Investigation of the Dynamic Systems’ Phase Portraits

Yıl 2019, Cilt: 7 , 338 - 345, 24.11.2019

İrina Andreeva Tatiana Efımova

Öz

Dynamic systems in a broad sense may be considered as mathematical models of processes and phenomena, where any statistical events we may disregard. The dynamic systems theory investigates curves, defined by differential equations. The same time the laws of Nature are written using the language of differential equations, as the great French mathematician and encyclopedist of the nineteenth and twentieth centuries J.H. Poincare has taught. Thus, these laws are written using dynamic systems. A study of a given family of dynamic systems depending on several parameters means splitting of a phase space belonging to the dynamic system under consideration into trajectories and investigation of the limit behavior of them with the aim to reveal and describe their positions of equilibrium, and to find the so-called sinks and sources. Also, very important are the question of the stability of equilibrium states and their types, as well as the question of a roughness of a system. Rough dynamic systems can preserve their qualitative character of motion under some considerably small changes in parameters of the system. The paper is devoted to the original investigation of a broad family of polynomial dynamic systems, depending on multiple parameters.

Anahtar Kelimeler

Dynamic systems, Trajectories, Phase portraits, Equilibrium states, Differential equations, Poincare sphere, Poincare disk, Singular points, Separatrices, Limit cycles

Kaynakça

• Andronov, A.A., Leontovich, E.A., Gordon, I.I., & Maier, A.G. (1973). Qualitative theory of second-order dynamic systems. New York, NY: Wiley. Andreev, A.F., & Andreeva, I.A. (1997). On limit and separatrix cycles of a certain quasi quadratic system. Differential Equations, 33 (5), 702 – 703. Andreev, A.F., & Andreeva, I.A. (2007). Local study of a family of planar cubic systems. Vestnik St. Petersburg University: Ser.1. Mathematics, Mechanics, Astronomy, 2, 11- 16. DOI: 10.3103/S1063454107020021, EID: 2-s2.0-84859730890. Andreev, A.F., Andreeva, I.A., Detchenya, L.V., Makovetskaya, T.V., & Sadovskii, A.P. (2017). Nilpotent Centers of Cubic Systems. Differential Equations, 53(8), 1003 - 1008. DOI: 10.1134/S0012266117080018, EID: 2-s2.0-85029534241. Andreev, A.F., & Andreeva, I.A. (2007). Phase flows of one family of cubic systems in a Poincare circle. I. Differential Equations and Control, 4, 17-26. Andreev, A.F., & Andreeva, I.A. (2008). Phase flows of one family of cubic systems in a Poincare circle. II. Differential Equations and Control, 1, 1 - 13. Andreev, A.F., & Andreeva, I.A. (2008). Phase flows of one family of cubic systems in a Poincare circle. III. Differential Equations and Control/, 3, 39 - 54. Andreev, A.F., & Andreeva, I.A. (2009). Phase flows of one family of cubic systems in a Poincare circle. Differential Equations and Control, 4, 181 - 213. Andreev, A.F., &Andreeva, I.A. (2010). Phase flows of one family of cubic systems in a Poincare circle. Differential Equations and Control, 4, 6- 17. Andreev, A.F., & Andreeva, I.A. (2017). Investigation of a Family of Cubic Dynamic Systems. Vibroengineering Procedia, 15, 88 – 93. DOI: 10.21595/vp.2017.19389. Andreev, A.F., & Andreeva, I.A. (2017). Phase Portraits of One Family of Cubic Systems in a Poincare Circle.I. Vestnik RAEN, 17, 4, 8 – 18. ISSN 1682-1696. Andreev, A.F., &Andreeva, I.A. (2018) On a Behavior of Trajectories of a Certain Family of Cubic Dynamic Systems in a Poincare Circle, IOP Journal of Physics, Conference Series, 1141. Andreeva, I.A. (2018) Phase Portraits of Cubic Dynamic Systems in a Poincare Circle. (A Chapter in the Monograph “Differential Equations- Theory and Current Research”. IntechOpen, UK, ISBN 978-953-51-6120-2. Andreev, A.F., & Andreeva, I.A. (2018). Phase Portraits of a Family of Cubic Dynamic Systems in a Poincare Circle.II. Vestnik RAEN, 18, 4, 11 – 15. ISSN 1682-1696. Andreeva, I.A., &Efimova, T.O. (2019) Phase Portraits of a Special Class of Dynamic Systems in a Poincare Circle, IOP Journal of Physics, Conference Series, 1236. Aksenova, O.A., & Khalidov, I.A. (2016). Analytic Model of the Effect of Poly-Gaussian Roughness on Rarefied Gas Flow near the Surface, Rarefied Gas Dynamics, AIP Conference Proceedings, 1786, American Institute of Physics, Melville, NY, 1000071 - 1000078. Aksenova, O.A., & Khalidov, I.A. (2016). Unstable Rarefied Gas Flow Conditions in a Channel, Rarefied Gas Dynamics, AIP Conference Proceedings 1786, American Institute of Physics, Melville, NY, 1000091-1000097.