A real autonomous quadratic system of three differential equations with an infinite number of limit cycles
https://doi.org/10.29235/1561-2430-2022-58-2-135-143
Abstract
In this paper, we consider the problem of construction of real autonomous quadratic systems of three differential equations with the nonlocal existence of an infinite number of limit cycles. This means that an infinite number of limit cycles, emerging from the focus due to the Andronov – Hopf bifurcation, can exist in the phase space not only in the vicinity of the focus and not only for parameter values close to the bifurcation value. To solve this problem we use the method of determination of limit cycles as the curves of intersection of an invariant plane with a family of invariant elliptic paraboloids. Then the study of the limit cycles of the constructed system of the third order is reduced to the study of the corresponding system of the second order on each of the invariant elliptic paraboloids. The proof of the nonlocal existence of the limit cycle and the investigation of its stability for such a second-order system is carried out by constructing a topographic system of Poincaré functions or by transforming to polar coordinates.
Keywords
autonomous quadratic system of the third order,
limit cycle,
invariant surface,
stationary point,
Andronov – Hopf bifurcation
Supporting agencies: The research is supported by the project Horizon 2020-2017-RISE-777911.
About the Authors
A. A. Hryn
Yanka Kupala State University of Grodno
Belarus
Belarus
Aliaksandr A. Hryn – Dr. Sc. (Physics and Mathematics), Professor, Head of the Department of Mathematical Analysis, Differential Equations and Algebra
22, Ozheshko Str., 230023, Grodno
E. V. Musafirov
Yanka Kupala State University of Grodno
Belarus
Belarus
Eduard V. Musafirov – Ph. D. (Physics and Mathe matics), Associate Professor of the Department of Technical Mechanics
1а, Kurchatov Str., 230005, Grodno
A. F. Pranevich
Yanka Kupala State University of Grodno
Belarus
Belarus
Andrei F. Pranevich – Ph. D. (Physics and Mathematics), Associate Professor, Vice-Rector for Research
23, Gaspadarchaya Str., 230005, Grodno
References
1. Shil′nikov L. P., Shil′nikov A. L., Turaev D. V., Chua L. Methods of Qualitative Theory in Nonlinear Dynamics. Moskow, Izhevsk, Institute for Computer Research, 2003. 428 p. (in Russian).
2. Hirsch M. W., Smale S., Devaney R. L. Differential Equations, Dynamical Systems, and an Introduction to Chaos. Amsterdam, Academic Press, 2013. 418 p. https://doi.org/10.1016/C2009-0-61160-0
3. Kuznetsov Y. A. Elements of Applied Bifurcation Theory, 2nd ed. Applied Mathematical Sciences, vol. 112. New York, Springer-Verlag, 1998. 593 p. https://doi.org/10.1007/978-1-4757-2421-9
4. Arnold V. I., Afraimovich V. S., Il′yashenko Y. S., Shil′nikov L. P. Bifurcation Theory and Catastrophe Theory. Dynamical Systems V. Encyclopaedia of Mathematical Sciences, vol. 5. New York, Springer-Verlag, 1994. 274 p. https://doi.org/10.1007/978-3-642-57884-7
5. Wiggins S. Introduction to Applied Non-linear Dynamical Systems and Chaos, 2nd ed. Texts in Applied Mathematics, vol. 2. New York, Springer-Verlag, 2003. 844 p.
6. Hassard B. D., Kazarinoff N. D., Wan Y.-H. Theory and Applications of Hopf Bifurcation. London Mathematical Society Lecture Note Series, vol. 41. Cambridge, Cambridge University Press, 1981. 320 p.
7. Reyn J. Phase Portraits of Planar Quadratic Systems. Mathematics and Its Applications, vol. 583. New York, Springer- Verlag, 2007. 334 p. https://doi.org/10.1007/978-0-387-35215-2
8. Bulgakov V. I., Grin A. A. On one bifurcation of a non-rough focus of a third-order autonomous system. Differential Equations, 1996, vol. 32, no. 12, pp. 1697–1698.
9. Romanovski V. G., Shafer D. S. Centers and limit cycles in polynomial systems of ordinary differential equations. Advanced Studies in Pure Mathematics, 2016, vol. 68, pp. 267–373. https://doi.org/10.2969/aspm/06810267
10. Bautin N. N., Leontovich E. A. Methods and Techniques for the Qualitative Study of Dynamical Systems on a Plane. Moscow, Nauka Publ., 1990. 486 p. (in Russian).
11. Tigan G., Llibre J., Ciurdariu L. Degenerate Fold–Hopf Bifurcations in a Rössler-Type System. International Journal of Bifurcation and Chaos, 2017, vol. 27, no. 5, pp. 1–8. https://doi.org/10.1142/S0218127417500687
12. Cherkas L. A., Grin A. A., Bulgakov V. I. Constructive Methods of Investigation of Limit Cycles of Second Order Autonomous Systems (Numerical-Algebraic Approach). Grodno, GrGU Publ., 2013. 489 p. (in Russian).
13. Bulgakov V. I. On the phase portrait of a third-order autonomous system. Differentsial′nye uravneniya [Differential Equations], 1988. vol. 24, no. 10, pp. 1821–1822 (in Russian).
14. Bulgakov V. I. About Bifurcations of Limit Cycles of a Quadratic Three-Dimensional System in a Neighborhood of a Non-Rough Focus. Doklady Akademii nauk BSSR [Doklady of the Academy of Sciences of BSSR], 1982. vol. 26, no. 108, pp. 681–684 (in Russian).
15. Bibikov Y. N. Local Theory of Nonlinear Analytic Ordinary Differential Equations. Lecture Notes in Mathematics, vol. 702. New York, Springer-Verlag, 1979. 150 p.
Review
Views: 595
Email this article 