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A wide range of dynamic models, including those of heating, evaporation and ignition processes in fuel sprays, is characterised by large differences in the rates of change of variables. Invariant manifold theory is an effective technique for investigation of these systems. In constructing the asymptotic expansions of slow invariant manifolds, it is commonly assumed that a limiting algebraic equation allows one to find a slow surface explicitly. This is not always possible due to the fact that the degenerate equation for this surface (small parameter equal to zero) is either a high degree polynomial or transcendental. In many problems, however, the slow surface can be described in a parametric form. In this case, the slow invariant manifold can be found in parametric form using asymptotic expansions. If this is not possible, it is necessary to use an implicit presentation of the slow surface and obtain asymptotic representations for the slow invariant manifold in an implicit form. The results of development of the mathematical theory of these approaches and the applications of this theory to some examples related to modelling combustion processes, including those in sprays, are presented.

Models of the spread of infectious diseases commonly have to deal with the problem of multiple timescales which naturally occur in the epidemic models. In the most cases, this problem is implicitly avoided with the use of the so-called constant population size assumption. However, applicability of this assumption can require a justification (which is typically omitted). In this paper we consider some multiscale phenomena that arise in a reasonably simple Susceptible- Infected-Removed (SIR) model with variable population size. In particular, we discuss examples of the canard cascades and a black swan that arise in this model. © 2020 the Author(s).

The paper is devoted to the application of methods for constructing invariant integrals of fast-slow systems to the problems of "nonholonomic mechanics".

The method of integral manifolds is used to study singularly perturbed systems of differential equations. The algorithms for the construction of the slow invariant manifolds in the case with different dimensions of the fast and slow variables was derived.

The method of decomposition of singularly perturbed differential systems is developed and application of the method to the problems of non-holonomic mechanics is considered.

We study an initial–boundary value problem for a singularly perturbed system of partial integro-differential equations. We prove a theorem on the passage to the limit. The result is used to decrease the dimension of a virus evolution model. We construct an asymptotic solution by the Tikhonov–Vasil’eva boundary function method. The analytic results obtained are compared with a numerical study of the system.

We consider a simple model of a passive dynamic biped robot with point feet and legs without knee. The model is a switched system, which includes an inverted double pendulum. We present an asymptotic solution of the model. The first correction to the zero order approximation is shown to agree with the numerical solution with high degree of accuracy for a limited parameter range.

We explore the possibility of applying the method of order reduction of the optimal estimation problem with low measurement noise for singularly perturbed systems. It is shown that matrix differential Riccati equation for the Kalman–Bucy filter has a periodic solution which may be used instead the solution of the original initial value problem for the matrix Riccati equation.

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.

The possibility of applying the method of integral manifolds to the reduction of optimal filtering problems for systems with low energy dissipation is explored. For such systems, it is shown that the slow subsystem of matrix Riccati differential equations turns out to have a higher dimension than expected, which leads to an increase in the dimension of the reduced problems. An optimal filter is constructed for the Langevin equation and for a dynamic model of a single-link flexible manipulator.

The problem of the decomposition of singularly perturbed differential systems by the method of integral manifolds is studied and the application of the method to the problems of enzyme kinetics is considered.

We use the method of integral manifolds to reduce optimal control and filtering problems in application to mechanical systems with low friction.

The mathematical model of the differential cascade with a symmetrical division of stem cells is considered in the paper. The using of the RouthHurwitz’s criterion and the Andronov-Hopf theorem [1] allows to find the conditions of bifurcation cycle in the model under consideration, and the phenomenon of "soft loss of stability" was stated.

The aim of the paper is to describe the main critical cases in the theory of singularly perturbed optimal control problems and to give examples which are typical for slow/fast systems. The theory has traditionally dealt only with perturbation problems near normally hyperbolic manifold of singularities and this manifold is supposed to isolated. We reduce the original singularly perturbed problem to a regularized one such that the existence of slow integral manifolds can be established by means of the standard theory. We illustrate our approach by several examples of control problems.

Questions of history, the theory, applications and tradition of use the slow integral manifolds in graduate educational programs are considered in given article with reference to universities of Samara.

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.

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.

The paper is devoted to the investigation of the slow integral manifolds of variable stability. The existence of non periodic canards, canard cascades and black swans is stated. The theoretical developments are illustrated by several examples. L'article est consacré à l'étude de variétés lentes intégrales de stabilité variable. L'existence de canards non périodiques, de cascades de canard et des cygnes noirs est établie. Les développements théoriques sont illustrés par plusieurs exemples.

The aim of the paper is to give examples of apparent disappearance phenomenon which is typical for microbiological systems. It is well known that canards play an important role in the theory of mixed−mode oscillations consisting of large amplitude, and followed by small amplitude oscillations. In a sense we consider limiting cases of mixed−mode oscillations, when large amplitude oscillations are followed by vanishingly small amplitude oscillations.

The basic elements of the theory of slow invariant manifolds and canard phenomena of singularly perturbed nonlinear differential equations in the context of thermal-explosion problems are outlined. The mathematical results are applied to the investigation of the critical phenomena in autocatalytic combustion models described by singularly perturbed differential equations with lumped and distributed parameters. Critical regimes are modeled by canards (one-dimensional stable-unstable slow invariant manifolds). The geometric approach in combination with asymptotic and numeric methods permits to explain the strong parametric sensitivity and to obtain asymptotic representations of the critical conditions of self-ignition.

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.

The work is devoted to the investigation of the integral manifolds of the nonautonomous slow–fast systems, which change their attractivity in time. The method used here is based on gluing attracting and repelling integral manifolds by using an additional control function.

A concept of stability of quasi-static paths is discussed in this paper that takes into account the existence of fast (dynamic) and slow (quasi-static) time scales in the evolution of many mechanical systems. The proposed concept is essentially a continuity property with respect to the smallness of the initial perturbations (as in Lyapunov stability) and the smallness of the quasi-static loading rate (that plays the role of the small parameter in singular perturbation problems). A related concept of attractiveness is also proposed. Several examples illustrate the relevance of the definitions. Sufficient conditions for attractiveness or for instability of quasi-static paths of smooth systems are proved. Copyright © 2005 John Wiley & Sons, Ltd.

In this paper we discuss some mathematical issues related to the stability of quasistatic paths of finite dimensional mechanical systems that have a smooth or an elastic-plastic behavior. The concept of stability of quasi-static paths used here is essentially a continuity property relatively to the size of the initial perturbations (as in Lyapunov stability) and to the smallness of the rate of application of the external forces (which here plays the role of the small parameter in singular perturbation problems). A related concept of attractiveness is also proposed. Sufficient conditions for attractiveness or for instability of quasi-static paths of smooth systems are presented. The Ziegler column and other examples illustrate these situations. Mathematical formulations (plus existence and uniqueness results) for dynamic and quasi-static elastic-plastic problems with linear hardening are recalled. A stability result is proved for the quasi-static evolution of these systems.

We study the existence of combustion waves for an autocatalytic reaction in the non-adiabatic case. Based on the fact that the reaction system has canard solutions separating the slow combustion regime from the explosive one, we prove by applying the geometric theory of singularly perturbed differential equations the existence of a new type of travelling waves solutions, the so-called canard travelling waves. Copyright © 2003 John Wiley & Sons, Ltd.

The aim of the paper is to describe the special critical case in the theory of singularly perturbed optimal control problems and to give an example which is typical for slow/fast systems. The theory has traditionally dealt only with perturbation problems near normally hyperbolic manifold of singularities and this manifold is supposed to be isolated. We reduce the original singularly perturbed problem to a regularized one such that the existence of slow integral manifolds can be established by means of the standard theory.

In many applications of the system order reduction models, including those focused on spray ignition and combustion processes, it is assumed that all functions in corresponding differential equations are Lipschitzian. This assumption has not been checked in most cases and the cases when these functions were non-Lipschitzian have sometimes been overlooked. This allows us to question the results of application of the conventional theory of integral manifolds to some such systems. The aim of this paper is to demonstrate that even in the case of singular perturbed systems with non-Lipschitzian nonlinearities the order reduction can be performed, using a new concept of positively invariant manifolds. This is illustrated by several examples including the problem of heating, evaporation, ignition and combustion of Diesel fuel sprays.

In constructing the asymptotic expansions of slow integral manifolds it is assumed that the degenerate equation (\(\varepsilon = 0\)) allows one to find the slow surface explicitly. In many problems this is not possible due to the fact that the degenerate equation is either a high degree polynomial or transcendental. In this situation many authors suggest the use of numerical methods. However, in many problems the slow surface can be described in parametric form, and then the slow integral manifold can be found in parametric form as asymptotic expansions. If this is not possible, it is necessary to use an implicit slow surface and obtain asymptotic representations for the slow integral manifold in an implicit form. Model examples, as well as examples borrowed from combustion theory, are treated.

In this chapter the first number in the title of a section denotes the dimension of the slow variable, the second one denotes the dimension of the fast variable. A series of examples, of increasing complexity, are given to illustrate the theoretical concepts. The main examples come from applications in enzyme kinetics. These examples illustrate the effectiveness of the order reduction method.

In the present chapter we use a method for the qualitative asymptotic analysis of singularly perturbed differential equations by reducing the order of the differential system under consideration. The method relies on the theory of integral manifolds. It essentially replaces the original system by another system on an integral manifold with a lower dimension that is equal to that of the slow subsystem. The emphasis in this chapter is on the study of autonomous systems.

In this chapter we consider singularly perturbed differential systems whose degenerate equations have an isolated but not simple solution. In that case, the standard theory to establish a slow integral manifold near this solution does not work. Applying scaling transformations and using the technique of gauge functions we reduce the original singularly perturbed problem to a regularized one such that the existence of slow integral manifolds can be established by means of the standard theory. We illustrate the method by several examples from control theory and chemical kinetics.

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.

The paper presents a geometric approach to a wide class of control problems with singular perturbation. It is shown that the boundary layer functions gor deterministic linear-quadratic control problems and for optimal linear estimation problems can be neglected. In the case of nonlinear optimal control problem the right boundary layer functions in solution of nonlinear boundary value problem can also be neglected. The consideration of some degenerate control problems requires the study of slow manifolds branching. In such cases we use the asymptotic expansions of slow manifolds with fractional exponents of small parameter.

In this paper we present sufficient conditions for stability of quasi-static paths of finite dimensional systems that have a smooth behavior. The concept of stability of quasi-static paths used here is essentially a continuity property relatively to the size of the initial perturbations (as in Lyapunov stability) and to the smallness of the rate of application of the external forces (which plays here the role of the small parameter in singular perturbation problems). These conditions are applied to a mechanical system with a convex potential energy.

Functional identity and significant similarities in cofactors and sequence exist between the L and M reaction center proteins of the photosynthetic bacteria and the D1 and D2 photosystem-II reaction center proteins of cyanobacteria, algae, and plants. A model of the quinone (QB) binding site of the D1 protein is presented based upon the resolved structure of the QB binding pocket of the L subunit, and introducing novel quantitative notions of complementarity and contact surface between atoms. This model, built without using traditional methods of molecular mechanics and restricted to residues in direct contact with QB, accounts for the experimentally derived functional state of mutants of the D1 protein in the region of QB. It predicts the binding of both the classical and phenol-type PSII herbicides and rationalizes the relative levels of tolerance of mutant phenotypes.

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

A method for all asymptotic separation of rapid and slow motions based on the concepts of the Bogolyubov-Mitropol'skii theory of integral manifolds, is proposed for use in analysing systems of differential equations with small parameters for the derivatives which arise in the course of solving dynamic problems. The motion of a gyroscope with contactless suspension in a magnetic field is studied. The method of separating the motions enables the problem to be reduced to that of investigating a regular, finite-dimensional system of ordinary differential equations.