Vladimir Sobolev

Criterion for Thermal Explosion with Reactant Consumption in a Dusty Gas

The thermal explosion problem in a dusty gas is investigated. Dynamical regimes of the system are classified: slow regimes, thermal explosion with delay, thermal explosion (without delay). The critical transition conditions for the different dynamical regimes are analysed. We emphasize that the critical condition for transition between slow regimes and explosion with delay is a thermal explosion limit. The thermal explosion limit is described in the phase space by a so-called duck-trajectory. (The notion of duck-trajectory was introduced by Cartier for detailed investigation of relaxation oscillations.)

Topological degree in analysis of canard-type trajectories in 3-D systems

Piecewise linear systems become increasingly important across a wide range of engineering applications spurring an interest in developing new mathematical models and methods of their analysis, or adapting methods of the theory of smooth dynamical systems. One such areas is the design of controllers which support the regimes of operation described by canard trajectories of the model, including applications to engineering chemical processes, flight control, electrical circuits design, and neural networks. In this article, we present a scenario which ensures the existence of a topologically stable periodic (cyclic) canard trajectory in slow-fast systems with a piecewise linear fast component. In order to reveal the geometrical structure responsible for the existence of the canard trajectory, we focus on a simple prototype piecewise linear nonlinearity. The analysis is based on application of the topological degree.

Canard doublet in a Lotka-Volterra type model

New types of canard solutions that arise in the modified Lotka-Volterra equations are discussed.

Self-ignition of dusty media

Thermal explosions of gaseous media containing inert dust particles are examined. The critical conditions for a thermal explosion are determined using integral manifolds. An asymptotic formula for calculating the critical conditions for thermal explosions with an autocatalytic combustion reaction is obtained.

Order reduction of a non-Lipschitzian model of monodisperse spray ignition

It is pointed out that the order reduction of singular perturbed systems with non-Lipschitzian nonlinearities can be performed, using the new concept of positively (negatively) invariant manifolds, if the five assumptions of the Tikhonov theorem are satisfied. Examples when these assumptions are satisfied and not satisfied are presented and discussed. The previously derived singularly perturbed system of three equations (droplet radii, gas temperature and fuel vapour concentration), describing evaporation and ignition of monodisperse fuel sprays, are shown to satisfy all five assumptions of the Tikhonov theorem. This allows us to reduce this system to a system of two equations describing the gas temperature and fuel concentration when the droplet evaporation process has been completed.

Asymptotic expansions of slow invariant manifolds and reduction of chemical kinetics models

Methods of the geometric theory of singular perturbations are used to reduce the dimensions of problems in chemical kinetics. The methods are based on using slow invariant manifolds. As a result, the original system is replaced by one on an invariant manifold, whose dimension coincides with that of the slow subsystem. Explicit and implicit representations of slow invariant manifolds are applied. The mathematical apparatus described is used to develop N.N. Semenov’s fundamental ideas related to the method of quasi-stationary concentrations and is used to study particular problems in chemical kinetics.

Invariant surfaces of variable stability

This paper presents a brief description of the theory of invariant manifolds of variable stability in the context of their connection with the theory of solutions that are bounded on the whole axis. This approach allows various generalizations both to the case of increasing of the dimension of the invariant manifolds and to the case of multiple change of their stability. The sufficient conditions for the existence of an invariant manifold of variable stability are revealed. The continuity condition for the invariant manifold yields the analytic representation of the gluing function. The theoretical developments are illustrated by several examples.

Asymptotic expansions of solutions in a singularly perturbed model of virus evolution

An initial boundary value problem for a singularly perturbed system of partial integro-differential equations involving two small parameters multiplying the derivatives is studied. The problem arises in a virus evolution model. An asymptotic solution of the problem is constructed by the Tikhonov-Vasil’eva method of boundary functions. The analytical results are compared with numerical ones.

Decomposition of a linear-quadratic optimal control problem with fast and slow variables

A linear-quadratic optimal control problem for a discrete different time-scale system is studied. The decomposition of the boundary value problem for the maximum principle is based on the geometric approach using the properties of invariant manifolds of slow and fast motions. This approach aids in constructing a transformation for reducing the initial problem to a boundary-value problem for slow variables and two initial-value problems for fast variables. The transformation is expressed as an asymptotic siries in powers of a small parameter.

Paradox of enrichment and system order reduction: bacteriophages dynamics as case study

The paradox of enrichment in a 3D model for bacteriophage dynamics, with a free infection stage of the phage and a bilinear incident rate, is considered. An application of the technique of singular perturbation theory allows us to demonstrate why the paradox arises in this 3D model despite the fact that it has a bilinear incident rate (while in 2D predator-prey models it is usually associated with the concavity of the attack rate). Our analysis demonstrates that the commonly applied approach of the model order reduction using the so-called quasi-steady-state approximation can lead to a loss of important properties of an original system.