Extended Abstracts Summer 2016

A model for the growth of lead sulphate particles in a gravity separation system from the crystal glassware industry is presented. The lead sulphate particles are an undesirable byproduct, and thus the model is used to ascertain the optimal system temperature configuration such that particle extraction is maximised. The model describes the evolution of a single, spherical particle due to the mass flux of lead particles from a surrounding acid solution. We divide the concentration field into two separate regions. Specifically, a relatively small boundary layer region around the particle is characterised by fast diffusion, and is thus considered quasi-static. In contrast, diffusion in the far-field is slower, and hence assumed to be time-dependent. The final system consisting of two nonlinear, coupled ordinary differential equations for the particle radius and lead concentration, is integrated numerically.

This note presents the results of [4]. It deals with the problem of location and existence of limit cycles for real planar polynomial differential systems. We provide a method to construct Poincaré–Bendixson regions by using transversal curves, that enables us to prove the existence of a limit cycle that has been numerically detected. We apply our results to several known systems, like the Brusselator one or some Liénard systems, to prove the existence of the limit cycles and to locate them very precisely in the phase space. Our method, combined with some other classical tools can be applied to obtain sharp bounds for the bifurcation values of a saddle-node bifurcation of limit cycles, as we do for the Rychkov system.

Various versions of Odd Number Theorem state necessary conditions for stabilizing an unstable periodic solution to a differential equation by Pyragas’ delayed feedback control. In this paper, we propose an equivariant counterpart of these conditions for systems with a finite symmetry group.

The present extended abstract summarizes our paper “A survey of the hysteretic Duhem model”, recently published in Archives of Computational Methods in Engineering.

In this paper we introduce a reasonably simple mechanistic model of aquatic bacteriophage evolution based upon the Beretta–Kuang bacteriophage dynamics model.

This paper discusses small amplitude, one-dimensional nonlinear hyperbolic waves in a medium of finite extent, and thus inherently involves the reflection of waves from the end boundaries. In each of the examples shown linear theory breaks down, either in the long term, as in a standing wave, or by the appearance of a singularity, as in resonance. Guided by experiments, see Saenger and Hudson (J Acoust Soc Am 32:961–970, 1960) [9], the fundamental hypothesis is that, for small amplitude disturbances, the interaction of these nonlinear waves is negligible in calculating the main features of the flow, such as the presence of shocks. We frame the fundamental hypothesis as: “the motion consists of non-interacting simple waves.”

The work in this paper concerns the axisymmetric pipe flow of a Herschel–Bulkley fluid, with the aim of determining a relation between the critical velocity (defining the transition between laminar and turbulent flow) and the pipe diameter in terms of the Reynolds number \(Re_3\). The asymptotic behaviour for large and small pipes is examined and simple expressions for the leading order terms are presented. Results are then compared with experimental data. A nonlinear regression analysis shows that for the tested fluids the transition occurs at similar values to the Newtonian case, namely in the range \(2100< Re_3 <2500\).

We present recent results on singularly perturbed reaction-advection-diffusion problems, which are based on a further development of the asymptotic comparison principle. We illustrate this approach applying it to new problems. We also give Theorems stating the existence of periodic solutions with an internal layer, providing their asymptotic approximation and establishing their Lyapunov stability for these problems. We discuss further development of the asymptotic method of differential inequalities (see [1‐5]) for the periodic parabolic problems and apply this method to new cases.

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.

Applying the technique of time-scale separation, we reduce a model of marine bacteriophage evolution to a system of two integro-differential equations and demonstrate the equivalence of the original and the reduced systems.

We propose an application of hysteresis models to explain non-smooth adjustments to individuals’ wage acceptance within organizations, and argue that individuals’ response to incentives can be described as a probabilistic “lazy relay”, which should have certain interesting non-linearities in aggregated settings (e.g., multi-person organizations).

The main ideas of simulation of two-phase flows, based on a combination of the conventional Lagrangian method or Osiptsov method for the dispersed phase and the mesh-free vortex and thermal blob methods for the carrier phase, are summarised. A meshless method for modelling of 2D transient, non-isothermal, gas-droplet flows with phase transitions, based on a combination of the viscous-vortex and thermal-blob methods for the carrier phase with the Lagrangian approach for the dispersed phase, is described. The one-way coupled, two-fluid approach is used in the analysis. The method makes it possible to avoid the ‘remeshing’ procedure (recalculation of flow parameters from Eulerian to Lagrangian grids) and reduces the problem to the solution of three systems of ordinary differential equations, describing the motion of viscous-vortex blobs, thermal blobs, and evaporating droplets. The gas velocity field is restored using the Biot–Savart integral. The numerical algorithm is verified against the analytical solution for a non-isothermal Lamb vortex. The method is applied to modelling of an impulse two-phase cold jet injected into a quiescent hot gas, taking into account droplet evaporation. Various flow patterns are obtained in the calculations, depending on the initial droplet size: (i) low-inertia droplets, evaporating at a higher rate, form ring-like structures and are accumulated only behind the vortex pair; (ii) large droplets move closer to the jet axis, with their sizes remaining almost unchanged; and (iii) intermediate-size droplets are accumulated in a curved band whose ends trail in the periphery behind the head of the cloud, with larger droplets being collected at the front of the two-phase region.

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.

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 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 this paper we introduce a simple mechanistic model of cancer evolution under immune response pressure and cancer evolutionary escape.

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 investigate a generalization of the nonlinear Poisson–Nernst–Planck system with respect to coupling phenomena, volume balance and positivity of species concentrations, and nonlinear interface conditions. We aim at existence, uniqueness and the Lyapunov stability of the solution. This system is motivated by applications to modeling of electro-kinetic phenomena in bio- and electro-chemistry.