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ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

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The Method of Transfer to the Derivative Space: 40 Years of Evolution

Author(s):

I. M. Burkin

Professor, head of department of mathematical analysis,
Department of Mechanics and Mathematics,
Tula State University,
Tula, Russia

i-burkin@yandex.ru

Abstract:

In 1975 the so called “method of transfer to the derivative space” was proposed. It is an efficiently verified frequency criterion of the existence of a nontrivial periodic solution in multidimensional models of automatic control systems with one differentiable nonlinear term. The method used the classical torus principle and refrained from any constructions in the phase space of the system under study. Moreover, the method allowed researchers to broaden the class of systems to that it may be applied. In this work we give a survey of the results presenting generalization and expansion of the method. We also show the connection between the method of transfer to the derivative space, well known the Poincare-Bendixon generalized principle proposed by R. A. Smith and the results of contemporary authors who are active in the theory of oscillations in multidimensional systems. During recent years the author obtained frequency criteria of the existence of orbitally stable cycles in multivariable automatic control systems (MIMO systems) and the methods of construction of multidimensional systems having the only equilibrium state and any given in advance number of orbitally stable cycles, which are described in the paper. The extension of the Poincare-Bendixon generalized principle to multidimensional systems with angular coordinate is presented. We show the application of described methods of investigation of oscillation processes in multidimensional dynamical systems to the solving S. Smale problem from the biological cells chemical kinetics theory and also to the finding hidden attractors of the Chua generalized system and minimal global attractor of a system with a polynomial nonlinear term. The publication is illustrated by numerous examples.

Keywords

References:

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