Perturbed Taylor expansion for bifurcation of solution of singularly parameterized perturbed ordinary differentia equations and differential algebraic equations
AbstractIn This paper deals with the study of singularity perturbed ordinary differential equation, and is considered the basis for obtaining the system of differential algebraic equations. In this study the we use implicit function theorem to solve for fast variable y to get a reduced model in terms of slow dynamics locally around x. It is well known that solving nonlinear algebraic equations analytical is quite difficult and numerical solution methods also face many uncertainties since nonlinear algebraic equations may have many solutions, especially around bifurcation points. We have used singularly perturbed ODE to study the bifurcation problem in Differential algebraic system. So for the first step we need to investigate the bifurcation problem in our original system when , for this purpose the known kinds bifurcations such as saddle node, transtritical and pitch fork has been studied by using Taylor expansion for one dimensional system. And for higher dimension we apply Sotomayor Theorem. The second step is going to study bifurcation problem in DAE: Where is bifurcation parameter. by converting such system to singularly perturbed ODE to make use the study in the first step: The method we used to convert DAEs to singular perturbed ODEs is PTE method. The bifurcation in index one DAEs is investigated by reduced the system to system with lower dimension by implicit function theorem. And for higher dimension index two DAEs we used Sotomayor Theorem. Also the singularity induced bifurcation for which this kind of bifurcation occurred in DAEs is studied by PTE method.
X. Song, “Dynamic modeling issues for power system applications,” Ph.D. dissertation, Texas A&M University, 2005.
V. Venkatasubramanian, H. Schattler, and J. Zaborszky, “Local bifurcations
and feasibility regions in differential-algebraic systems,” IEEE Transactions on Automatic Control, vol. 40, no. 12, pp. 1992–2013, 1995.
S.-N. Chow and J. K. Hale, Methods of bifurcation theory. Springer
Science & Business Media, 2012, vol. 251.
A. J. Tamraz, “Bifurcation of periodic solutions of singularly perturbed
delay differential equation,” 1988.
V. Venkatasubramanian, “Singularity induced bifurcation and the van
der pol oscillator,” IEEE Transactions on Circuits and Systems I:
Fundamental Theory and Applications, vol. 41, no. 11, pp. 765–769,
S. Ayasun, C. O. Nwankpa, and H. G. Kwatny, “An efficient method
to compute singularity induced bifurcations of decoupled parameter dependent
differential-algebraic power system model,” Applied mathematics
and computation, vol. 167, no. 1, pp. 435–453, 2005.
C. McCann, “Bifurcation analysis of non-linear differential equations.”
C. C¸. Karaaslanlı, Bifurcation analysis and its applications. INTECH
Open Access Publisher, 2012.
J. D. Crawford, “Introduction to bifurcation theory,” Reviews of Modern
Physics, vol. 63, no. 4, p. 991, 1991.
L. Perko, Differential equations and dynamical systems. Springer
Science & Business Media, 2013, vol. 7.
Copyright (c) 2021 Journal of Education for Pure Science- University of Thi-Qar
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
The Authors submitting a manuscript do so on the understanding that if accepted for publication, copyright of the article shall be assigned to Journal of education for Pure Science (Jeds), University of Thi-Qar as publisher of the journal.
Copyright encompasses exclusive rights to reproduce and deliver the article in all form and media, including reprints, photographs, microfilms and any other similar reproductions, as well as translations. The reproduction of any part of this journal, its storage in databases and its transmission by any form or media, such as electronic, electrostatic and mechanical copies, photocopies, recordings, magnetic media, etc. , will be allowed only with a written permission from Journal of education for Pure Science (Jeds), University of Thi-Qar.
Journal of education for Pure Science (Jeds), University of Thi-Qar, the Editors and the Advisory International Editorial Board make every effort to ensure that no wrong or misleading data, opinions or statements be published in the journal. In any way, the contents of the articles and advertisements published in the Journal of education for Pure Science (Jeds), University of Thi-Qar are sole and exclusive responsibility of their respective authors and advertisers.