Vol. 5, No. 1 (2016): Regular Issue

Two nonlinear modifications of the Structural Dynamic Economic Model SDEM-2 are developed and studied analytically and numerically. In the first model version described in the present paper the production function is assumed to be nonlinear that leads to increasing returns to scale, while the second model version proposed describes endogenous technological progress by treating the technology parameter of the production function as an additional state variable. Dependent on the values of model parameters and on initial conditions, both modifications of SDEM considered demonstrate two different dynamic regimes: either an explosive economic growth or the collapse of the economy.

Optimal control problems for a class of semilinear distributed systems unsolved with respect to the times derivative are studied. Two types of initial condition for the system state and various cost functionals are considered in the problems. Abstract results are illustrated by examples of the start control problems for the quasistationary system of phase field equations.

The electric manipulation of magnetic textures in nanostructures, important for applications in spintronics, can be realized through the spin-transfer torque mechanism: a spin-polarized current can modify the magnetization of skyrmions and magnetic vortices, and eventually change the topology of the magnetization. The spin-transfer torque and the intrinsic space and time scales of the topological changes are essentially quantum mechanical. We model the interaction between itinerant and fixed spins with a simple tightbinding hamiltonian in a square lattice. The dynamics is described by the Schrödinger equation for the electrons and the Landau-Lifshitz equation for the evolution of the magnetic texture. We investigate the phenomenology of the topological change of a Belavin-Polyakov skyrmion under the action of a spin-polarized current and show that adding an exchange dissipation term, regularizes the transition towards a ferromagnetic state.

We study the equilibrium states for an energy functional with a parametric force field on a region of a surface under a constraint of geometrical character. We use an improved method, based in Skrypnik’s variational theories [10]. In local coordinates, equilibrium points satisfy an elliptic boundary value problem. This model can be described as the deformation of the elastic medium and membranes.

In this paper, by using the concept of differential equations with piecewise constant argument, the model of Hopfield neural networks with constant delay is developed. Sufficient conditions for the existence of an equilibrium as well as its global exponential stability by means of Lyapunov functionals and a linear matrix inequality (LMI) are obtained. An example is given to illustrate our results.

We present our investigation on vibrational resonance in a system with a signum nonlinearity. We construct an exact analytical solution of the system in the presence of an external biharmonic force with two frequencies ω and Ω, Ω≥ω and use it for the computation of the response amplitude Q at the low-frequency ω. We analyse the effect of the strength of the signum nonlinearity on vibrational resonance for the cases of the potential with a single-well, a double-well and a single-well with a double-hump. An interesting feature of vibrational resonance in the system is that Q does not decay to zero for g (the amplitude of the high-frequency force) → ꝏ. We compare the features of the vibrational resonance of these two systems, since the potential of the system with the signum nonlinearity and that of the Duffing oscillator show ssimilar forms. The strength of the nonlinearity in these two systems is found to give rise distinct effects on resonance.

In this paper we establish the sufficient conditions for controllability of nonlinear fractional delay integrodifferential systems. The results are obtained by using the solution representation of fractional delay differential equations and the application of Schauder’s fixed point theorem. Examples are provided to illustrate the results.

The article deals with the existence of solutions of a system of nonlocal reaction-diffusion equations which appears in population dynamics. The proof relies on a fixed point technique. Solvability conditions for elliptic operators in unbounded domains which fail to satisfy the Fredholm property are being used.

In this paper we employ three recent analytical approaches to investigate several classes of traveling wave solutions of the so-called extendedreduced Ostrovsky Equation (exROE). A recent extension of phase-plane analysis is first employed to show the existence of breaking kink wave solutions and smooth periodic wave (compacton) solutions. Next, smooth traveling waves are derived using a recent technique to derive convergent multi-infinite series solutions for the homoclinic orbits of the travelingwave equations for the exROE equation. These correspond to pulse solutions respectively of the original PDEs. We perform many numerical tests in different parameter regime to pinpoint real saddle equilibrium points of the corresponding traveling-wave equations, as well as ensure simultaneous convergence and continuity of the multi-infinite series solutions for the homoclinic orbits anchored by these saddle points. Unlike the majority of unaccelerated convergent series, high accuracy is attained with relatively few terms. And finally, variational methods are employed to generate families of both regular and embedded solitary wave solutions for the exROE PDE. The technique for obtaining the embedded solitons incorporates several recent generalizations of the usual variational technique and it is thus topical in itself. One unusual feature of the solitary waves derived here is that we are able to obtain them in analytical form (within the assumed ansatz for the trial functions). Thus, a direct error analysis is performed, showing the accuracy of the resulting solitary waves. Given the importance of solitary wave solutions in wave dynamics and information propagation in nonlinear PDEs, as well as the fact that not much is known about solutions of the family of generalized exROE equations considered here, the results obtained are both new and timely.