The generalized M–J sets for bicomplex numbers

• Published in: Nonlinear dynamics 2013-04, Vol.72 (1-2), p.17-26
• Main Authors: Wang, Xing-yuan, Song, Wen-jing
• Format: Article
• Language: English
• Subjects: Automotive Engineering; Classical Mechanics; Control; Cross-sections; Dynamical Systems; Engineering; Mapping; Mechanical Engineering; Multiplication; Original Paper; Set theory; Vibration
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-012-0686-6

We explained the theory about bicomplex numbers, discussed the precondition of that addition and multiplication are closed in bicomplex number mapping of constructing generalized Mandelbrot–Julia sets (abbreviated to M–J sets), and listed out the definition and constructing arithmetic of the generalized Mandelbrot–Julia sets in bicomplex numbers system. And we studied the connectedness of the generalized M–J sets, the feature of the generalized Tetrabrot, and the relationship between the generalized M sets and its corresponding generalized J sets for bicomplex numbers in theory. Using the generalized M–J sets for bicomplex numbers constructed on computer, the author not only studied the relationship between the generalized Tetrabrot sets and its corresponding generalized J sets, but also studied their fractal feature, finding that: (1) the bigger the value of the escape time is, the more similar the 3-D generalized J sets and its corresponding 2-D J sets are; (2) the generalized Tetrabrot set contains a great deal information of constructing its corresponding 3-D generalized J sets; (3) both the generalized Tetrabrot sets and its corresponding cross section make a feature of axis symmetry; and (4) the bigger the value of the escape time is, the more similar the cross section and the generalized Tetrabrot sets are.