複素写像関数とマンデルブロ集合

概要

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The characteristics of complex dynamical system are investigated by using our original map functions where a parameter m is contained. The map functions are of (m-1) order and parabolic. The diagram of the map function in case of m=3, which agrees with the logistic equation, is symmetrical. The presented complex map function has a zero as one of the fixed point and the (m-2) fixed points which are spaced equally round the circumference. The regions of A=a+bi in which the attractors are generated are divided into two and they form circles. One is generated by the fixed point for Z=0 and the other is generated by the fixed points for Z≠0. These circles are inscribed to each other in case of m<2 and circumscribed in case of m>2. Thus, such as behaviors of a bifurcation for one dimensional chaos, the circle of attractor alters under the standard of value of m=2. The two intersections of the circle and the real axis, and the center of the circle are agreed with A_0, A_1 and A_<t1> points decided by the bifurcation. MANDELBROT sets show the patterns formed by the numbers of (m-2) petals. The patterns of MANDELBROT sets in case of integers of m≧3 are symmetry for the real axis, but it is not symmetry in case of integers of m<3. Furthermore it is also not symmetry when m is not integer.
明治大学の論文