Mathematics

2000, No. 4

V.A. Alyakin, S.Y. Novikov, S.V. Dvoryaninov

TO THE MEMORY OF VICTOR KLIMKIN

The article is devoted to the memory of Prof. V.M. Klimkin, the Dean of the Mechanics and Mathemetics department, the Chief of the Chair of Functional Analysis and Functions Theory in Samara State University.

S.V. Bogatyrev

БИФУРКАЦИИ СТАЦИОНАРНЫХ РЕШЕНИЙ В ОДНОЙ МОДЕЛИ ХИЩНИК--ЖЕРТВА С ПЕРЕКЛЮЧЕНИЕМ

В рамках теории оптимальной трофической стратегии строится модель, описывающая взаимодействие популяции хищника с двумя популяциями жертв. Модель представляет собой систему из трех обыкновенных дифференциальных уравнений с разрывными правыми частями. Рассматриваются случаи линейной, квадратичной и рациональной зависимостей частоты встреч хищника с жертвой. Изучается устойчивость нетривиального положения равновесия модели как функции одного из параметров. Находятся условия смены устойчивости положения равновесия при пересечении им поверхности переключения.

R.E. Popovich, V.N. Boiko

DIFFERENTIAL INVARIANTS OF ONE-PARAMETER GROUP OF LOCAL TRANSFORMATION AND INTEGRABLE RICCATI EQUATIONS

A new tool to construction of differential invariants for a oneparameter group of local transformations in the space of one independent and m dependent variables is proposed. It is proved that if its universal invariant is known then the complete set of functionally independent differential invariants can be constructed via one quadrature and differentiation. Relation of first-order differential invariants to Riccati equations is analysed.

I.S. Orlova

ON ONE METHOD OF AN EXACT LINEARIZATION OF NONLINEAR DIFFERENTIAL EQUATIONS FIFTH AND SIXTH ORDERS

This article is a direct continuing of work [1], in which the method of an exact linearization [2] was applied. In [1] the non-linear autonomous ordinary differential equations 2-nd, 3-rd and 4-th orders were considered. In the present work the talk goes about equations 5-th and 6-th is ordinal. The illustrative examples are reduced. Some from obtained results were presented in [4].

H.A. Chihanov

KAMPE DE FERIETS FUNCTION AND FUNDAMENTAL SOLUTIONS OF ONE CLASS OF PARTIAL DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS

It is presented a new way of building of fundamental solutions for one class of differential equations with partial derivations with constant coeficients, avoiding some algebraic manipulations and calculation of integrals. It is considered the generalised Laurichella series, their classifications and problem of convergencies.