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

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

Invariant Surfaces of Two-dimensional Standard Systems at First Approximation with Nine Steady Points


Vladimir V. Basov

Universitetsky prospekt, 28,
198504, Peterhof, St. Petersburg, Russia,
Saint-Petersburg State University,
The Faculty of Mathematics and Mechanics,
Differential Equations Department

Artyom Zhukov

PhD student
The Faculty of Mathematics and Mechanics, Differential Equations Department
Saint-Petersburg State University
Universitetsky prospekt, 28, 198504, Peterhof, St. Petersburg, Russia,


We study two classes of two-dimensional time-periodic systems of ODEs with a small positive parameter --- systems with "slow" and "fast" time. Right-hand sides are assumed to be three times continuously differentiable on the phase variables and the parameter, and the corresponding unperturbed systems are autonomous, conservative and have nine steady points such that both coordinates take values -1,0 or 1. For parameter-independent perturbations of a system explicit conditions guaranteeing the existence of a certain number of invariant two-dimensional surfaces homeomorphic to tori for all small enough values of positive parameter have been obtained. Formulae of these surfaces are given. As an example of practical usage of the obtained result the~class of systems with three invariant surfaces enclosing some of steady points has been found in Refs 10.



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