Vol. 5, No. 2 (2016): Regular Issue

We propose a novel approach to generate chaotic business cycles in a deterministic setting. Rather than producing chaos endogenously, we consider aggregate economic models with limit cycles and equilibriums, subject them to chaotic exogenous shocks and obtain chaotic cyclical motions. Thus, we emphasize that chaotic cycles, which are inevitable in economics, are not only interior properties of economic models, but also can be considered as a result of interaction of several economical systems. This provides a comprehension of chaos (unpredictability, lack of forecasting) and control of chaos as a global economic phenomenon from the deterministic point of view. We suppose that the results of our paper are contribution to the mixed exogenous-endogenous theories of business cycles in classification by P.A. Samuelson [1]. Moreover, they demonstrate that the irregularity of the extended chaos can be structured, and this distinguishes them from the generalized synchronization. The advantage of the knowledge of the structure is that by applying instruments, which already have been developed for deterministic chaos, one can control the chaos, emphasizing a parameter or a type of motion. For the globalization of cyclic chaos phenomenon we utilize new mechanisms such as entrainment by chaos, attraction of chaotic cycles by equilibriums and bifurcation of chaotic cycles developed in our earlier papers.

The noncommutative Bell polynomials and their dual Bell polynomials are presented, respectively, which are extensively applied to mathematics and physics. We make use of them to exhibit a method for generating integrable hierarchies of evolution equations. As applications, we obtain the Burgers hierarchy and a convection-diffusion equation which can be applied to fluid mechanics, specially, be used to represent mass transformations in fluid systems under some constrained conditions. As reduced cases, the Burgers equation which has extensive applications in physics is followed to produce. Furthermore, we obtain a set of nonlinear evolution equations with four potential functions which reduces to a new nonlinear equation similar to the Calogero-Degasperis-Fokas equation. Finally, we discrete the convection-diffusion equation and obtain its three kinds of finite-difference schemes, that is,the weighted implicit difference scheme and the Lax difference scheme. Their some properties including truncation errors, compatibilities and stabilities based on the Von Neumann condition are discussed in detail.

In this paper, a finite difference method for a non-linear reaction diffusion equation defined on a circular domain is presented. A simple second-order finite difference treatment of polar coordinate singularity for Laplacian operator, the centered difference approximations, the treatments for Neumann boundary problems are used to discretize this equation. By using this method, numerical solutions can be computed. In the end, we give two applications of reaction diffusion predator-prey models with modified Leslie-Gower and Holling type II functional responses.

We study the asymptotic behavior of the two-point integral boundary value problem for third order integro-differential equations with the small parameter at two highest derivatives. The asymptotic estimations of the solution of the integral boundary value problem is obtained. The obtained results shown that the solution of integral boundary value problem on both sides of given segment has the initial jumps with different orders.

A nonlocal boundary value problem for a system of impulsive hyperbolic equations at the fixed times is considered. The questions of existence, uniqueness, and construction of algorithms for finding the solutions to this problem are studied. By introducing the additional parameters as values of solutions on specific lines the considered problem is reduced to the problem consisting of the Goursat problem for a system of hyperbolic equations and the Cauchy problem for ordinary differential equations. The algorithms for finding the approximate solutions of latter problem are obtained and their convergence to the solution of original problem is proved. Conditions for existence of a unique solution to the nonlocal boundary value problem with impulse effects are set in the terms of initial data.

We prove the global well-posedness in H1(R2,CN) for certain systems of the critical Nonlinear Schrodinger equations coupled linearly or nonlinearly with nonlinear supercritical dissipation terms, generalizing the previous result of [1] obtained for a single equation of this kind.

In this paper we consider discrete-time dynamical systems generated by gonosomal evolution operators of sex linked inheritance. Mainly we study dynamical systems of a hemophilia, which biologically is a group of hereditary genetic disorders that impair the body’s ability to control blood clotting or coagulation, which is used to stop bleeding when a blood vessel is broken. We give an algebraic model of the biological system corresponding to the hemophilia. The evolution of such system is studied by a nonlinear (quadratic) gonosomal operator. In a general setting, this operator is considered as a mapping from Rn, n ≥ 2 to itself. In particular, for a gonosomal operator at n = 4 we explicitly give all (two) fixed points. Then limit points of the trajectories of the corresponding dynamical system are studied. Moreover we consider a normalized version of the gonosomal operator. In the case n = 4, for the normalized gonosomal operator we show uniqueness of fixed point and study limit points of the dynamical system.

Combinations of truncated Painlevé expansions, invariant Painlevé analysis, and generalized Hirota series are used to solve (’partially reduce to quadrature’) the integrable Boussinesq and the cubic and quintic generalized Pochammer-Chree (GPC) equation families. Although the multisolitons of the Boussinesq equation are very well-known, the solutions obtained here for all the three NLPDEs are novel, and non-trivial. All of the solutions obtained via invariant Painlevé analysis are complicated rational functions, with arguments which themselves are trigonometric functions of various distinct traveling wave variables. This is reminiscent of doublyperiodic elliptic function solutions when nonlinear ODE systems are reduced to quadratures. The solutions obtained using recently-generalized Hirota-type expansions are closer in functional form to conventional hyperbolic secant solutions, although with non-trivial traveling-wave arguments which are distinct for the two GPC equations.