The lead article for this issue of EJAM is Prof. Bernard J. Matkowsky's personalized survey-style article on the theory and application of singular perturbation methods to noisy dynamical systems in the limit of small noise. This article is based on his John von Neumann Prize lecture presented at the Society of Industrial and Applied Mathematics (SIAM) Annual Meeting in July 2017. The John von Neumann Lecture is awarded by SIAM for outstanding and distinguished contributions to the field of applied mathematical sciences and for the effective communication of these ideas to the community. From 1990–1996, Matkowsky was an inaugural editorial board member for EJAM.

Biofilms are colonies of microorganisms, usually growing on solid-liquid interfaces, consisting of cells and a matrix of extracellular polymeric substances (EPS). Such colonies are often elaborately structured and highly dynamic, expanding through cell division and recruitment of cells from outside, and contracting via individual cells or flocs (groups of cells and biofilm matrix) detachment from the biofilm surface. Even amongst simplest single species bacterial biofilms, the behaviour (phenotype) of individual cells is highly heterogenous across the biofilm due to microenvironment variation (e.g. nutrient concentration, pH) and cell-cell signalling (quorum sensing); consequently, many researchers consider biofilms as more akin to multi-cellular organisms rather than a colony of autonomous individual cells.

We first investigate spectral properties of the Neumann–Poincaré (NP) operator for the Lamé system of elasto-statics. We show that the elasto-static NP operator can be symmetrized in the same way as that for the Laplace operator. We then show that even if elasto-static NP operator is not compact even on smooth domains, it is polynomially compact and its spectrum on two-dimensional smooth domains consists of eigenvalues that accumulate to two different points determined by the Lamé constants. We then derive explicitly eigenvalues and eigenfunctions on discs and ellipses. Using these resonances occurring at eigenvalues is considered. We also show on ellipses that cloaking by anomalous localized resonance takes place at accumulation points of eigenvalues.

In this paper, we propose an integral equation approach for pricing an American-style Parisian up-and-out call option under the Black–Scholes framework. The main difficulty of pricing this option lies in the determination of its optimal exercise price, which is a three-dimensional surface, instead of a two-dimensional (2-D) curve as is the case for a “one-touch” barrier option. In our approach, we first reduce the 3-D pricing problem to a 2-D one by using the “moving window” technique developed by Zhu and Chen (2013, Pricing Parisian and Parasian options analytically. Journal of Economic Dynamics and Control, 37(4): 875-896), then apply the Fourier sine transform to the 2-D problem to obtain two coupled integral equations in terms of two unknown quantities: the option price at the asset barrier and the optimal exercise price. Once the integral equations are solved numerically by using an iterative procedure, the calculation of the option price and the hedging parameters is straightforward from their integral representations. Our approach is validated by a comparison between our results and those of the trusted finite difference method. Numerical results are also provided to show some interesting features of the prices of American-style Parisian up-and-out call options and the behaviour of the associated optimal exercise boundaries.

We study a problem related to the spin-coating process in which a fluid coats a rotating surface. Our interest lies in the contact-line region for which we propose a simplified travelling wave approximation. We construct solutions to this problem by a shooting method that matches solution branches in the contact-line region and in the interior of the droplet. Furthermore, we prove uniqueness and qualitative properties of the solution connected to the fourth-order nature of the equation, such as a global maximum in the film height close to the contact line, elevated from the average height of the film.

This issue is dedicated to nonsmooth dynamics, particularly the widening applications where dynamics is modelled by nonsmooth differential equations. The modeling of electrical or mechanical switches as nonsmooth can be traced throughout the (at least) 90-year history in which nondifferentiable terms, such as sign or step functions, have been turning up in differential equations. In recent decades, nonsmooth models have found increasing use in areas like contact mechanics, climate modeling, and the life sciences, among others, with a wealth of new theory and novel dynamical phenomena discovered along the way. Our aim here is to give just a partial snapshot of the current landscape of research topics in the field. We open with Paul Glendinning's article extending a classic phenomenon of nonlinear dynamics — a form of chaos introduced by Leonid Pavlovich Shilnikov — to nonsmooth systems. The scenario introduces a fundamental notion of nonsmooth dyamics, that of trajectories (in this case the crucial homoclinic orbit) that can ‘slide’ along a discontinuity. Shilnikov's scenario is moreover shown to occur naturally as an equilibrium hits a discontinuity, helping with a fundamental yet complex problem, namely that of extending the notion of boundary equilibrium bifurcations beyond systems of two variables.

In the 1960s, L.P. Shilnikov showed that certain homoclinic orbits for smooth families of differential equations imply the existence of chaos, and there are complicated sequences of bifurcations near the parameter value at which the homoclinic orbit exists. We describe how this analysis is modified if the differential equations are piecewise smooth and the homoclinic orbit has a sliding segment. Moreover, we show that the Shilnikov mechanism appears naturally in the unfolding of boundary equilibrium bifurcations in $\mathbb{R}^3$.

We revisit the well-known chemostat model, considering that bacteria can be attached together in aggregates or flocs. We distinguish explicitly free and attached compartments in the model and give sufficient conditions for coexistence of these two forms. We then study the case of fast attachment and detachment and show how it is related to density-dependent growth functions. Finally, we give some insights concerning the cases of multi-specific flocs and different removal rates.

The high Reynolds-number structure of the laminar, chaotic and turbulent attractors is investigated in a two-dimensional Kolmogorov flow. The laminar attractors include the families of multi-phased travelling waves and quasi-periodic standing waves both of which form the backbone of the transition to a turbulent flow. At leading order, each laminar attractor under study is obtained by solving the Euler equations on a manifold subject to the appropriate periodicity and symmetry conditions. The manifold is determined by a finite number of vorticity equations, these being required to suppress the secular terms at the next order. Our results show that, for the multi-phased travelling waves, the first phase velocity can be determined by an integral conservation law for kinetic energy and the subsequent phase velocities can be evaluated by a non-linear eigenvalue problem. The results also reveal that whereas viscosity determines the smallest scales and controls the amplitude of the flow, the inertial terms govern the shape and form of the flow. The comparison of our analytical predictions for evaluating the stable single-phased travelling wave with the direct numerical simulation of the Navier–Stokes equations has been undertaken, the agreement being excellent. For sufficiently high Reynolds number, after the bifurcation to chaotic flow, all of the multi-phased travelling waves and quasi-periodic standing waves become unstable non-wandering sets. Based on the above new findings for these unstable non-wandering sets and other travelling and standing waves of this kind in phase space, necessary conditions for the invariant manifolds of the chaotic and turbulent attractors are obtained, these necessary conditions being conjectured to be also sufficient.

In the present article, we show that there exists a critical number that stabilizes the Reissner–Mindlin–Timoshenko system with frictional dissipation acting on rotation angles. We identify two speed characteristics v12:=K/ρ1 and v22:=D/ρ2, and we show that the system is exponentially stable if and only if

\begin{equation*} v_{1}^{2}=v_{2}^{2}. \end{equation*}

Consider a deterministic dynamical system in a domain containing a stable equilibrium, e.g., a particle in a potential well. The particle, independent of initial conditions, eventually reaches the bottom of the well. If however, the particle is subjected to white noise, due, e.g., to collisions with a population of smaller, lighter particles comprising the medium through which the particle travels, a dramatic difference in the behaviour of the Brownian particle occurs. The particle will exit the well. The natural questions then are how long will it take for it to exit and from where on the boundary of the domain of attraction of the deterministic equilibrium (the rim of the well) will it exit. We compute the mean first passage time to the boundary and the mean probabilities of the exit positions. When the noise is small each quantity satisfies a singularly perturbed deterministic boundary value problem. We treat the problem by the method of matched asymptotic expansions (MAE) and generalizations thereof. MAE has been used successfully to solve problems in many applications. However, there exist problems for which MAE does not suffice. Among these are problems exhibiting boundary layer resonance, i.e., the problem of ‘spurious solutions’, which led some to conclude that this was ‘the failure of MAE’. We present a physical argument and four mathematical arguments to modify or augment MAE to make it successful. Finally, we discuss applications of the theory.

In this paper, we present numerical simulations that demonstrate the effect of the particular choice of the equation of state (EoS) relating the surfactant concentration to the surface tension in surfactant-driven thin liquid films. Previous choices of the model EoS have been an ad-hoc decreasing function. Here, we instead propose an empirically motivated EoS; this provides a route to resolve some discrepancies and raises new issues to be pursued in future experiments. In addition, we test the influence of the choice of initial conditions and values for the non-dimensional groups. We demonstrate that the choice of EoS improves the agreement in surfactant distribution morphology between simulations and experiments, and influences the dynamics of the simulations. Because an empirically motivated EoS has regions with distinct gradients, future mathematical models may be improved by considering more than one timescale. We observe that the non-dimensional number controlling the relative importance of gravitational versus capillary forces has a larger influence on the dynamics than the other non-dimensional groups, but is nonetheless not a likely cause of discrepancy between simulations and experiments. Finally, we observe that the experimental approach using a ring to contain the surfactant could affect the surfactant and fluid dynamics if it disrupts the intended initial surfactant distribution. However, the fluid meniscus itself does not significantly affect the dynamics.

In industrialized nations, potable water is often provided through sophisticated water distribution systems. If pathogenic bacteria are introduced into the water distribution network, the presence of a biofilm can lead to biofilm-assisted retention of the pathogens, affecting the potability of the water. To study the dynamics of planktonic and biofilm-bound pathogens within the large network of pipes in a water distribution system, we develop a network model governing the concentration of introduced pathogens within the bulk fluid and the biofilms lining the pipes. Under time-constant flow regimes within the network, we prove that the long-time behaviour of the entire network is dependent on the Lyapunov exponents for each connection in the network when viewed in isolation and the network connectivity. An efficient algorithm is developed for predicting the long-time behaviour of the pathogen population within large networks using the network's topological ordering. The algorithm's predictions are validated using numerical simulations of the full non-linear system on a range of water distribution network sizes.

We present a detailed derivation and analysis of a model consisting of seven coupled delay differential equations for louse-borne relapsing fever (LBRF), a disease transmitted from human to human by the body louse Pediculus humanus humanus. Delays model the latency stages of LBRF in humans and lice, which vary in duration from individual to individual, and are therefore modelled using distributed delays with relatively general kernels. A particular feature of the transmission of LBRF to a human is that it involves the death of the louse, usually by crushing which has the effect of releasing the infected body fluids of the dead louse onto the hosts skin. Careful attention is paid to this aspect. We obtain results on existence, positivity, boundedness, linear and nonlinear stability, and persistence. We also derive a basic reproduction number R0 for the model and discuss its dependence on the model parameters. Our analysis of the model suggests that effective louse control without crushing should be the best strategy for LBRF eradication. We conclude that simple measures and precautions should, in general, be sufficient to facilitate disease eradication.

On the seventh line of the fifth page of the article the Backward Kolmogorov equation should read pt = Lp, and not ps = Lp.

The author apologises for the error.

We investigate a degenerate parabolic variational inequality arising from optimal continuous exercise perpetual executive stock options. It is also shown in Qin et al. (Continuous-Exercise Model for American Call Options with Hedging Constraints, working paper, available at SSRN: http://dx.doi.org/10.2139/ssrn.2757541) that to make this problem non-trivial the stock's growth rate must be no smaller than the discount rate. Well-posedness of the problem is established in Lai et al. (2015, Mathematical analysis of a variational inequality modeling perpetual executive stock options, Euro. J. Appl. Math., 26 (2015), 193–213), Qin et al. (2015, Regularity free boundary arising from optimal continuous exercise perpetual executive stock options, Interfaces and Free Boundaries, 17 (2015), 69–92), Song & Yu (2011, A parabolic variational inequality related to the perpetual American executive stock options, Nonlinear Analysis, 74 (2011), 6583-6600) for the case when the underlying stock's expected return rate is smaller than the discount rate. In this paper, we consider the remaining case: the discount rate is bigger than the growth rate but no bigger than the return rate. The existence of a unique classical solution as well as a continuous and strictly decreasing free boundary is proved.

The dynamics of moving solids with unilateral contacts are often modelled by assuming rigidity, point contacts, and Coulomb friction. The canonical example of a rigid rod with one endpoint slipping in two dimensions along a fixed surface (sometimes referred to as Painlevé rod) has been investigated thoroughly by many authors. The generic transitions of that system include three classical transitions (slip-stick, slip reversal, and liftoff) as well as a singularity called dynamic jamming, i.e., convergence to a codimension 2 manifold in state space, where rigid body theory breaks down. The goal of this paper is to identify similar singularities arising in systems with multiple point contacts, and in a broader setting to make initial steps towards a comprehensive list of generic transitions from slip motion to other types of dynamics. We show that – in addition to the classical transitions – dynamic jamming remains a generic phenomenon. We also find new forms of singularity and solution indeterminacy, as well as generic routes from sliding to self-excited microscopic or macroscopic oscillations.

This paper is concerned with the invisibility cloaking in acoustic wave scattering from a new perspective. We are especially interested in achieving the invisibility cloaking by completely regular and isotropic mediums. It is shown that an interior transmission eigenvalue problem arises in our study, which is the one considered theoretically in Cakoni et al. (Transmission eigenvalues for inhomogeneous media containing obstacles, Inverse Problems and Imaging, 6 (2012), 373–398). Based on such an observation, we propose a cloaking scheme that takes a three-layer structure including a cloaked region, a lossy layer and a cloaking shell. The target medium in the cloaked region can be arbitrary but regular, whereas the mediums in the lossy layer and the cloaking shell are both regular and isotropic. We establish that if a certain non-transparency condition is satisfied, then there exists an infinite set of incident waves such that the cloaking device is nearly invisible under the corresponding wave interrogation. The set of waves is generated from the Herglotz approximation of the associated interior transmission eigenfunctions. We provide both theoretical and numerical justifications.

Biofilms, and collections of embedded microbial communities, present structural heterogeneities with functional consequences for important processes, such as transport. The origin of such structures has been unclear. Here, we propose that they can arise as a consequence of diffusive transport limitation. To illustrate, a model allowing internal heterogeneity is developed. Linear analysis is applied to a simplified version of the model suggesting that heterogeneity forms on (or below) the active layer length, a length scale that may not be suitable for homogenization, with non-trivial implications for system scale properties such as reduction in system-wide diffusive transport efficiency. Numerics suggest that the simplified model provides useful insight into behaviour of the full model. We then show examples based on microcolony formation in host domains and argue that internal heterogeneity can be related to community function.

On a sufficiently soft substrate, a resting fluid droplet will cause significant deformation of the substrate. This deformation is driven by a combination of capillary forces at the contact line and the fluid pressure at the solid surface. These forces are balanced at the surface by the solid traction stress induced by the substrate deformation. Young's Law, which predicts the equilibrium contact angle of the droplet, also indicates an a priori radial force balance for rigid substrates, but not necessarily for soft substrates that deform under loading. It remains an open question whether the contact line transmits a non-zero force tangent to the substrate surface in addition to the conventional normal (vertical) force. We present an analytic Fourier transform solution technique that includes general interfacial energy conditions, which govern the contact angle of a 2D droplet. This includes evaluating the effect of gravity on the droplet shape in order to determine the correct fluid pressure at the substrate surface for larger droplets. Importantly, we find that in order to avoid a strain singularity at the contact line under a non-zero tangential contact line force, it is necessary to include a previously neglected horizontal traction boundary condition. To quantify the effects of the contact line and identify key quantities that will be experimentally accessible for testing the model, we evaluate solutions for the substrate surface displacement field as a function of Poisson's ratio and zero/non-zero tangential contact line forces.