Fractal property of generalized M-set with rational number exponent
Article
Liu, S., Cheng, X., Lan, C., Fu, W., Zhou, J., Li, Q. and Gao, G. 2013. Fractal property of generalized M-set with rational number exponent. Applied Mathematics and Computation. 220, pp. 668-675. https://doi.org/10.1016/j.amc.2013.06.096
| Type | Article |
|---|---|
| Title | Fractal property of generalized M-set with rational number exponent |
| Authors | Liu, S., Cheng, X., Lan, C., Fu, W., Zhou, J., Li, Q. and Gao, G. |
| Abstract | Dynamic systems described by fc(z) = z2 + c is called Mandelbrot set (M-set), which is important for fractal and chaos theories due to its simple expression and complex structure. fc(z) = zk + c is called generalized M set (k–M set). This paper proposes a new theory to compute the higher and lower bounds of generalized M set while exponent k is rational, and proves relevant properties, such as that generalized M set could cover whole complex number plane when k < 1, and that boundary of generalized M set ranges from complex number plane to circle with radius 1 when k ranges from 1 to infinite large. This paper explores fractal characteristics of generalized M set, such as that the boundary of k–M set is determined by k, when k = p/q, where p and q are irreducible integers, (GCD(p, q) = 1, k > 1), and that k–M set can be divided into |p–q| isomorphic parts. |
| Keywords | Fractals; Mandelbrot set; Generalized Mandelbrot set; Bound; Rational exponent |
| Research Group | Artificial Intelligence group |
| Publisher | Elsevier |
| Journal | Applied Mathematics and Computation |
| ISSN | 1873-5649 |
| Electronic | 0096-3003 |
| Publication dates | |
| Online | 07 Aug 2013 |
| 01 Sep 2013 | |
| Publication process dates | |
| Deposited | 22 Nov 2013 |
| Accepted | 01 Jan 2013 |
| Output status | Published |
| Publisher's version | License |
| Digital Object Identifier (DOI) | https://doi.org/10.1016/j.amc.2013.06.096 |
| Web of Science identifier | WOS:000324558600066 |
| Language | English |
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