Symmetry structure of the hyperbolic bifurcation without reflection of periodic orbits in the standard map |
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| Institution: | 1. National Institute for Fusion Science, Toki 509-5292, Japan;2. College of Engineering, Chubu University, Kasugai 487-8501, Japan;1. School of Cyberspace Security, Beijing University of Posts and Telecommunications, Beijing 100876, China;2. Beijing Electronic Science and Technology Institute, Beijing 100070, China;3. Xidian University, Xi’an, Shaanxi 710071, China;1. School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China;2. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China |
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| Abstract: | For the area preserving maps, the linearized tangent map determines the stability of the fixed point. When the trace of the tangent map is less than −2, the fixed point is inversion hyperbolic, thus the subsequent points of mapping alternate across the destabilized fixed point. That is to say, the fixed point undergoes periodic doubling bifurcation. While for the trace of the tangent map is larger than +2, the fixed point undergoes the hyperbolic bifurcation without reflection. Here, the processes of the hyperbolic bifurcation without reflection in the standard map have been examined in terms of the higher order symmetry in the momentum inversion. It is shown that the higher order symmetry lines approach asymptotically to the separatrix of the hyperbolic fixed point, and the existing symmetry lines cannot determine the structure of the periodic islands born after the hyperbolic bifurcation without reflection. |
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