Icon对称群映射的广义M集与混沌吸引子的研究

Research on General M Set and Chaotic Attractor from Icon Symmetric Group

目的旨在解决构造Icon对称群映射广义M集计算量大,耗时长的问题.方法运用无约束最优化理论中的单纯形替换算法,求解使其雅科比矩阵行列式为零的点集作为选定参数下动力平面上的初始点集,考察这个初始点集中轨道有界的点的李雅谱诺夫指数值,构造Dn或Zn对称模型的广义M集.结果运用新算法大大降低了广义M集的构造时间,通过在参数断面λ与ω上构造的Icon对称群广义M集,生成了内部结构完全不同的具有Z5对称的混沌吸引子.结论通过提出新的构造M集的方法更好地描述了I con对称群映射的动力学特性,提高了作图效率.

With the purpose of reducing the calculating time and improve the efficiency for constructing M sets, the research applies the simplex method of the unbound optimization theory for the set of initial iterating points, which make the determinant of Jacobin matrix of the mapping equal to zero. The solution set is the set of initial points with selected parameter on the dynamic plane. According to the Lyapunov exponent of some points in the set, whose orbits are bounded, the general M sets of symmetry mappings Dn or Zn are constructed. The result shows that the new method can greatly decrease the drawing time, and chaotic attractors with the symmetry of Z5 that have different inner structure are created by the parameters chosen from the general M sets, which is constructed on the parameter sectional plane. The authors therefore draw the conclusion that the new method can better describe the dynamic characteristics of the mapping, and greatly improve the efficiency for constructing M sets.