Homoclinic orbits for second order self-adjoint difference equations

AbstractIn this paper we discuss how to use variational methods to study the existence of nontrivial homoclinic orbits of the following nonlinear difference equationsΔ[p(t)Δu(t−1)]+q(t)u(t)=f(t,u(t)),t∈Z, without any periodicity assumptions on p(t), q(t) and f, providing that f(t,x) grows superlinearly both at origin and at infinity or is an odd function with respect to x∈R, and satisfies some additional assumptions

Dynamical Behavior of a New Epidemiological Model

A new epidemiological model is introduced with nonlinear incidence, in which the infected disease may lose infectiousness and then evolves to a chronic noninfectious disease when the infected disease has not been cured for a certain time τ. The existence, uniqueness, and stability of the disease-free equilibrium and endemic equilibrium are discussed. The basic reproductive number R0 is given. The model is studied in two cases: with and without time delay. For the model without time delay, the disease-free equilibrium is globally asymptotically stable provided that R0≤1; if R0>1, then there exists a unique endemic equilibrium, and it is globally asymptotically stable. For the model with time delay, a sufficient condition is given to ensure that the disease-free equilibrium is locally asymptotically stable. Hopf bifurcation in endemic equilibrium with respect to the time τ is also addressed

Global Stability for a Viral Infection Model with Saturated Incidence Rate

A viral infection model with saturated incidence rate and viral infection with delay is derived and analyzed; the incidence rate is assumed to be a specific nonlinear form βxv/(1+αv). The existence and uniqueness of equilibrium are proved. The basic reproductive number R0 is given. The model is divided into two cases: with or without delay. In each case, by constructing Lyapunov functionals, necessary and sufficient conditions are given to ensure the global stability of the models

Comparison of a Material Point Method and a Galerkin meshfree method for the simulation of cohesive-frictional materials

The simulation of large deformation problems, involving complex history-dependent constitutive laws, is of paramount importance in several engineering fields. Particular attention has to be paid to the choice of a suitable numerical technique such that reliable results can be obtained. In this paper, a Material Point Method (MPM) and a Galerkin Meshfree Method (GMM) are presented and verified against classical benchmarks in solid mechanics. The aim is to demonstrate the good behavior of the methods in the simulation of cohesive-frictional materials, both in static and dynamic regimes and in problems dealing with large deformations. The vast majority of MPM techniques in the literature are based on some sort of explicit time integration. The techniques proposed in the current work, on the contrary, are based on implicit approaches, which can also be easily adapted to the simulation of static cases. The two methods are presented so as to highlight the similarities to rather than the differences fromPeer ReviewedPostprint (published version

Reversible Transition Between Thermodynamically Stable Phases with Low Density of Oxygen Vacancies on SrTiO3_3(110) Surface

The surface reconstruction of SrTiO$_3$(110) is studied with scanning tunneling microscopy and density functional theory (DFT) calculations. The reversible phase transition between (4$\times$1) and (5$\times$1) is controlled by adjusting the surface metal concentration [Sr] or [Ti]. Resolving the atomic structures of the surface, DFT calculations verify that the phase stability changes upon the chemical potential of Sr or Ti. Particularly, the density of oxygen vacancies is low on the thermodynamically stabilized SrTiO$_3$(110) surface.Comment: Accepted by Physical Review Letter

Contagion processes on the static and activity driven coupling networks

The evolution of network structure and the spreading of epidemic are common coexistent dynamical processes. In most cases, network structure is treated either static or time-varying, supposing the whole network is observed in a same time window. In this paper, we consider the epidemic spreading on a network consisting of both static and time-varying structures. At meanwhile, the time-varying part and the epidemic spreading are supposed to be of the same time scale. We introduce a static and activity driven coupling (SADC) network model to characterize the coupling between static (strong) structure and dynamic (weak) structure. Epidemic thresholds of SIS and SIR model are studied on SADC both analytically and numerically with various coupling strategies, where the strong structure is of homogeneous or heterogeneous degree distribution. Theoretical thresholds obtained from SADC model can both recover and generalize the classical results in static and time-varying networks. It is demonstrated that weak structures can make the epidemics break out much more easily in homogeneous coupling but harder in heterogeneous coupling when keeping same average degree in SADC networks. Furthermore, we show there exists a threshold ratio of the weak structure to have substantive effects on the breakout of the epidemics. This promotes our understanding of why epidemics can still break out in some social networks even we restrict the flow of the population

Numerical simulation of explosive fracturing with smoothed particle hydrodynamics

In this paper we study explosive fracturing with smoothed particle hydrodynamics (SPH). As a particle based Lagrangian method, SPH is particularly suited to the analysis of fracture due to its full Lagrangian frame and capacity to model large deformation. We adopt the Jones-Wilkins-Lee equation as equation of state of the trinitrotoluene (TNT) explosive and a continuum elasto-damage model to predict the fracture of the rock. We predict the evolution of damage using the strain history of each particle. To strengthen the interaction of coupling interfaces we use a penalty function to avoid penetration between different material particles