Table of contents

Foreword

MURPHYS-HSFS-2014 was the 7th International Workshop on MUlti-Rate Processes & HYSteresis (MURPHYS) in conjunction with the 2nd International Workshop on Hysteresis and Slow-Fast Systems (HSFS). It took place at the Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin, Germany, from April 7 to April 11 in 2014.

The international workshop on "Multi-Rate Processes & Hysteresis" continued a series of biennial conferences (Cork, Ireland, 2002-2008; Pecs, Hungary, 2010; Suceava, Romania, 2012) and the international workshop on "Hysteresis and Slow-Fast Systems" was the follow-up of the HSFS-workshop that had taken place in Lutherstadt Wittenberg, Germany, in 2011.

More then 60 scientists from nine European countries and from the USA participated in MURPHYS-HSFS-2014. The program of the workshop featured 49 talks, including 15 main lectures and 15 invited talks. Recent mathematical results for systems with hysteresis operators, multiple scale systems, rate-independent systems, systems with energetic solutions, singularly perturbed systems, and systems with stochastic effects were presented. The considered applications included magnetization dynamics, biological systems, smart materials, networks, ferroelectric and ferroelastic hysteresis, fatigue in materials, market models with hysteresis, biomedical applications, chemical reactions, noise-induced phenomena, partially saturated soils, colloidal films and evaporation of automotive fuel droplets.

Statement of Peer Review: 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.

International steering committee: E. Benoit (France), M. Brokate (Germany), R. Cross (UK), K. Dahmen (USA), M. Dimian (Romania), M. Eleuteri (Italy), G. Friedman (USA), P. Gurevich (Germany), A. Ivanyi (Hungary), L. Kalachev (USA), O. Klein (Germany), D. Knees (Germany), P. Krejcl (Czech Republic), A. Neishtadt (UK), R. O'Malley (USA), N. Popovic (UK), D. Rachinskii (Ireland/USA), S. Sazhin (England), A. Shchepakina (Russia), V. Sobolev (Russia), S. Tikhomirov (Germany), C. Visone (Italy)

International scientific program committee: S. Alonso (Germany), A. Amann (Ireland), P. Andrei (USA), Z. Balanov (USA), M. Brpns (Denmark), D. Flockerzi (Germany), V. Goldshtein (Israel), R. Iyer (USA), A. Korobeinikov (Spain), I. Mayergoyz (USA), A. Mielke (Germany), P. O'Kane (Ireland), J. Sprekels (Germany), A. Stancu (Romania), M. Thomas (Germany), A. Visintin (Italy)

Organization committee: O. Klein (WIAS Berlin, Germany), M. Dimian (Stefan cel Mare University, Suceava, Romania), P. Gurevich (Free University Berlin, Germany), D. Knees (WIAS Berlin/ University of Kassel, Germany), D. Rachinskii (University College Cork, Ireland/ University of Texas, Dallas, USA), S. Tikhomirov (Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany).

Financial support: The workshop was supported by the DFG Research Center Matheon "Mathematics for key technologies", by the DFG Collaborative Research Center 910: "Control of self-organizing nonlinear systems: Theoretical methods and concepts of application" and by the Weierstrass Institute for Applied Analysis and Stochastics, Berlin.

Weierstrass Institute for Applied Analysis and Stochastics

DFG Research Center MATHEON

Mathematics for key technologies

Collaborative Research Center 910: Control of self-organizing nonlinear systems: Theoretical methods and concepts of application

Stefan cel Mare University of Suceava 1

O. Klein, M. Dimian, P. Gurevich, D. Knees, D. Rachinskii, S. Tikhomirov

Guest Editors for Journal of Physics: Conference Series

¹Some logos are omitted because of licensing issues.

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.

It is well known that the capillary pressure curve of partially saturated soils exhibits a hysteresis. For the same degree of saturation it has different values depending on the initial state of the soil, thus for drying of a wet soil or wetting of a dry soil. The influence of these different values of the capillary pressure on the propagation of sound waves is studied by use of a linear hyperbolic model. Even if the model does not contain a hysteresis operator, the effect of hysteresis in the capillary pressure curve is accounted for. In order to obtain the limits of phase speeds and attenuations for the two processes the correspondent values for main drying and main wetting are inserted into the model separately. This is done for two examples of soils, namely for Del Monte sand and for a silt loam both filled by an air-water mixture. The wave analysis reveals four waves: one transversal wave and three longitudinal waves. The waves which are driven by the immiscible pore fluids are influenced by the hysteresis in the capillary pressure curve while the waves which are mainly driven by the solid are not.

Reentries are reexcitations of cardiac tissue after the passing of an excitation wave which can cause dangerous arrhythmias like tachycardia or life-threatening heart failures like fibrillation. The heart is formed by a network of cells connected by gap junctions. Under ischemic conditions some of the cells lose their connections, because gap junctions are blocked and the excitability is decreased. We model a circular region of the tissue where a fraction of connections among individual cells are removed and substituted by non-conducting material in a two-dimensional (2D) discrete model of a heterogeneous excitable medium with local kinetics based on electrophysiology. Thus, two neighbouring cells are connected (disconnected) with a probability ϕ (1 – ϕ). Such a region is assumed to be surrounded by homogeneous tissue. The circular heterogeneous area is shown to act as a source of new waves which reenter into the tissue and reexcitate the whole domain. We employ the Fenton-Karma equations to model the action potential for the local kinetics of the discrete nodes to study the statistics of the reentries in two dimensional networks with different topologies. We conclude that the probability of reentry is determined by the proximity of the fraction of disrupted connections between neighboring nodes ("cells") in the heterogeneous region to the percolation threshold.

Capillary action or Capillarity is the ability of a liquid to flow in narrow spaces without the assistance of, and in opposition to, external forces like gravity. Three effects contribute to capillary action, namely, adhesion of the liquid to the walls of the confining solid; meniscus formation; and low Reynolds number fluid flow. We investigate the dissipation of energy during one cycle of capillary action, when the liquid volume inside a capillary tube first increases and subsequently decreases while assuming quasi-static motion. The quasi-static assumption allows us to focus on the wetting phenomenon of the solid wall by the liquid and the formation of the meniscus. It is well known that the motion of a liquid on an non-ideal surface involves the expenditure of energy due to contact angle hysteresis. In this paper, we derive the equations for the menisci and the flow rules for the change of the contact angles for a liquid column in a capillary tube at a constant temperature and volume by minimizing the Helmholtz free energy using calculus of variations. We describe the numerical solution of these equations and present results from computations for the case of a capillary tube with 1 mm diameter.

A new method to model the phenomena 'bursting' and 'buffering' in neural systems is represented. Namely, a singularly perturbed nonlinear scalar differential difference equation with two delays is introduced, which is a mathematical model of a single neuron. It is shown that for suitably chosen parameters this equation has a stable periodic solution with an arbitrary prescribed number of asymptotically high impulses (spikes) on a period interval. It is also shown that the buffering phenomenon occurs in a one-dimensional chain of diffusively coupled neurons of this type: as the number of components in the chain grows in a way compatible with a decrease of the diffusion coefficient, the number of co-existing stable periodic motions increases indefinitely.

We present a simple model of experimental setup for in vitro study of drug release from drug eluting stents and drug propagation in artificial tissue samples representing blood vessels. The model is further reduced using the assumption on vastly different characteristic diffusion times in the stent coating and in the artificial tissue. The model is used to derive a relationship between the times at which the measurements have to be taken for two experimental platforms, with corresponding artificial tissue samples made of different materials with different drug diffusion coefficients, to properly compare the drug release characteristics of drug eluting stents.

Sweeping processes are a class of evolution differential inclusions arising in elastoplasticity and were introduced by J.J. Moreau in the early seventies. The solution operator of the sweeping processes represents a relevant example of rate independent operator. As a particular case we get the so called play operator, which is a typical example of a hysteresis operator. The continuity properties of these operators were studied in several works. In this note we address the continuity with respect to the strict metric in the space of functions of bounded variation with values in the metric space of closed convex subsets of a Hilbert space. We provide counterexamples showing that for all BV-formulations of the sweeping process the corresponding solution operator is not continuous when its domain is endowed with the strict topology of BV and its codomain is endowed with the L¹-topology. This is at variance with the play operator which has a BV-extension that is continuous in this case.

A mathematical or computational model in evolutionary biology should necessary combine several comparatively fast processes, which actually drive natural selection and evolution, with a very slow process of evolution. As a result, several very different time scales are simultaneously present in the model; this makes its analytical study an extremely difficult task. However, the significant difference of the time scales implies the existence of a possibility of the model order reduction through a process of time separation. In this paper we conduct the procedure of model order reduction for a reasonably simple model of RNA virus evolution reducing the original system of three integro-partial derivative equations to a single equation. Computations confirm that there is a good fit between the results for the original and reduced models.

A steady state (or equilibrium point) of a dynamical system is hyperbolic if the Jacobian at the steady state has no eigenvalues with zero real parts. In this case, the linearized system does qualitatively capture the dynamics in a small neighborhood of the hyperbolic steady state. However, one is often forced to consider non-hyperbolic steady states, for example in the context of bifurcation theory. A geometric technique to desingularize non-hyperbolic points is the blow-up method. The classical case of the method is motivated by desingularization techniques arising in algebraic geometry. The idea is to blow up the steady state to a sphere or a cylinder. In the blown-up space, one is then often able to gain additional hyperbolicity at steady states. The method has also turned out to be a key tool to desingularize multiple time scale dynamical systems with singularities. In this paper, we discuss an explicit example of the blow-up method where we replace the sphere in the blow-up by hyperbolic space. It is shown that the calculations work in the hyperbolic space case as for the spherical case. This approach may be even slightly more convenient if one wants to work with directional charts. Hence, it is demonstrated that the sphere should be viewed as an auxiliary object in the blow-up construction. Other smooth manifolds are also natural candidates to be inserted at steady states. Furthermore, we conjecture several problems where replacing the sphere could be particularly useful, i.e., in the context of singularities of geometric flows, for avoiding compactification, and generating 'interior' steady states.

This note deals with the analysis of a model for partial damage, where the rate- independent, unidirectional flow rule for the damage variable is coupled with the rate-dependent heat equation, and with the momentum balance featuring inertia and viscosity according to Kelvin-Voigt rheology. The results presented here combine the approach from Roubicek [1, 2] with the methods from Lazzaroni/Rossi/Thomas/Toader [3]. The present analysis encompasses, differently from [2], the monotonicity in time of damage and the dependence of the viscous tensor on damage and temperature, and, unlike [3], a nonconstant heat capacity and a time-dependent Dirichlet loading.

Several mechanical systems are modeled by the static momentum balance for the displacement u coupled with a rate-independent flow rule for some internal variable z. We consider a class of abstract systems of ODEs which have the same structure, albeit in a finite-dimensional setting, and regularize both the static equation and the rate-independent flow rule by adding viscous dissipation terms with coefficients ε^α and ε, where 0 < ε « 1 and α > 0 is a fixed parameter. Therefore for α ≠ 1 u and z have different relaxation rates.

We address the vanishing-viscosity analysis as ε ↓ 0 of the viscous system. We prove that, up to a subsequence, (reparameterized) viscous solutions converge to a parameterized curve yielding a Balanced Viscosity solution to the original rate-independent system, and providing an accurate description of the system behavior at jumps. We also give a reformulation of the notion of Balanced Viscosity solution in terms of a system of subdifferential inclusions, showing that the viscosity in u and the one in z are involved in the jump dynamics in different ways, according to whether α > 1, α =1, and α є (0,1).

In this work we discuss the further development of the general scheme of the asymptotic method of differential inequalities to investigate stability and motion of sharp internal layers (fronts) for nonlinear singularly perturbed parabolic equations, which are called in applications reaction-diffusion-advection equations. Our approach is illustrated for some new important cases of initial boundary value problems. We present results on stability and on the motion of the fronts.

We construct examples of robust homoclinic orbits for systems of ordinary differential equations coupled with the Preisach hysteresis operator. Existence of such orbits is demonstrated for the first time. We discuss a generic mechanism that creates robust homoclinic orbits and a method for finding them. An example of a homoclinic orbit in a population dynamics model with hysteretic response of the prey to variations of the predator is studied numerically.

We derive quantitative error estimates for coupled reaction-diffusion systems, whose coefficient functions are quasi-periodically oscillating modeling the microstructure of the underlying macroscopic domain. The coupling arises via nonlinear reaction terms and we allow for different diffusion length scales, i.e. whereas some species have characteristic diffusion length of order 1 other species may diffuse with the order of the characteristic microstructure-length scale. We consider an effective system, which is rigorously obtained via two-scale convergence, and we derive quantitative error estimates.

Frictional hysteresis at relative motion in the pre-sliding range is considered. This effect is characterized by an elasto-plastic interaction, and that on the micro-scale, between two rubbing surfaces in contact that gives rise to nonlinear friction force. The pre-sliding friction force yields hysteresis in displacement. In this study, the damping characteristics of frictional hysteresis are analyzed. It is worth noting that we exclude the viscous damping mechanisms and focus on the pure hysteresis damping to be accounted in the friction modeling. The general properties of pre-sliding friction hysteresis are demonstrated and then compared with the limit case of discontinuous Coulomb friction. Further we consider two advanced dynamic friction models, LuGre and Maxwell-slip, so as demonstrate their damping properties and convergence of the motion system to equilibrium state. Experimental observations of the free motion in pre-sliding range are also shown and discussed.

New mathematical tools and approximations developed for the analysis of automotive fuel droplet heating and evaporation are summarised. The approach to modelling biodiesel fuel droplets is based on the application of the Discrete Component Model (DCM), while the approach to modelling Diesel fuel droplets is based on the application of the recently developed multi-dimensional quasi-discrete model. In both cases, the models are applied in combination with the Effective Thermal Conductivity/Effective Diffusivity model and the implementation in the numerical code of the analytical solutions to heat transfer and species diffusion equations inside droplets. It is shown that the approximation of biodiesel fuel by a single component leads to under-prediction of droplet evaporation time by up to 13% which can be acceptable as a crude approximation in some applications. The composition of Diesel fuel was simplified and reduced to only 98 components. The approximation of 98 components of Diesel fuel with 15 quasi-components/components leads to under-prediction of droplet evaporation time by about 3% which is acceptable in most engineering applications. At the same time, the approximation of Diesel fuel by a single component and 20 alkane components leads to a decrease in the evaporation time by about 19%, compared with the case of approximation of Diesel fuel with 98 components. The approximation of Diesel fuel with a single alkane quasi-component (C_14.763H_31.526) leads to under-prediction of the evaporation time by about 35% which is not acceptable even for qualitative analysis of the process. In the case when n-dodecane is chosen as the single alkane component, the above-mentioned under-prediction increases to about 44%.

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.

We consider singularly perturbed differential systems in cases where the standard theory to establish a slow integral manifold existence does not work. The theory has traditionally dealt only with perturbation problems near normally hyperbolic manifold of singularities and this manifold is supposed to isolated. Applying transformations 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.

We find chimera states with respect to amplitude dynamics in a network of Stuart- Landau oscillators. These partially coherent and partially incoherent spatio-temporal patterns appear due to the interplay of nonlocal network topology and symmetry-breaking coupling. As the coupling range is increased, the oscillations are quenched, amplitude chimeras disappear and the network enters a symmetry-breaking stationary state. This particular regime is a novel pattern which we call chimera death. It is characterized by the coexistence of spatially coherent and incoherent inhomogeneous steady states and therefore combines the features of chimera state and oscillation death. Additionally, we show two different transition scenarios from amplitude chimera to chimera death. Moreover, for amplitude chimeras we uncover the mechanism of transition towards in-phase synchronized regime and discuss the role of initial conditions.