Enviar artigo por via de e-mail SPECIFIED GLOBAL POINCARE–BENDIXSON ANNULUS WITH THE LIMIT CYCLE OF THE RAYLEIGH SYSTEM
- Autores: Li Y.1, Grin A.A2, Kuzmich A.V2
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Afiliações:
- Lanzhou City University
- Yanka Kupala State University of Grodno
- Edição: Volume 60, Nº 6 (2024)
- Páginas: 736-746
- Seção: ORDINARY DIFFERENTIAL EQUATIONS
- URL: https://ogarev-online.ru/0374-0641/article/view/265610
- DOI: https://doi.org/10.31857/S0374064124060023
- EDN: https://elibrary.ru/KWOKWN
- ID: 265610
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Somente assinantes
Resumo
In the work of A. Grin and K. Schneider [1] two algebraic transversal ovals, which form the Poincare–Bendixson annulus 𝐴(𝜆), are analytically constructed. This annulus contains the unique limit cycle of the Rayleigh equation ¨ 𝑥+𝜆( ˙ 𝑥2/3−1) ˙ 𝑥+𝑥 = 0 for all values of the parameter 𝜆 > 0. In the constructed annulus 𝐴(𝜆) inner boundary consists of zero-level set of the Dulac–Cherkas function and outer boundary represents the diffeomorphic image corresponding boundary of such an annulus for the van der Pol system. In this paper new methods of construction two Dulac–Cherkas functions are worked out. With help of these functions a better inner boundary of the Poincare–Bendixson annulus 𝐴(𝜆) depending on the parameter is found. Also, for the Rayleigh system, a procedure for direct finding a polynomial whose zero-level set contains a transversal oval used as the outer boundary 𝐴(𝜆) is proposed. Then we determine the interval for 𝜆 where the better outer boundary of the annulus is a closed contour, which composed from two arcs of the constructed oval and two arcs of non-closed curves belonging to zerolevel set of one from Dulac–Cherkas functions. Thus, finally the specified global Poincare–Bendixson annulus for the limit cycle of the Rayleigh system is presented.
Sobre autores
Y. Li
Lanzhou City University
Email: li_liyong120@163.com
China
A. Grin
Yanka Kupala State University of Grodno
Email: grin@grsu.by
Belarus
A. Kuzmich
Yanka Kupala State University of Grodno
Email: kuzmich_av@grsu.by
Belarus
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