SPECIFIED GLOBAL POINCARE–BENDIXSON ANNULUS WITH THE LIMIT CYCLE OF THE RAYLEIGH SYSTEM

Мұқаба

Дәйексөз келтіру

Толық мәтін

Аннотация

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.

Авторлар туралы

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

Әдебиет тізімі

  1. Bautin, N.P. and Leontovich, E.A., Metody i priemy kachestvennogo issledovaniia dinamicheskikh sistem na ploskosti (Methods and techniques for qualitative research of dynamic systems on a plane), Moscow: Bukinist, 1990.
  2. Andronov, A.A., Vitt, A.A., and Hajkin, S.E., Teoriia kolebanii (Oscillation theory), Moscow: Fizmatgiz, 1959.
  3. Perko, L. Differential Equations and Dynamical Systems / L. Perko. — New York ; Berlin ; Heidelberg : Springer, 2001. — 555 p.
  4. Rejssig, R., Sansone, G., and Konti, R., Non-Linear Differential Equations of Higher Order, Dordrecht: Springer, 1974.
  5. Lynch, S. Dynamical systems with Applications using Mathematica / S. Lynch. — Boston : Birkh¨auser, 2007. — 585 p.
  6. Flanders, D.A. The limit case of relaxation oscillations / D.A. Flanders, J.J. Stoker // Studies in Nonlinear Vibration Theory ; ed. Howard J. Eckweiler. — New York : New York University, 1946. — P. 51–64.
  7. Schneider, K.R. New approach to study the Van der Pol equation for large damping / K.R. Schneider // Electron. J. Qual. Theory Differ. Equat. — 2018. — V. 8. — P. 1–10.
  8. Gasull, A. Effective construction of Poincar´e–Bendixson regions / A. Gasull, H. Giacomini, M. Grau // J. Appl. Anal. Comp. — 2017. — V. 7. — P. 1549–1569.
  9. Giacomini, H. Transversal conics and the existence of limit cycles / H. Giacomini, M. Grau // J. Math. Anal. Appl. — 2015. — V. 428. — P. 563–586.
  10. Grin, A.A. and Schneider, K.R., Global algebraic Poincar´e–Bendixson annulus for the Van der Pol equation, Differ. Equat., 2022, vol. 58, no. 3, pp. 285–295.
  11. Cherkas, L.A., Dulac function for polynomial autonomous systems on a plane, Differ. Equat., 1997, vol. 33, pp. 692–701.
  12. Cherkas, L.A., Grin, A.A., and Bulgakov, V.I., Konstruktivnye metody issledovaniia predel’nykh tsiklov avtonomnykh sistem vtorogo poriadka (chislenno-algebraicheskii podkhod) (Constructive methods for studying limit cycles of second-order autonomous systems (numerical-algebraic approach)), Grodno: Grodnen. gos. un-t im. Yanki Kupaly, 2013.
  13. Grin, A.A. Location of the limit cycle for a class of Lienard systems by means of Dulac–Cherkas functions / A.A. Grin, K.R. Schneider // Memoirs on Differ. Equat. and Math. Phys. — 2023. — V. 90. — P. 1–11.
  14. Grin, A.A. Global algebraic Poincar´e–Bendixson annulus for the Rayleigh equation / A.A. Grin, K.R. Schneider // Electron. J. Qual. Theory Differ. Equat. — 2023. — V. 35. — P. 1–12.
  15. Birkhoff, G. Ordinary Differential Equations / G. Birkhoff, G.-C. Rota. — New York : John Wiley & Sons, 1989. — 416 p.
  16. Rayleigh, J. The Theory of Sound / J. Rayleigh. — New York, 1945. — 520 p.
  17. Georgescu, A. Approximate limit cycles for the Rayleigh model / A. Georgescu, P. Bazavan, M. Sterpu // ROMAI J. — 2008. — V. 4, № 2. — P. 73–80.
  18. Ghaffari, A. On Rayleigh’s nonlinear vibration equation / A. Ghaffari // Proc. Int. Sympos. Non-linear Vibrations. Kiev, 1963. — V. 2. — P. 131–133.
  19. Lopez, M.A. A note on the generalized Rayleigh equation: limit cycles and stability / M.A. Lopez, R. Martinez // J. Math. Chem. — 2013. — V. 51. — P. 1164–1169.
  20. Palit, A. Comparative study of homotopy analysis and renormalization group methods on Rayleigh and Van der Pol equations / A. Palit, D.P. Datta // Differ. Equat. Dynan. Syst. — 2016. — V. 24. — P. 417–443.
  21. Saha, S. Systematic designing of bi-rhythmic and tri-rhythmic models in families of Van der Pol and Rayleigh oscillators / S. Saha, G. Gangopadhyay, R.D. Shankar // Commun. Nonlin. Sci. Numer. Simul. — 2020. — V. 85. — P. 12.
  22. Tliachev, V.B., Usho, A.D., and Usho, D.S., On periodic solutions of the Rayleigh equation, Izvestiia Saratovskogo universiteta. Novaia seriia. Seriia: Matematika. Mekhanika. Informatika, 2021, vol. 21, no. 2, pp. 173–181.
  23. Grin, A.A. On some classes of limit cycles of planar dynamical systems / A.A. Grin, K.R. Schneider // Dyn. Contin. Discrete Impuls. Syst. Ser. A. Math. Anal. — 2007. — V. 14. — P. 641–656.

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