Vol. 5, No. 3 (2016): Regular Issue

An unsteady flow of rheological complex fluid with fractal structure in various pipe geometries was studied when the pressure is a harmonic function of time at a given frequency at the initial cross section of the pipe. An integral type rheological equation by fractional derivative is applied for the first time to model the thixotropic rheological type oils with a high content of wax, resins and asphaltene. The obtained results show that, the relaxation time and the fractal parameters can significantly influence the process of damping pressure along the pipe. It is shown that depending on these parameters the attenuation process may increase and decrease compared to the processes in a viscous damping fluid pressure. The results of simulations are compared to those for resin-asphalt-paraffin oils of two types.

An analytical approach to homoclinic bifurcations at a saddle fixed point is developed in this paper based on high-order, high-accuracy approximations of the stable periodic orbit created at a supercritical Hopf bifurcation of a neighboring fixed point. This orbit then expands as the Hopf bifurcation parameter(s) is(are) varied beyond the bifurcation value, with the analytical criterion proposed for homoclinic bifurcation being the merging of the periodic orbit with the neighboring saddle. Thus, our approach is applicable in any situation where the homoclinic bifurcation at any saddle fixed point of a dynamical system is associated with the birth or death of a periodic orbit. We apply our criterion to two systems here. Using approximations of the stable, post-Hopf periodic orbits to first, second, and third orders in a multiple-scales perturbation expansion, we find that, for both systems, our proposed analytical criterion indeed reproduces the numerically-obtained parameter values at the onset of homoclinic bifurcation very closely.

In this work, we establish a Freidlin-Wentzell type large deviation principle for a diffusive predator-prey modelwith hyperbolic mortality rate perturbed by multiplicative type Gaussian noise. We implement the variational representation developed by Budhiraja and Dupuis to establish the large deviation principle for the solution processes.

A one-layer and two-layer zero-dimensional (0D) energy balance models (EBMs) of the global climate system with different approximations for parametrization of outgoing longwave radiation (OLR) are considered. Three alternative approximations for parametrizing the OLR are explored in detail: (i) the (conventional) linear approximation, (ii) the quadratic approximation, and (iii) the ‘exact’ (power 4) model. In case of one-layer 0D EBM, exact analytical solutions are derived in closed form for all three alternative approximations for parametrizing the OLR. In the numerical examples provided, the deviations of the linear approximation from the ‘exact’ model are visible, while the quadratic approximation is virtually indistinguishable from the ‘exact’ model.

Capital dynamics are calculated for (i) the AK model with output reduced by climate damages, (ii) the AK model with climate-dependent depreciation rate, and (iii) the Solow–Swan model with output given either by the Cobb–Douglas production function or by the constant elasticity of substitution (CES) production function and reduced by climate damages. The climate projections used as model inputs are exogenous. Simple analytical parametrisations for temperature dynamics are assumed (either linear or exponential temperature growth). The quadratic and the Nordhaus climate damage functions are considered. Exact analytical solutions for capital dynamics are derived in closed form (with the exception of the Solow–Swan model with CES production function). Numerical examples are provided for illustrative purposes. As the unabated climate change with unlimited temperature growth is assumed, the long-runmodel dynamics are dramatic: the capital converges to zero at infinite time, and the economy collapses.

For a symmetric strictly non-Volterra quadratic stochastic operator on the three-dimensional simplex it is proved that this operator has a unique fixed point. A sufficient condition of attractiveness for the unique fixed point is found. For such operators we describe the set of ω− limit points. We proved that some classes of such operators have infinitely many periodic points. Also it is shown that there are trajectories which are asymptotically cyclic with period two.

The article is devoted to the proof of the existence of solutions of a system of integro-differential equations appearing in the case of anomalous diffusion when the negative Laplacian is raised to some fractional power. The argument relies on a fixed point technique. Solvability conditions for elliptic operators without Fredholm property in unbounded domains along with the Sobolev inequality for a fractional Laplace operator are being used.

In this paper, sufficient conditions are given to investigate the existence of mild solutions on a semi-infinite interval for first order semi linear impulsive neutral functional differential evolution inclusions with infinite delay using a recently developed nonlinear alternative for contractivemultivalued maps in Frechet spaces due to Frigon combined with semigroup theory. The existence result has been proved without assumption of compactness of the semigroup. We study a new phase space for impulsive system with infinite delay.

As Physics did in previous centuries, there is currently a common dream of extracting generic laws of nature in economics, sociology, neuroscience, by focalising the description of phenomena to a minimal set of variables and parameters, linked together by causal equations of evolution whose structure may reveal hidden principles. This requires a huge reduction of dimensionality (number of degrees of freedom) and a change in the level of description. Beyond the mere necessity of developing accurate techniques affording this reduction, there is the question of the correspondence between the initial system and the reduced one. In this paper, we offer a perspective towards a common framework for discussing and understanding multi-level systems exhibiting structures at various spatial and temporal levels. We propose a common foundation and illustrate it with examples from different fields. We also point out the difficulties in constructing such a general setting and its limitations.

A transient free surface flow past a two-dimensional semi-infinite flat plate in the fluid of a finite depth is considered in the linear approximation. It is assumed that the fluid is inviscid and incompressible and the flow is irrotational. The plate is suddenly submerged at relatively small depth below the free surface into the fluid uniformly moving with a constant velocity. The linearized problem is solved for relatively small Froude numbers F < 1 using the Laplace and Fourier transforms, as well as the Wiener– Hopf technique. It is shown that eventually at large time, the transient solution approaches asymptotically the steady-state solution. Peculiarities of the solution obtained are discussed and illustrated graphically.

Two parallel plates with shear moving velocity in opposite direction is introduced as external excitations to induce cavitating flow between them, and a developed scheme based on Lattice Boltzmann method is used to simulate and analyze the evolution of the cavitation or phase transition. First, the principles and simulation process of Lattice Boltzmann method and potential models for single component multiphase flow are introduced, including a special model to the moving boundary conditions. Then, the numerical simulations of evolution of phase transition, induced by shear motions of two parallel plates, are carried out in detail, and the complicated pattern formation of cavitating flows are analyzed in such micro- and multiphase dynamic system and some new results are obtained. In particular, the influences of main parameters, such as initial density and moving velocity, on the cavitation and flow pattern are studied further. The results show that the shear moving motion of two parallel plates could induce the cavitation, and the cavitation and cavitating flow pattern could be controlled availably and efficiently by the main parameters listed above. Further, the method and analysis could be extended to flowing liquid, and an idea of drag reduction utilizing the cavitation due to phase transition in such liquid is proposed.