Schemes of analysis for systems with different forms of aftereffect

Vladimir V.Malanin, Igor E.Poloskov

Perm State University, Russia

Abstract

A phenomenon of different delays is an essential element of functioning of objects. Therefore deterministic and stochastic differential equations with divergent arguments and integro-differential equations have been attracting added interest both from theoretical and practical viewpoints. Such equations are encountered in those areas, where the properties of an object depend of a hereditary effect, and serve as models for different processes, viz. an automatic control for technical devices and engineering procedure, development of economic and social systems and so forth.

A set of equations with divergent arguments under our researches includes deterministic and stochastic, linear and nonlinear ordinary differential equations with single and multiple constant and variable delays, neutral differential equations, systems excited by continuous and discrete fluctuations, stochastic integro-differential equations with additive and multiplicative noises, and others.

Our schemes for study of such systems are based on extensions of phase spaces of systems and implemented due to Mathematica code programs. These schemes allow to transform equations with divergent arguments into equations without delays and to use standard Mathematica tools such as function NDSolve.

The techniques are applied to study a number of linear and nonlinear models.