Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
7155171 | Communications in Nonlinear Science and Numerical Simulation | 2016 | 20 Pages |
Abstract
For the fold bifurcation with an eigenvalue +1, there are three types of potential solutions from saddle-node bifurcation, transcritical bifurcation and pitchfork bifurcation. In the existing analysis methods for high maps, there is a problem that for the fold bifurcation, saddle-node bifurcation and transcritical bifurcation cannot be distinguished by the center manifold without bifurcation parameter. In this paper, a parameterized center manifold has been derived to unfold the solutions of the fold bifurcation with an eigenvalue +1, which is used to reduce a general n-dimensional map to one-dimensional map. On the basis of the reduced map, the conditions of the fold bifurcations including saddle-node bifurcation, transcritical bifurcation and pitchfork bifurcation are established for general maps, respectively. We show the applications of the proposed bifurcation conditions by three four-dimensional map examples to distinguish saddle-node bifurcation, transcritical bifurcation and pitchfork bifurcation.
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Authors
Wen Guilin, Yin Shan, Xu Huidong, Zhang Sijin, Lv Zengyao,