A Class of Quintic Kolmogorov Systems with Explicit Non-algebraic Limit Cycle
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https://elib.sfu-kras.ru/handle/2311/110229Author:
Bendjeddou, Ahmed
Grazem, Mohamed
Бенджедду, Ахмед
Грэм, Мохамед
Date:
2019-06Journal Name:
Журнал Сибирского федерального университета. Математика и физика. Journal of Siberian Federal University. Mathematics & Physics; 2019 12 (3)Abstract:
Various physical, ecological, economic, etc phenomena are governed by planar differential systems. Sub-
sequently, several research studies are interested in the study of limit cycles because of their interest in
the understanding of these systems. The aim of this paper is to investigate a class of quintic Kolmogorov
systems, namely systems of the form
x=xP4 (x;y); y= y Q4 (x; y) ;
where P4 and Q4 are quartic polynomials. Within this class, our attention is restricted to study the limit
cycle in the realistic quadrant {(x; y) 2 R2; x > 0; y > 0}. According to the hypothesises, the existence
of algebraic or non-algebraic limit cycle is proved. Furthermore, this limit cycle is explicitly given in
polar coordinates. Some examples are presented in order to illustrate the applicability of our result Различные физические, экологические, экономические и т.д. явления перекрываются планарными
дифференциальными системами. Впоследствии, некоторые исследования привлекут внимание к
изучению предельных циклов из-за их интереса к пониманию этих систем. Целью данной работы
является исследование одного класса квинтических колмогоровских систем, а именно систем вида :x=xP4 (x;y); y= y Q4 (x; y) ;
где P4 и Q4 — квартичные полиномы. В этом классе наше внимание ограничено изучением пре
дельного цикла в реалистическом квадранте {(x; y) 2 R2; x > 0; y > 0}. Согласно гипотезам дока-
зано существование алгебраического или неалгебраического предельного цикла. Кроме того, этот
предельный цикл явно задан в полярных координатах. Некоторые примеры представлены для то-
го, чтобы проиллюстрировать возможности применения нашего результата
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