A. A. Davydov

• Doctor of Science
• Chair at Vladimir State University

The paper is devoted to the theory of normal forms of main symbols for linear second order partial differential equations on the plane. We discuss the results obtained in the last decades and some problems, which are important both for the development of this theory and the applications. The reduction theorem, which was used to obtain many of recen...

We consider a periodic impulse harvesting of a renewable resource distributed in a domain of the arithmetic space and obeying the logistic growth low. We prove that, for a given harvesting effort, there exists an appropriate stationary distribution of the effort in the resource domain providing the maximum efficiency of the harvesting at infinite h...

The paper obtains existence of a solution and necessary optimality conditions for a problem of optimal (long run averaged) periodic extraction of a renewable resource distributed along a circle. The resource grows according to the logistic law, and is harvested by a single harvester periodically moving around the circle.

For a nonlinear model of the dynamics of a size-structured (exploited) population with asymmetric form of competition, we prove a uniqueness theorem for a positive stationary solution under sufficiently general assumptions on the parameters of the model.

The paper contributes to the topic of optimal utilization of spatially distributed renewable resources. Namely, a problem of “sustainable” optimal cyclic exploitation of a renewable resource with logistic law of recovery is investigated. The resource is distributed on a circle and is collected by a single harvester moving along the circle. The reco...

We consider the optimization problem that consists in maximizing the time averaged profit for a motion of a smooth polydynamical system on the circle in the presence of a smooth profit density. When the problem depends on a k-dimensional parameter the optimal averaged profit is a function of the parameter. It is known that an optimal motion can alw...

For a given exploitation intensity of size-structured population with asymmetric competition form we prove the existence of a nontrivial stationary state in the population dynamics. Bibliography: 7 titles.

In this study we present a dynamic model evaluation of the chemistry transport model LOTOS-EUROS to analyse the ability of the model to reproduce observed non-linear responses to emission changes and interannual variabiliy of secondary inorganic aerosol (SIA) and its precursors over Europe from 1990 to 2009. The 20 year simulation was performed usi...

We consider the optimization problem of maximizing the time-averaged profit for the motion of a smooth polydynamical system on the circle in the presence of a smooth profit density. If the problem depends on a k-dimensional parameter, then the optimal averaged profit is a function of the parameter. It is known from [4] that an optimal motion can al...

The prominent Russian mathematician Vladimir Mikhailovich Zakalyukin died suddenly in Moscow on 30 December 2011.

In this paper, we investigate the relationship between the production and ecological carrying capacity of shellfish cultured species (mussels, oysters, clams, etc.) and the spatial distribution of shellfish density within a licensed area. In order to achieve this goal, we used an analytical model for simulating the impact of a shellfish farm on the...

We classify generic local controllability bifurcations in two-parameter families of bidynamical systems on the plane at points with nonzero velocity indicatrix.

On the circle for smooth one-parametric families of pairs of profit densities and control systems with positive velocities only, we classify generic singularities of the profit along a cyclic motion with prescribed period. The classification includes six singularities and is different from the respective one for the general case when the period of...

For a 2D system of ordinary differential equations that gives a qualitative description of the thermohaline circulation in the ocean, we prove the existence of a limit cycle for a large class of transfer functions. We show that this cycle arises in the system as a result of the soft loss of stability of a steady state when a step transfer function...

For a smooth one-parameter family of pairs of control systems and profit densities on a circle, the generic transitions between optimal rotations and stationary strategies are studied in the problem of maximization of the time-averaged profit on the infinite horizon. It is shown that there are only two types of such transitions, the corresponding s...

Structural stability of a smooth control system in general position on a smooth orientable compact surface is proved. This result is an analogue for control systems of the theorem on structural stability of a smooth vector field in general position on the sphere.

The structural stability of families of orbits is proved for the simplest generic smooth dynamical inequality in the plane with bounded complement of the domain of complete controllability. Typical singularities of the boundaries of nonlocal transitivity zones for such inequalities are found. The stability of these singularities under small perturb...

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Using singularity theory tools we study the time averaged problem for control systems and profit densities on the circle. We show that the optimal strategy always can be selected within periodic cyclic motions and stationary strategies. When the problem depends additionally on a one dimensional parameter, we analyze generic transitions between thes...

An optimal strategy which exists among periodic rotations or stationery strategies in Arnold's model is described. The study of periodic phenomena of various natures is reduced to analyze the averaged profit of such processes. The infinite horizon averaged profit can be maximized by a level motion or a suitable stationary point for a continuous con...

Sufficient conditions for small time local transitivity and for the absence of local controllability of a generic dynamic inequality with locally bounded derivatives are presented.

Smooth orbital normal forms of generic one-parametric families of slow motion of relaxation-type equations with two-dimensional slow variable are obtained near a singular point of the type Whitney-fold of the equation folding when the slow velocity is not tangent to the set of critical values of the folding. For example, a generic family for a gene...

Generic singularities of the maximum time-averaged profit are classified for a one-parameter smooth pair of families of profit densities and control systems with positive velocities on the circle. For example, a typical passage of this profit through a local nondegenerate extremum of the profit density under the change of the parameter yields the s...

For the implicit systems of first order ordinary differential equations on the plane there is presented the complete local classification of generic singularities of family of its phase curves up to smooth orbital equivalence. Besides the well known singularities of generic vector fields on the plane and the singularities described by a generic fir...

In this paper the authors obtained some sufficient conditions under which one has the small local controllability property for a general nonlinear control system described by a system of nonlinear inequalities.

We deal with one-parametric families of optimization problems in finite dimension with a finite number of (in-)equality constraints. Its set of generalized critical points can be classified according to five types (cf. ). In this paper the corresponding normal forms of the problem near each of these five types of singular points are presented.

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In this work the smooth and topological normal forms of the first-order implicit differential equations in the plane near its folded degenerate elementary singular point are found, and thereby, the smooth and topological classification of folded elementary singular points of these equations is completed. It is proved, for example, that the equation...

The singularities of the fields of limiting directions of two-dimensional control systems in general position are classified, and it is proved that these singularities are stable under small perturbations of such systems.Bibliography: 18 titles. Bibtex entry for this abstract Preferred format for this abstract (see Preferences) Find Similar Abstrac...

Normal forms of fibrations of binomial surfaces are found. The results are applied to the study of slow motions of equations of relaxation type.Figures: 2. Bibliography: 7 titles. Bibtex entry for this abstract Preferred format for this abstract (see Preferences) Find Similar Abstracts: Use: Authors Title Abstract Text Return: Query Results Return...

We consider dynamic inequalities with locally bounded derivatives on the circle and optimize the time averaged profit along admissible motions which are defined for all nonegative time. The nature of optimal strategies is discribed.