Bifurcation analysis of a three dimensional system
In order to enrich the stability and bifurcation theory of the three dimensional chaotic systems, taking a quadratic truncate unfolding system with the triple singularity equilibrium as the research subject, the existence of the equilibrium, the stability and the bifurcation of the system near the e...
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doaj-bce6074236ea4d6294c9576b4cdb5a3e2020-11-24T21:52:09ZzhoHebei University of Science and TechnologyJournal of Hebei University of Science and Technology1008-15422018-04-0139213514110.7535/hbkd.2018yx02006b201802006Bifurcation analysis of a three dimensional systemYongwen WANG0Zhiqin QIAO1Yakui XUE2School of Science, North University of China, Taiyuan, Shanxi 030051, ChinaSchool of Science, North University of China, Taiyuan, Shanxi 030051, ChinaSchool of Science, North University of China, Taiyuan, Shanxi 030051, ChinaIn order to enrich the stability and bifurcation theory of the three dimensional chaotic systems, taking a quadratic truncate unfolding system with the triple singularity equilibrium as the research subject, the existence of the equilibrium, the stability and the bifurcation of the system near the equilibrium under different parametric conditions are studied. Using the method of mathematical analysis, the existence of the real roots of the corresponding characteristic equation under the different parametric conditions is analyzed, and the local manifolds of the equilibrium are gotten, then the possible bifurcations are guessed. The parametric conditions under which the equilibrium is saddle-focus are analyzed carefully by the Cardan formula. Moreover, the conditions of codimension-one Hopf bifucation and the prerequisites of the supercritical and subcritical Hopf bifurcation are found by computation. The results show that the system has abundant stability and bifurcation, and can also supply theorical support for the proof of the existence of the homoclinic or heteroclinic loop connecting saddle-focus and the Silnikov's chaos. This method can be extended to study the other higher nonlinear systems.http://xuebao.hebust.edu.cn/hbkjdx/ch/reader/create_pdf.aspx?file_no=b201802006&flag=1&journal_stability theorysaddle-focusHopf bifurcationsupercriticalsubcritical |
collection |
DOAJ |
language |
zho |
format |
Article |
sources |
DOAJ |
author |
Yongwen WANG Zhiqin QIAO Yakui XUE |
spellingShingle |
Yongwen WANG Zhiqin QIAO Yakui XUE Bifurcation analysis of a three dimensional system Journal of Hebei University of Science and Technology stability theory saddle-focus Hopf bifurcation supercritical subcritical |
author_facet |
Yongwen WANG Zhiqin QIAO Yakui XUE |
author_sort |
Yongwen WANG |
title |
Bifurcation analysis of a three dimensional system |
title_short |
Bifurcation analysis of a three dimensional system |
title_full |
Bifurcation analysis of a three dimensional system |
title_fullStr |
Bifurcation analysis of a three dimensional system |
title_full_unstemmed |
Bifurcation analysis of a three dimensional system |
title_sort |
bifurcation analysis of a three dimensional system |
publisher |
Hebei University of Science and Technology |
series |
Journal of Hebei University of Science and Technology |
issn |
1008-1542 |
publishDate |
2018-04-01 |
description |
In order to enrich the stability and bifurcation theory of the three dimensional chaotic systems, taking a quadratic truncate unfolding system with the triple singularity equilibrium as the research subject, the existence of the equilibrium, the stability and the bifurcation of the system near the equilibrium under different parametric conditions are studied. Using the method of mathematical analysis, the existence of the real roots of the corresponding characteristic equation under the different parametric conditions is analyzed, and the local manifolds of the equilibrium are gotten, then the possible bifurcations are guessed. The parametric conditions under which the equilibrium is saddle-focus are analyzed carefully by the Cardan formula. Moreover, the conditions of codimension-one Hopf bifucation and the prerequisites of the supercritical and subcritical Hopf bifurcation are found by computation. The results show that the system has abundant stability and bifurcation, and can also supply theorical support for the proof of the existence of the homoclinic or heteroclinic loop connecting saddle-focus and the Silnikov's chaos. This method can be extended to study the other higher nonlinear systems. |
topic |
stability theory saddle-focus Hopf bifurcation supercritical subcritical |
url |
http://xuebao.hebust.edu.cn/hbkjdx/ch/reader/create_pdf.aspx?file_no=b201802006&flag=1&journal_ |
work_keys_str_mv |
AT yongwenwang bifurcationanalysisofathreedimensionalsystem AT zhiqinqiao bifurcationanalysisofathreedimensionalsystem AT yakuixue bifurcationanalysisofathreedimensionalsystem |
_version_ |
1725876566772154368 |