Using distributed contacts in DEM

Granular flows in shear cells have been extensively studied using the Discrete Element Method (DEM) over the last two decades. These studies have typically been performed using the soft-sphere approach where deformation is assumed elastic and small relative to the characteristic grain scale. Consequently internal stresses and strains are not able to be modelled. As a first step towards addressing these limitations, we introduce a variant of DEM, the Distributed Contact DEM (DCDEM). This method models distributed normal and frictional contacts. In this initial implementation plastic deformation is not simulated and elastic deformation is simulated by permitting overlap as in traditional DEM. The method is compared against traditional DEM for a normal and oblique impact and a granular shear cell in the small deformation limit.     

Vibrational resonance in biological nonlinear maps

We investigate vibrational resonance in two different nonlinear maps driven by a biharmonic force: the Bellows and the Rulkov map. These two maps possess dynamical features of particular interest for the study of these phenomena. In both maps, the resonance occurs at the low-frequency of the biharmonic signal as the amplitude of the high-frequency signal is varied. We also consider an array of unidirectionally coupled maps with the forcing signal applied to the first unit. In this case, a signal propagation with several interesting features above a critical value of the coupling strength is found, while the response amplitude of the ith unit is greater than the first one. This response evolves in a sigmoidal fashion with the system number i, meaning that at some point the amplitudes saturate. The unidirectional coupling acts as a low-pass filter for distant units. Moreover, the analysis of the mean residence time of the trajectory in a given region of the phase space unveils a multiresonance mechanism in the coupled map system. These results point at the relevance of the discrete-time models for the study of resonance phenomena, since analyses and simulations are much easier than for continuous-time models.     

Oscillations in multi-stable monotone systems with slowly varying feedback

The study of dynamics of gene regulatory networks is of increasing interest in systems biology. A useful approach to the study of these complex systems is to view them as decomposed into feedback loops around open loop monotone systems. Key features of the dynamics of the original system are then deduced from the input-output characteristics of the open loop system and the sign of the feedback. This paper extends these results, showing how to use the same framework of input-output systems in order to prove existence of oscillations, if the slowly varying strength of the feedback depends on the state of the system.     

Harvesting a logistic population in a slowly varying environment

The classic problem for a logistically evolving single species population being harvested involves three parameters: rate constant, carrying capacity and harvesting rate, which are taken to be positive constants. However, in real world situations, these parameters may vary with time. This paper considers the situation where these vary on a time scale much longer than that intrinsic to the population evolution itself. Application of a multiple time scale approach gives approximate explicit closed form expressions for the changing population, that compare favorably with those generated from numerical solutions.     

Localized vibration in elastic structures with slowly varying thickness

High frequency localized vibration modes for linear isotropicelastic plates and rods of slowly varying thickness are shownto occur within the vicinity of maximal or minimal cross-sections.Long wave asymptotic integration close to the eigenmodes ofsuch cross-sections is provided. The cases of a one integerparameter of thickness resonances (for a plate in plane stress)and two integer parameters (for a rod of circular cross-section)are fully investigated. One-dimensional Hermite-type equationsfor the eigenmodes are derived and associated frequency shiftsfrom those of the thickness resonances of the maximal or minimalcross-section obtained. It is established that the existenceof this type of localization phenomena is strongly dependenton the sign of the group velocity of the associated uniformthickness structure this being the same as that of the localgroup velocity calculated at the maximal or minimal cross-section.It is confirmed that positive group velocity allows localizationfor elevated structures with negative group velocity allowinglocalization only for ligaments. In each case intervals of thePoisson's ratio providing localization are given for the firstfew modes.     

Fast-slow dynamics in Logistic models with slowly varying parameters

Fast-slow behaviors in the Logistic models with slowly varying parameters are revealed by using singular perturbation method. We first rewrite the Logistic models with slowly varying parameters in the form of singularly perturbed systems and separate their fast and slow limits. Then we apply matching to obtain the approximate solutions, which are explicit and analytical and compare very well with the numerically integrated ones. More importantly, we prove the uniform validity of the approximate solutions rigorously and give the error estimate between the approximate solutions and the exact solutions via the way of upper and lower solutions.     

High-order sliding modes in control systems

The article considers the properties of a new type of solutions that arise in discontinuous dynamic systems. A specific feature of these solutions is the tangency of the phase velocity vectors to the manifold of the right-hand side discontinuities, instead of the transversal intersection of the manifold typically observed for ordinary sliding modes. The solutions identified in this case are high-order sliding modes, and the order of the mode is determined by the smoothness of tangency of the sliding manifold. Second-order sliding modes are considered in detail. Examples of systems with such modes are given; application of the theory to stabilization of uncertain dynamic systems is described. It is shown that the sensitivity of high-order sliding modes to small variations in the right-hand side of the discontinuous system is an order of magnitude higher than for ordinary sliding modes.Translated from Nelineinye Dinamicheskie Sistemy: Kachestvennyi Analiz i Upravlenie Sbornik Trudov, No. 2, pp. 39–70, 1993.     

Vibrational frequency shifts in the stretched polypropylene molecule

The GF matrix method is used to compute the changes in the skeletal vibration frequencies for a simplified model of isotactic polypropylene stretched by a loadf. The series of frequencies obtained atf=0 are identified with those experimentally observed. A linear approximation is used for the deformation potential energy atf=0–2·10^–4 dyneperchain. The results of the calculations are presented in graph form.A. F. Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, Vol. 4, No. 4, pp. 586–594, July–August, 1968.     

Vibrational resonance in a time-delayed genetic toggle switch

Biological oscillators can respond in a surprising way when they are perturbed by two external periodic forcing signals of very different frequencies. The response of the system to a low-frequency signal can be enhanced or depressed when a high-frequency signal is acting. This is what is known as vibrational resonance (VR). Here we study this phenomenon in a simple time-delayed genetic toggle switch, which is a synthetic gene-regulatory network. We have found out how the low-frequency signal changes the range of the response, while the high-frequency signal influences the amplitude at which the resonance occurs. The delay of the toggle switch has also a strong effect on the resonance since it can also induce autonomous oscillations.     

Vibrational resonance in neuron populations with hybrid synapses

In this paper, vibrational resonance in excitable neuron populations with synapses is investigated by numerical simulation. In particular, the effect of the hybrid synapses on the signal detection and transmission in neural system is studied. Different topologies from regular and random networks to small-world networks are considered to analyze the dependence of vibrational resonance on the network structure and parameters. It is shown that there exists an optimal amplitude of high-frequency driving, enhancing the response of coupled neuron populations to a subthreshold signal. We find that chemical synaptic coupling is more efficient than the electrical coupling in signal detection and electrical synaptic coupling is better in signal transmission. Neuron populations with hybrid synapses compromise the merits of the two types of coupling and have an advantage in information communication.     

Peak congestion in multi-server service systems with slowly varying arrival rates

In this paper we consider the M _t queueing model with infinitely many servers and a nonhomogeneous Poisson arrival process. Our goal is to obtain useful insights and formulas for nonstationary finite-server systems that commonly arise in practice. Here we are primarily concerned with the peak congestion. For the infinite-server model, we focus on the maximum value of the mean number of busy servers and the time lag between when this maximum occurs and the time that the maximum arrival rate occurs. We describe the asymptotic behavior of these quantities as the arrival changes more slowly, obtaining refinements of previous simple approximations. In addition to providing improved approximations, these refinements indicate when the simple approximations should perform well. We obtain an approximate time-dependent distribution for the number of customers in service in associated finite-server models by using the modified-offered-load (MOL) approximation, which is the finite-server steady-state distribution with the infinite-server mean serving as the offered load. We compare the value and lag in peak congestion predicted by the MOL approximation with exact values for M _t/M/s delay models with sinusoidal arrival-rate functions obtained by numerically solving the Chapman–Kolmogorov forward equations. The MOL approximation is remarkably accurate when the delay probability is suitably small. To treat systems with slowly varying arrival rates, we suggest focusing on the form of the arrival-rate function near its peak, in particular, on its second and third derivatives at the peak. We suggest estimating these derivatives from data by fitting a quadratic or cubic polynomial in a suitable interval about the peak. This revised version was published online in June 2006 with corrections to the Cover Date.     

Bursting mechanism in a time-delayed oscillator with slowly varying external forcing

This paper investigates the generation of complex bursting patterns in the Duffing oscillator with time-delayed feedback. We present the bursting patterns, including symmetric fold–fold bursting and symmetric Hopf–Hopf bursting when periodic forcing changes slowly. We make an analysis of the system bifurcations and dynamics as a function of the delayed feedback and the periodic forcing. We calculate the conditions of fold bifurcation and Hopf bifurcation as well as its stability related to external forcing and delay. We also identify two regimes of bursting depending on the magnitude of the delay itself and the strength of time delayed coupling in the model. Our results show that the dynamics of bursters in delayed system are quite different from those in systems without any delay. In particular, delay can be used as a tuning parameter to modulate dynamics of bursting corresponding to the different type. Furthermore, we use transformed phase space analysis to explore the evolution details of the delayed bursting behavior. Also some numerical simulations are included to illustrate the validity of our study.     

Survival to extinction in a slowly varying harvested logistic population model

This work considers a harvested logistic population for which birth rate, carrying capacity and harvesting rate all vary slowly with time. Asymptotic results from earlier work, obtained using a multiscaling technique, are combined to construct approximate expressions for the evolving population for the situation where the population initially survives to a slowly varying limiting state, but then, due to increasing harvesting, is reduced to extinction in finite time. These results are shown to give very good agreement with those obtained from numerical computation.     

The cyclic sliding operation in Garside groups

We present a new operation to be performed on elements in a Garside group, called cyclic sliding, which is introduced to replace the well known cycling and decycling operations. Cyclic sliding appears to be a more natural choice, simplifying the algorithms concerning conjugacy in Garside groups and having nicer theoretical properties. We show, in particular, that if a super summit element has conjugates which are rigid (that is, which have a certain particularly simple structure), then the optimal way of obtaining such a rigid conjugate through conjugation by positive elements is given by iterated cyclic sliding.