Parameterized center manifold for unfolding bifurcations with an eigenvalue +1 in n-dimensional maps

• Published in: Communications in nonlinear science & numerical simulation 2016-10, Vol.39, p.495-503
• Main Authors: Wen, Guilin, Yin, Shan, Xu, Huidong, Zhang, Sijin, Lv, Zengyao
• Format: Article
• Language: English
• Subjects: Bifurcations; Computer simulation; Eigenvalues; Fold bifurcation; High-dimensional maps; Manifolds; Mathematical models; Nonlinearity; Parameterized center manifold
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2016.04.002

•A parameterized center manifold is derived to unfold the solutions of the foldbifurcation with an eigenvalue +1.•The conditions of potential bifurcation solutions are established for the fold bifurcation of general maps.•The numerical simulations of three four-dimensional maps validate the corresponding theory analysis. 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.