a School of Mechanical Engineering, Anhui University of Technology, Maanshan 243002, China; b School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110004, China
Abstract A simple branch of solution on a bifurcation diagram, which begins at static bifurcation and ends at boundary crisis (or interior crisis in a periodic window), is generally a period-doubling cascade. A domain of solution in parameter space, enclosed by curves of static bifurcation and that of boundary crisis (or the interior of a periodic window), is the trace of branches of solution. A P-n branch of solution refers to the one starting from a period-n (n≥1) solution, and the corresponding domain in parameter space is named the P-n domain of solution. Because of the co-existence of attractors, there may be several branches within one interval on a bifurcation diagram, and different domains of solution may overlap each other in some areas of the parameter space. A complex phenomenon, concerned both with the co-existence of attractors and the crises of chaotic attractors, was observed in the course of constructing domains of steady state solutions of the Hénon map in parameter space by numerical methods. A narrow domain of period-m solutions firstly co-exists with (lies on) a big period-n (m≥3n) domain. Then it enters the chaotic area of the big domain and becomes period-m windows. The co-existence of attractors disappears and is called the landing phenomenon. There is an interaction between the two domains in the course of landing: the chaotic area in the big domain is enlarged, and there is a crisis step near the landing area.
Received: 27 August 2003
Revised: 20 October 2003
Accepted manuscript online:
Fund: Project supported by the National Natural Science Foundation of China (Grant No 50275024) and the Research Project of Science of the Education Department of Anhui Province (Grant No 2003kj013zd).
Cite this article:
Xu Pei-Min (徐培民), Wen Bang-Chun (闻邦椿) A new type of global bifurcation in Hénon map 2004 Chinese Physics 13 618
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