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Improved generalized cell mapping for global analysis of dynamical systems

中国科学:技术科学 2010年第3期 348-348页

作者： ZOU HaiLin & XU JianXue Institute of Nonlinear Dynamics,School of Aerospace,Xi’an Jiaotong University,Xi’an 710049,China

Three main parts of generalized cell mapping are improved for global analysis.A simple method,which is not based on the theory of digraphs,is presented to locate complete self-cycling sets that correspond to attractors and unstable invariant sets involving saddle,unstable periodic orbit and chaotic *** for complete self-cycling sets is developed to locate attractors and unstable invariant sets with high degree of accuracy,which can start with a coarse cell structure.A nonuniformly interior-and-boundary sampling technique is used to make the refinement *** homeomorphic dissipative dynamical systems,a controlled boundary sampling technique is presented to make generalized cell mapping method with refinement extremely accurate to obtain invariant *** laws of group absorption probability and expected absorption time are introduced into generalized cell mapping,and then an optimal order for quantitative analysis of transient cells is established,which leads to the minimal computational *** improved method is applied to four examples to show its effectiveness in global analysis of dynamical systems.

关键词： global analysis generalized cell mapping invariant set chaotic saddle crisis

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Study of the Wada fractal boundary and indeterminate crisis

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Chinese Physics B 2002年第11期11卷 1115-1123页

作者： Hong Ling(洪灵) Xu Jian-Xue(徐健学) Institute of Nonlinear Dynamics Xi'an Jiaotong University Xi'an 710049 China

By using the generalized cell mapping digraph (GCMD) method, we study bifurcations governing the escape ofperiodically forced oscillators in a potential well, in which a chaotic saddle plays an extremely important role. In thispaper, we find the chaotic saddle, and we demonstrate that the chaotic saddle is embedded in a strange fractal boundarywhich has the Wada property, that any point on the boundary of that basin is also simultaneously on the boundary ofat least two other basins. The chaotic saddle in the Wada fractal boundary, by colliding with a chaotic attractor, leadsto a chaotic boundary/crisis with a global indeterminate outcome which presents an extreme form of indeterminacy ina dynamical system ./We also investigate the origin and evolution of the chaotic saddle in the Wada fractal boundary,particularly concentrating on its discontinuous bifurcations (metamorphoses). We demonstrate that the chaotic saddlein the Wada fractal boundary is created by the collision between two chaotic saddles in different fractal *** a final escape bifurcation, there only exists the attractor at infinity; a chaotic saddle with a beautiful pattern isleft behind in phase space.

关键词： global analysis generalized cell mapping indeterminate chaotic boundary crisis chaoticsaddle Wada fractal boundary

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