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The perturbed mK(3,1) equation is restudied to further explore the dynamics of solitary wave solutions by combining the geometric singular perturbation theorem and bifurcation analysis in this paper. Besides the solitary waves presented in literature [1–3], we show that this equation possesses a family of solitary waves which decay to some constants determined by their wave speeds and a parameter. It is shown that a portion of the solitary wave solutions to the mK(3,1) equation will persist under small perturbations and the wave speed selection principle is presented as well. In addition to the solitary waves, each of which has only one crest or trough and approximates to a solitary wave of the unperturbed equation as the perturbation parameter tends to zero, we theoretically prove the existence of a new type of solitary waves with coexisting crest and trough. The numerical simulations are carried out, and the results are in complete agreement with our theoretical analysis.

A previously developed model for mono-component droplet evaporation is revisited using new mathematical tools for its analysis. The analysis is based on steady-state mass, momentum, and energy balance equations for the vapor and air mixture surrounding a droplet. The previously obtained solution to these equations was based on the assumption that the parameter ε (proportional to the squared ratio of the diffusion coefficient and droplet radius) is equal to zero. The analysis presented in the paper is based on the method of integral manifolds, and it allowed us to present the droplet evaporation rate as the sum of the evaporation rate predicted by the model based on the assumption that [Formula: see text] and the correction proportional to ε. The correction is shown to be particularly important in the case of small water and methanol droplets (diameters less than 5 μm) evaporating in air at low pressure (0.1 atm.). In this case, this correction could reach 35% of the original evaporation rate. In the case of evaporation of relatively large droplets (with radii more than 10 μm) in air at atmospheric and higher pressures, these corrections are shown to be small (less than [Formula: see text] of the evaporation rate predicted by the model based on the assumption that [Formula: see text]).

The perturbed mK(3,1) equation is restudied to further explore the dynamics of solitary wave solutions by combining the geometric singular perturbation theorem and bifurcation analysis in this paper. Besides the solitary waves presented in literature, we show that this equation possesses a family of solitary waves which decay to some constants determined by their wave speeds and a parameter. It is shown that a portion of the solitary wave solutions to the mK(3,1) equation will persist under small perturbations and the wave speed selection principle is presented as well. In addition to the solitary waves, each of which has only one peak or valley and approximates to a solitary wave of the unperturbed equation as the perturbation parameter tends to zero, we theoretically prove the existence of a new type of solitary waves with the coexisting peak and valley. The numerical simulations are carried out, and the results are in complete agreement with our theoretical analysis.

The paper deals with the canard traveling waves in a reaction–diffusion model. The use of the geometric theory of invariant manifolds allows us to simplify the traveling wave problem of the original PDE system. We are focused on point-to-periodic traveling waves. It is shown that the profile of such waves is a canard. Canard traveling waves are critical waves since they separate waves with qualitatively different behaviors.

The paper deals with a special type of traveling waves in a combustion model. A non–adiabatic case of autocatalytic combustion reaction in one spatial dimension is considered. The existence of critical traveling waves is stated. The critical traveling waves play the role of an intermediate form, they separate the waves of slow burnout and waves corresponding to the thermal explosion. It is shown that the profiles of the critical waves are canards. An asymptotic representation for the speed of the critical traveling wave is obtained.

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.

In most papers focused on the system order reduction models, describing processes of heating, evaporation and ignition in fuel sprays, it is assumed that all functions in corresponding differential equations are sufficiently smooth and consequently Lipschitzian. In many cases, however, these functions are non-Lipschitzian. This means that the conventional approach to system order reduction, based on the theory of integral manifolds, cannot be applied. It is pointed out that the order reduction of systems with non-Lipschitzian non-linearities can be performed, using a concept of positively invariant manifolds. This concept is discussed and applied to the analysis of spray ignition based on five ODEs (for gas temperature, fuel vapour and oxygen concentrations, and droplet temperatures and radii). This system is reduced to single ordinary differential equations for the gas temperature or fuel concentrations. It is shown that the equation for gas temperature predicts an increase in gas temperature up to its limiting value during finite time. The reaching of this temperature is accompanied by the complete depletion of either fuel vapour or oxygen depending on their initial concentrations, as follows from the analysis of the equations for gas temperature and fuel concentration.

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 deals with the critical phenomena in the dynamic model of an electrochemical reactor. We consider the case of an external resistance. We use asymptotic and geometrical approaches to obtain the conditions for the critical regime. The critical regime plays a bordering role between the slow processes and modes with self-acceleration. We show that the critical regime can be modelled by a canard. Asymptotic formulae for the canard and for the corresponding value of the control parameter of the system are obtained. The multidimensional invariant surface of variable stability (so-called black swan) is constructed. The black swan consists entirely of canards that model critical modes with various initial data. The black swan existence permits us to realize the critical regime with taking into account small perturbations in the chemical system.

A new approach to the canards chase in 3D for some class of singularly perturbed systems is suggested. The proposed approach is discussed by the use of a competitive model of population dynamics. The presence of an exact black swan (a stable/unstable slow invariant manifold) makes it possible to find a new kind of trajectories with multiple stability changes.

The effect of Gaussian white noise on a canard cycle in dynamical model of an electrochemical reaction is analyzed. A critical noise intensity, at which the small-amplitude oscillations are transformed into mixed-mode oscillations, is obtained.

Air pollution is a major concern of recent decades, which has a serious toxicological impact on human health and the environment. It has a number of different emission sources, but one of the main sources of environmental pollutions is transported systems, in particular aviation gas turbine engines (GTE). Currently environmental issues of GTE are mainly solved by using semi-empirical techniques and experimental refinement of prototypes. In this paper we presented an algorithm for simulating the emission of CO and CO2 in a model combustion chamber under various initial conditions and compared the results, validated with an experimental data.

The review covers publications concerned with propagation of laser beams through turbulent media described by the Kolmogorov theory and generalizations thereof to describe signal transmission in optical communications and detection systems. In this case, the turbulent medium is interpreted as an optical channel with random parameters. Various optical signals considered include partially coherent beams, non-uniformly polarized vector beams, as well as specifically configured spatial laser beams. Special attention is given to vortex laser beams. The latter are shown to have a number of remarkable properties that give them an advantage over conventional Gaussian beams.

The paper deals with the study of an electrocatalytic reaction mechanism underlying an electrochemical reactor. The analysis is based on the theory of integral manifolds of the singularly perturbed systems. This approach allows us to define the different types of chemical regimes including the critical mode. The relation between the critical regime and the phenomenon of delayed loss of stability in the dynamic model is shown.

The paper deals with the critical cases causing the loss of stability of slow integral manifold of singularly perturbed systems. In addition to the well-known critical cases, when the equilibrium of the fast subsystem loses its stability with the passage of one real or a pair of complex conjugate eigenvalues through the imaginary axis, we consider the case when the real parts as well as the imaginary parts of a pair of complex conjugate eigenvalues become zero simultaneously.

Some new approaches to the solution of complex problems, focused on spray modelling, using relatively simple mathematical tools are briefly summarised. The following problems are considered: modelling of spray drying with pharmaceutical applications, modelling of heating and evaporation of suspended droplets with applications to water sprays for fire suppression, modelling of micro-explosions with automotive applications, and parameterisations of slow invariant manifolds with applications to spray ignition modelling. The modelling of spray drying is based on the assumption that the array of small suspended solid particles inside droplets can be treated as a non-evaporating liquid component. The modelling of the effect of a support rod on droplet heating and evaporation is based on the assumption that this heat is homogeneously distributed inside the whole volume of a droplet. In the modelling of a puffing/micro-explosion it is assumed that a water sub-droplet is located exactly at the centre of a fuel droplet and the temperature at the surface of the fuel droplet is fixed. In the presentation of slow invariant manifolds in parametric form these manifolds are found using asymptotic expansions.

In the paper, we consider a new type of travelling waves, a profile of which is a pure slow heteroclinic canard. To construct the profile we use the so-called canard cascade. Such travelling waves are critical because they are a watershed between travelling waves of different types and simulate critical phenomena in various applied problems.

This paper deals with the modelling of oscillations in a class of population dynamics models with slow and fast variables. The approach discussed is based on the geometric method of invariant manifolds, and the techniques of canards and black swans.

The paper is devoted to the study of the dynamic model of the autocatalytic combustion reaction in an inert medium with partial heat removal from the reaction phase to the environment. We pay particular attention to modelling of the critical regime, which is a kind of a watershed between the slow burning regimes and explosion modes. New algorithm for computing a critical value of the control parameter is presented.

A detailed analysis of the dynamic model of the tuberculosis epidemic was conducted. It is shown that the dynamic model contains several time scales and can be represented in a singularly perturbed form. With help of the integral manifolds theory and the reduction principle, the reduction of the modeling system was justified and carried out. This approach allow us to replace the original system by another system of a lower order on an integral manifold whose dimension is equal to that of the slow subsystem and which, at the same time, preserves the essential properties of the original system. The conditions for the stabilization of the epidemiology based on the selection of necessary treatment and preventive measures are determined.

The paper deals with the study of the relation between the Andronov--Hopf bifurcation, the canard explosion and the critical phenomena for the van der Pol's type system of singularly perturbed differential equations. Sufficient conditions for the limit cycle birth bifurcation in the case of the singularly perturbed systems are investigated. We use the method of integral manifolds and canards techniques to obtain the conditions under which the system possesses the canard cycle. Through the application to some chemical and optical models it is shown that the canard point should be considered as the critical value of the control parameter.

The scintillation index of plane and spherical light waves as well as of a Gaussian beam, propagating in the clear-water weakly turbulent ocean, is revealed. The results are of utmost importance for underwater optical communications and sensing. An analysis of the threshold between the weak and strong regimes of oceanic turbulence is made, with the accent on the contribution from salinity-induced turbulence. It is found that strong oceanic turbulence can occur at distances as short as several meters, in striking contrast with atmospheric studies for which the typical distances are on the order of a kilometer.

The behavior of the spectral composition of a typical stochastic beam in a turbulent ocean environ is revealed. The analysis is based on the extended Huygens–Fresnel principle and the coherence theory in the space-frequency domain. The optical source is assumed to be of the Gaussian Schell-model type with a single narrow Gaussian spectral line in the visible region. Optical turbulence in the ocean is assumed to be driven by temperature and salinity fluctuations. It is found that the well-known source correlation-induced spectral shift is compensated by turbulence at sufficiently large distances.

In this paper, we show how canards can be easily caught in a class of 3D systems with an exact black swan (a slow invariant manifold of variable stability). We demonstrate this approach to a canard chase via the two predator – one prey model. It is shown that the technique described allows us to get various 3D oscillations by changing the shape of the trajectories of two 2D-projections of the original 3D system.

This paper deals with a special kind of trajectories of a singularly perturbed system which includes the so-called canard doublets. We call these trajectories the cascades of canard doublets. By use of a biological model a new approach to the canard constructing in 3D is described.

The relation between the phenomenon of delayed loss of stability in singularly perturbed differential equations and the estimation of the maximum temperature of safe combustion is investigated. Using the methods of the qualitative theory of singular perturbations and the theory of canards, we determine the maximum temperature during the transition regime between the slow combustion regimes and the thermal explosion.

The ignition of combustible gas containing liquid fuel droplets is investigated. The analysis is based on the theory of integral manifolds of singularly perturbed systems. This approach allows us to define different types of chemical regimes including the critical mode. The relation between the critical regime and the phenomenon of delayed loss of stability in the dynamical model is shown.

The paper is devoted to the study of the dynamical model of HIV. An application of the technique of singular perturbation theory allows us to calculate the conditions for the stabilization of the HIV status of the patient depending on the features that reflect the different nature of the intervention of doctors in the treatment process.

The paper deals with the study of the features of oscillatory processes in the electrocatalytic reaction mechanism underlying the electrochemical reactor. The conditions for the existence of several types of oscillations (small amplitude harmonic oscillations, relaxation oscillations and oscillations of transitional type) is obtained.

The paper deals with the specific cases of the changing of stability of slow integral manifold of singularly perturbed systems of ODE via the dynamic model of nerve conduction. It is shown that the proper choice of the additional parameters of the system allows us to construct the slow integral manifold with multiple change of its stability.

This work is concerned with the phenomenon of delayed loss of stability and the estimation of the maximal temperature of safe combustion. Using the qualitative theory of singular perturbations and canard techniques we determine the maximal temperature on the trajectories located in the transition region between the slow combustion regime and the explosive one. This approach is used to estimate the maximal temperature of safe combustion in multi–phase combustion models.

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.

The mathematical model of a 3-D autocatalator is studied using the geometric theory of singular perturbations, namely, the black swan and canard techniques. Critical regimes are modeled by canards (one-dimensional stable–unstable slow integral manifolds). The meaning of criticality here is as follows. The critical regime corresponds to a chemical reaction which separates the domain of self-accelerating reactions from the domain of slow reactions. A two-dimensional stable–unstable slow integral manifold (black swan) consisting entirely of canards, which simulate the critical phenomena for different initial data of the dynamical system, is constructed. It is shown that this procedure leads to the phenomenon of auto-oscillations in the chemical system. The geometric approach combined with asymptotic and numerical methods permits us to explain the strong parametric sensitivity and to obtain asymptotic representations of the critical behavior of the chemical system.

The paper is devoted to the investigation of thermal explosion (self-ignition) of reaction mixture situated in an inert porous medium. The analysis is based on qualitative theory of singularly perturbed systems, canards and black swans techniques.

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 paper is devoted to the modelling of the critical phenomena in multiscale combustion models. Such models are usually described by singularly perturbed systems of differential equations to reflect the significant distinction in characteristic relaxation times of different physicochemical processes. The paper proposes an approach for modelling of critical phenomena on the basis of the geometric asymptotic method of invariant manifolds. The critical phenomenon means as a sharp change in the dynamics of the process under consideration. As an illustration of this approach a dynamic model of fuel spray ignition and combustion is considered. The realizability conditions for the critical regime is obtained in the form of the asymptotic expression for the control parameter. The main feature of the critical regime is that during it the temperature of the combustible mixture can reach a high value within the framework of a safe process. It is shown that the critical regime plays the role of a watershed between the slow combustion regimes and the thermal explosion.

The autoignition of flammable liquid in an inert porous medium are studied. In this paper we concentrated on the critical case which is concerned with the phenomenon of delayed loss of stability in the dynamical model. The realizability conditions for the critical regime are obtained. It is shown that critical regime is modelled by a canard − a trajectory of slow–fast system, which first move near the stable part of the slow invariant manifold, then move near the unstable part of it.