Table of contents

The current issue of Journal of Physics: Conference Series offers to your attention a selection of papers presented at the joint 8th International Workshop on MUlti-Rate Processes and HYSteresis (MURPHYS) and 3rd International Workshop on Hysteresis and Slow-Fast Systems (HSFS) that was hosted by the Centre de Recerca Matemàtica (www.crm.cat), Barcelona in June 13 to 17 of 2016. This meeting, MURPHYS-HSFS-2016, continued a successful series of biennial multidisciplinary conferences on Multi-Rate Processes and Hysteresis, that previously took place in Cork (Ireland, 2002 to 2008), Pécs (Hungary, 2010), Suceava (Romania, 2012) and Berlin (Germany, 2014), as well as the series of workshops on Hysteresis and Slow-Fast Systems in Lutherstadt, Wittenberg and Berlin.

MURPHYS-HSFS-2016 workshop brought together over 30 researchers working on hysteresis and multi-scale phenomena from Europe, Russia and US. Participants shared and discussed recent developments of analytical techniques in several areas of common interest. Topics of this volume include analysis of hysteresis phenomena, multiple scale systems, self-organizing nonlinear systems, singular perturbations and critical phenomena, as well as applications of the hysteresis and the theory of singularly perturbed systems to fluid dynamics, chemical kinetics, cancer modeling, population modeling, mathematical economics and control. More information about the Workshop can be found at:

http://www.crm.cat/en/Activities/Curs_2015-2016/Pages/MURPHYS.aspx

All papers published in this volume of Journal of Physics: Conference Series have been peer reviewed through processes administered by the Editors. Reviews were conducted by expert referees to the professional and scientific standards expected of a proceedings journal published by IOP Publishing.

We are happy to acknowledge support of the Workshop by AGAUR of the Generalitat de Catalunya, CERCA Programme of the Generalitat de Catalunya, the Centre de Recerca Matemàtica, Collaborative Research Center 910: Control of Self-organizing Nonlinear Systems (Germany), the Samara National Research University (Russia) and Drexel University (USA).

Pavel Gurevich, Andrei Korobeinikov, Dmitrii Rachinskii and Vladimir Sobolev, Editors

All papers published in this volume of Journal of Physics: Conference Series have been peer reviewed through processes administered by the proceedings Editors. Reviews were conducted by expert referees to the professional and scientific standards expected of a proceedings journal published by IOP Publishing.

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.

In the paper, a detailed study of the critical phenomena mechanism in a model of an electrocatalytic reaction is carried out. The use of the geometric theory of integral manifolds allows us to show the relationship between the critical regime and the stability loss delay of the slow integral manifold of the corresponding differential system. The asymptotic formula for the critical value of the control parameter is obtained.

We prove that the differential equation xẍ + 1 = 0 has continuous weak periodic solutions and compute their periods. Then, we use the Harmonic Balance Method until order six to approximate these periods and to illustrate how the accuracy of the method increases with the order. Our computations rely on the Gröbner basis approach.

It can be expected that adequate immune response should be able to annihilate cancer at a very early stage of its appearance. However, in some cases cancer is able to persist avoiding immune response. One can conject that cancer is able to avoid immune response control due to a succession of mutations leading to the development of immune-resistant cells. In order to illustrate this possibility, in this paper we present a 2n–dimensional mathematical model that describes interaction of n subtypes of tumor cells with corresponding genotype-specific immune response. The model postulates that there is a probability for tumor cells of each of n subtype to produce offsprings of other types. Each of the subtypes activates the genotype-specific immune response with a possibility of suppressing cancer cells of other genotypes (the cross-immunity). Numerical simulations show that if cancer cells are able to mutate comparatively fast and if immune response is not strong enough, then, despite immune system pressure, cancer is able to persist.

It was suggested that the ability of cancer to avoid immune response pressure (that should be expected to be capable to annihilate cancer at its early stage) can be attributed to the ability of the cancer cells to evolve. The goal of this notice is to illustrate this possibility by the means of mathematical modelling. In this notice, we construct a simple mechanistic model of cancer evolution, which is based upon a classical model of cancer-immune response interaction. Numerical simulations confirm the hypothesis that if cancer mutates fast enough and if immune response is not sufficiently strong, then cancer is able to avoid immune response pressure by evolution.

We consider problems such as a standing wave in a closed straight tube, a self-sustained oscillation, damped resonance, evolution of resonance and resonance between concentric spheres. These nonlinear problems, and other similar ones, have been solved by a variety of techniques when it is seen that linear theory fails.

The unifying approach given here is to initially set up the appropriate linear difference equation, where the difference is the linear travel time. When the linear travel time is replaced by a corrected nonlinear travel time, the nonlinear difference equation yields the required solution.

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 Re₃. 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 < Re₃ < 2500.

This is an extended variant of the paper presented at MURPHYS-HSFS 2016 conference in Barcelona. We discuss further development of the asymptotic method of differential inequalities to investigate existence and stability of sharp internal layers (fronts) for nonlinear singularly perturbed periodic parabolic problems and initial boundary value problems with blow-up of fronts for reaction-diffusion-advection equations. In particular, we consider periodic solutions with internal layer in the case of balanced reaction. For the initial boundary value problems we prove the existence of fronts and give their asymptotic approximation including the new case of blowing-up fronts. This case we illustrate by the generalised Burgers equation.

We explore the possibility of applying the method of order reduction of optimal estimation problem for singularly perturbed systems with low measurement noise. It is shown that matrix Riccati equation for the Kalman-Bucy filter has a periodic solution. An optimal filter is constructed for a dynamic model of a crank mechanism.

A typical mechanistic model of viral evolution necessary includes several time scales which can differ by orders of magnitude. Such a diversity of time scales makes analysis of these models difficult. Reducing the order of a model is highly desirable when handling such a model. A typical approach applied to such slow-fast (or singularly perturbed) systems is the time scales separation technique. Constructing the so-called quasi-steady-state approximation is the usual first step in applying the technique. While this technique is commonly applied, in some cases its straightforward application can lead to unsatisfactory results.

In this paper we construct the quasi-steady-state approximation for a model of evolution of marine bacteriophages based on the Beretta-Kuang model. We show that for this particular model the quasi-steady-state approximation is able to produce only qualitative but not quantitative fit.

This paper subscribes to the view that a key distinguishing feature of a firm is its social nature. We present a model in which hysteresis arises from the social interactions between employees. Employees have a simple response to incentives in the form of the pay available outside the firm relative to that available within the firm. Allowing for social interaction, whereby employees are influenced by the effort levels of fellow employees, leads to the distinctive effects, such as lazy relay responses to incentives, associated with hysteresis.

In the social sciences there are plausible reasons to postulate that hysteresis effects are important. The available evidence, however, is predominantly at the macro level. In this paper we review the evidence regarding hysteresis in the neural processes underlying human behavior. We argue that there is a need for experimental and neuroimaging studies to fill the gap in knowledge about hysteresis processes at the micro level in the social sciences.

Prandtl-Ishlinskii stop-type hysteresis operators allow for modeling elasto-plasticity in the relative stress-strain coordinates including the saturation level of the residual constant-tension flow. This lies in direct equivalence to the force-displacement characteristics of nonlinear Coulomb friction, whose constant average value at unidirectional motion depends on the motion sign only, after the transient presliding phase at each motion reversal. In this work, we analyze and demonstrate the use of Prandtl-Ishlinskii operators for modeling the Coulomb friction with presliding phase. No viscous i.e. velocity-dependent component is considered at this stage, and the constant damping rate of the Coulomb friction is combined with the rate-independent losses of presliding hysteresis. The general case of Prandtl-Ishlinskii operator with a continuous distribution function is considered together with a finite elements case, which is useful for implementation in multiple applications. Finally, identification of parameters is addressed and discussed along with two experimental examples.

The main ideas of simulation of two-phase flows, based on a combination of the conventional Lagrangian or fully Lagrangian (Osiptsov) approaches for the dispersed phase and the mesh-free vortex and thermal-blob methods for the carrier phase, are summarised. In the approach based on a combination of the fully Lagrangian approach for the dispersed phase and the vortex blob methods for the carrier phase the problem of calculation of all parameters in both phases (including particle concentration) is reduced to the solution of a high-order system of ordinary differential equations, describing transient processes in both carrier and dispersed phases. It contrast to this approach, in the approach based on a combination of the conventional Lagrangian approaches for the dispersed phase and the vortex and thermal-blob methods for the carrier phase the non-isothermal effects in the two-phase flow were taken into account. The one-way coupled, two-fluid approach was used in the analysis. The gas velocity field was restored using the Biot-Savart integral. Both these approaches were applied to modelling of two processes: the time evolution of a two-phase Lamb vortex and the development of an impulse two-phase jet. Various flow patterns were obtained in the calculations, depending on the initial droplet size.

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.

The paper deals with the problem of a construction of global stable/unstable slow integral manifolds of the singularly perturbed systems in critical cases. In addition to the well-known critical cases a novel scenario of the stability change of the slow integral manifold is considered. All three critical cases leading to the change of the stability are discussed via the Hindmarsh-Rose dynamic model. It is shown that the suitable choice of the additional parameters of the system yields the slow integral manifold with multiple change of its stability.

The aim of the paper is to describe the special critical case in the theory of singularly perturbed optimal control problems. 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 an example of control problem.

We consider a simple model of a passive dynamic biped robot walker with point feet and legs without knee. The model is a switched system, which includes an inverted double pendulum. Robot's gait and its stability depend on parameters such as the slope of the ramp, the length of robot's legs, and the mass distribution along the legs. We present an asymptotic solution of the model. The first correction to the zero order approximation is shown to agree with the numerical solution for a limited parameter range.