System Of Nonlinear Differential-difference Equations Research Articles

Dynamic Model of Economic Agents Including Interaction and Delay Effects

Abstract: The dynamic model of economic growth of several agents is presented. This is a system of non-linear differential-difference equations. The model takes into account the interaction of economic agents and the delay effect. Some simple analytic particular solutions are shown. These analytical solutions allow some clear economic interpretation. The comparative scenario quantitative calculations are given. The calculations are presented for time period of a decade for the typical global development rate of 3% per year. There are two main scenarios. The first scenario of cooperative or isolated agent development is considered. There is the scenario of the agent development due to use the other partners. The consequence of isolation is the backlog of 60% per decade. The second scenario is the time delay-lead effect to use the achievements of other agents. A lead of one - two years is considered. In this case, ceteris paribus, the dynamic growth of the agent is 1.5 times ahead of the others. More effective growth occurs with the faster and advance reaction rate to partner achievements. In this case there is the effect of the multiplicative accumulation of advantage. This scenario can be interpreted as the illustration of the accelerated economic growth of post-war Japan.

DISCRETE MATURITY AND DELAY DIFFERENTIAL-DIFFERENCE MODEL OF HEMATOPOIETIC CELL DYNAMICS WITH APPLICATIONS TO ACUTE MYELOGENOUS LEUKEMIA

In the last few years, many efforts were oriented towards describing the hematopoiesis phenomenon in normal and pathological situations. This complex biological process is organized as a hierarchical system that begins with primitive hematopoietic stem cells (HSCs) and ends with mature blood cells: red blood cells, white blood cells and platelets. Regarding acute myelogenous leukemia (AML), a cancer of the bone marrow and blood, characterized by a rapid proliferation of immature cells, which eventually invade the bloodstream, there is a consensus about the target cells during the HSCs development which are susceptible to leukemic transformation. We propose and analyze a mathematical model of HSC dynamics taking into account two phases in the cell cycle, a resting and a proliferating one, by allowing just after division a part of HSCs to enter the resting phase and the other part to come back to the proliferating phase to divide again. The resulting mathematical model is a system of nonlinear differential-difference equations. Due to the hierarchical organization of the hematopoiesis, we consider n stages of HSCs characterized by their maturity levels. We obtain a system of 2n nonlinear differential-difference equations. We study the existence, uniqueness, positivity, boundedness and unboundedness of the solutions. We then investigate the existence of positive and axial steady states for the system, and obtain conditions for their stability. Sufficient conditions for the global asymptotic stability of the trivial steady state as well as conditions for its instability are obtained. Using neutral differential equation associated to the differential-difference system, we also obtain results on the local asymptotic stability of the positive steady state. Numerical simulations are carried out to show the influence of variations of the differentiation rates and self-renewal coefficients of the HSCs on the behavior of the system. In particular, we show that a blocking of differentiation at an early stage of HSC development can lead in an overexpression of very immature cells. Such situation corresponds to the observation in the case of AML.

طريـقة ميرسن لحل معادلات الفـروق- التفاضلية اللاخطية من الرتبة n

يقدم البحث طريقة مقترحة مع اشتقاق خوارزميات تمت برمجتها بلغة (Matlab) لإيجاد الحل العددي لمعادلات الفروق التفاضلية اللاخطية باستخدام طريقة ميرسن من الرتبة الرابعة. حيث تمت معالجة معادلة الفروق- التفاضلية اللاخطية من الرتبة الأولى عددياً مثلما لمنظومة من معادلات الفروق- التفاضلية اللاخطية باستخدام طريقة ميرسن من الرتبة الرابعة. ومن ناحية اخرى طورت هذه الطريقة لإيجاد النتائج العددية لمعادلات الفروق- التفاضلية اللاخطية من الرتبة n حيث من الممكن ملاحظة كفاءة الطريقة و دقة حساباتها في حل المعادلات اللاخطية. كما تمت مقارنة النتائج العددية و الحقيقية لأنواع مختلفة من معادلات الفروق- التفاضلية اللاخطية من خلال ثلاثة امثلة وقد تم الحصول على نتائج دقيقة.

Nonlinear orientational dynamics of a molecular chain

We investigate the nonlinear rotational dynamics of a molecular chain with quadrupole interaction in both the discrete and the continuous cases. Based on a system of nonlinear differential-difference equations, we obtain approximate equations describing the chain excitations and preserving the initial symmetry. We introduce an effective potential and normal coordinates, using which allows decoupling the system into linear and nonlinear parts. As a result of a strong anisotropy of the potential, narrow “valleys” occur in the angle plane. Motion along a valley corresponds to a softer interaction (nonlinear equations). Linear equations describe motion across a valley (hard interaction). We consider cases where the derived nonlinear equations reduce to the sine-Gordon equation. We find integrals of motion and exact solutions of our approximate equations. We uniformly describe the energy interval encompassing the domains of order, of orientational melting, and of rotational motion of the molecules in the chain.

Symbolic Computation of Recursion Operators for Nonlinear Differential-Difference Equations

An algorithm for the symbolic computation of recursion operators for systems of nonlinear differential-difference equations (DDEs) is presented. Recursion operators allow one to generate an infinite sequence of generalized symmetries. The existence of a recursion operator therefore guarantees the complete integrability of the DDE. The algorithm is based in part on the concept of dilation invariance and uses our earlier algorithms for the symbolic computation of conservation laws and generalized symmetries. The algorithm has been applied to a number of well-known DDEs, including the Kacvan Moerbeke (Volterra), Toda, and Ablowitz-Ladik lattices, for which recursion operators are shown. The algorithm has been implemented in Mathematica, a leading computer algebra system. The package DDERecursionOperator.m is briefly discussed.

Analytical treatment of the delayed feedback controlled Lorenz system close to a subcritical Hopf bifurcation

We develop an analytical approach for the delayed feedback control of the Lorenz system close to a subcritical Hopf bifurcation. The periodic orbits arising at this bifurcation have no torsion and cannot be stabilized by a conventional delayed feedback control technique. We utilize a modication based on an unstable delayed feedback controller. The analytical approach employs the centre manifold theory, the near identity transformation, and averaging. We derive the characteristic equation for the Floquet exponents of the controlled orbit in an analytical form and obtain simple expressions for the threshold of stability as well as for an optimal value of the control gain. The analytical results are supported by numerical analysis of the original system of nonlinear differential-difference equations.

Delayed feedback control of the Lorenz system: An analytical treatment at a subcritical Hopf bifurcation

We develop an analytical approach for the delayed feedback control of the Lorenz system close to a subcritical Hopf bifurcation. The periodic orbits arising at this bifurcation have no torsion and cannot be stabilized by a conventional delayed feedback control technique. We utilize a modification based on an unstable delayed feedback controller. The analytical approach employs the center manifold theory and the near identity transformation. We derive the characteristic equation for the Floquet exponents of the controlled orbit in an analytical form and obtain simple expressions for the threshold of stability as well as for an optimal value of the control gain. The analytical results are supported by numerical analysis of the original system of nonlinear differential-difference equations.

Asymptotic Solution of Some Systems of Nonlinear Differential-Difference Equations

Asymptotic methods developed by Feshchenko, Shkil', and their disciples for the integration of differential equations with slowly varying coefficients are used for the solution of certain nonlinear differential-difference equations.

Computation of conserved densities for systems of nonlinear differential-difference equations

A new method for the computation of conserved densities of nonlinear differential-difference equations is applied to Toda lattices and discretizations of the Korteweg-de Vries and nonlinear Schrödinger equations. The algorithm, which can be implemented in computer algebra languages such as Mathematica, can be used as an indicator of integrability.

Existence and uniqueness of bounded (on the whole axis) solutions of systems of nonlinear differential-difference equations of neutral type

The existence and uniqueness of the bounded (on the whole axis) solution of a certain class of systems of nonlinear differential-difference equations of neutral type are proved.

Existence and uniqueness of bounded (on the whole axis) solutions of systems of nonlinear differential-difference equations of neutral type

Existence and uniqueness of bounded (on the whole axis) solutions of systems of nonlinear differential-difference equations of neutral type