one of the possible mechanisms for avoiding chaos in gene networks can be a negative evolutionary selection, which prevents fixation or realization of regulatory circuits, creating too mild, from the biological point of view, conditions for the emergence of chaos. It has been shown that one of such circuits may be a combination of negative autoregulation of expression of transcription factors at the level of their synthesis and degradation. The presence of such a circuit results in formation of multiple branches of chaotic solutions as well as formation of hyperchaos with equal and sufficiently low values of the delayed argument that can be implemented not only in eukaryotic, but in prokaryotic cells.