中文模板

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1 ISSN , CODEN RUXUEW Journal of Software, Vol17, Supplement, November 2006, pp70 77 http//wwwjosorgcn 2006 by Journal of Software All rights reserved Tel/Fax ,2+, 1,2,3 1,2, 1 (, ) 2 ( ), ) 3 (, ) Iterated Fractal Based on Distance Ratio ZHANG Xi-Zhe 1,2+, LÜ Tian-Yang 1,2,3, WANG Zheng-Xuan 1,2 1 (College of Computer Science and Technology, Jilin University, Changchun , China) 2 (Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education (Jilin University), Changchun , China) 3 (Department of Computer Science and Technology, Harbin Engineering University, Harbin , China) + Corresponding author Phn , zxzok@163com Zhang XZ, Lü TY, Wang ZX Iterated fractal based on distance ratio Journal of Software, 2006,17(Suppl) http//wwwjosorgcn/ /17/s70htm Abstract The escape time algorithm cannot render the convergence region of mapping, so there are some black regions in escape time fractal In this paper, a novel method is presented to construct fractal image, which is named the distance ratio iteration method This method performs iteration on two points and render fractal image by using their distance ratio convergence times Taking complex mapping z z α +c as example, the generalized Mandelbrot and Julia sets are constructed based on distance ratio and their visual properties are analyzed The result fractal image has complex and self-similarity structure in inner convergence region It is proved that the boundary of distance ratio fractal is the same as M-J set when α>0, and some visual structure of it with various exponent α are discussed When α<0, the generalized Mandelbrot and Julia set based on distance ratio have some complex structures which M-J set does not have Key words fractal; complex mapping; Julia set; Mandelbrot set; distance ratio,, z z α +c M-J M-J, α>0 M-J, α<0 M-J ; ;Julia ;Mandelbrot ; Mandelbrot [1] M, [2] Mandelbrot Julia [3 5] Received ; Accepted

2 71 M-J [6,7] [8] ; M-J [9,10] [11] M-J,, 0,,,, M-J,Hooper [12] ε M ;Philip [13], Mandelbrot, [6] M-J [14] ; Lyapunov M [15],,,,,,, M-J, Mandelbrot Julia M J, 1 2, M-J, M-J,,,, z z α +c, J Julia ; M, M Fig1 DRJ for z z i Fig2 DRM for z z 3 +c 1 z z i J 2 z z 3 +c M 1 2 J, Julia 3 M, M 4 1 fc C, f n (x)=f o f n 1 (x), f 0 (x)=x,n z 1 z 2 C, L(z 1,z 2 ) Lz (, z) = ( ) ( ) f z1 f z 2 z z z 1,z 2, (1)

3 72 Journal of Software Vol17, Supplement, November 2006 n L ( z, z ) = n f ( z ) f ( z ) n n 1 n 1 f z1 f z2 ( ) ( ) f, z z,, 1 f z, A(z ), z, z A( z ), lim L k ( z, z ) = f ( ) k z 2 ε, z, z A( z ), k k>k, L k ( z, z ) f ( z ) < ε 1 ; 2 ε,,,,, z Julia, J (distance ratio Julia set, DRJ), k { 1, 2 { lim ( 1, 2) ( )}} k DRJ = z z C L z z f z (3), (3), z, (3) { { lim k (, }} ) ( ) k DRJ = z C L z z f z (4) z,ε, z k, z L k 1 k ( z, z ) f ( z ) > ε, L ( z, z ) f ( z ) ε k k k, k, zone(k), J k DRJ = Uzone( k) 3 f, zone(k+1) z, z zone(k), f 1 ( z ) = k = 1 z zone(k+1), k k L ( z, z ) f ( z ) = L ( f 1 ( z ), z ) f ( z ) = L k 1 ( z, z ) f ( z ) > ε, z (2) (5) (6) L k + 1 ( z, z ) f ( z ) = L k+ 1 ( f 1 ( z ), z ) f ( z ) k = L ( z, z ) f ( z ) ε,z zone(k), z zone(k) 1 f 1 (zone(k))=zone(k+1) 3 zone(k) zone(k+1),, zone(k) zone(k+1) 1, k, 2 J 3 1, zone(k) zone(k+1) zone(1),, zone(2),zone(3),, zone(n),n, J f ( z) f ( z ) (5),zone(1) f ( z ) ε, zone(1) z z, zone(1), zone(k),

4 73, (1) k=1,n=10 000, zone(k); (2) zone(k) zone(k+1); (3) k<n, k=k+1, (2); (4) zone(1),zone(2),,zone(n 1),,zone(n), J 3(c) z z i, 3(a) 3(b) 5(a) Julia,DRJ Julia, Julia, (a) (b) Fig3 (c) DRJ for complex mapping f (z)=z i 3 f (z)=z i J 3,DRJ,,, 3(b),, 3(c), zone(k), k 1, DRJ Julia, 4 α>0,drj f, Julia,zone( )=J( f ),,,,,,DRJ Julia, DRJ Julia,DRJ,, α>0, Julia [12 14], Julia α<0, z z α +c, [16,17] α<0 Julia,,, Julia α<0, 4 z z i J 5(b) Julia,, DRJ,, Julia,DRJ Julia

5 74 Journal of Software Vol17, Supplement, November 2006, Julia,, Fig4 DRJ for z z i and zoomed image 4 z z i J b 3 M (a) z z i (b) z z i Fig5 Julia sets generated by escape time algorithm 5 Julia f c, c f f c c, M (distance ratio Mandelbrot set, DRM), { { lim k (0, }} ) ( ) c k DRM = c C L z f z (7) (7) (0,0) z, (0,0) f c (z)=z α +c, z, M, DRM { {, k+ }} 1 (0, ) k (0, ) c c DRM = c C k N L z L z ε (8) (8) DRM,, k, k, D, 1 n=10000; ε= ;k=1 2 D c, f(z,c); z 1 =(0,0),z 2 f(z,c) z 3 k<n, f k (z 1 )= f ( f k 1 (0,c)), f k (z 2 )= f ( f k 1 (z,c)), 6 k+ 1 k+ 1 f (0) f ( z ) k 4 L ( z1, z2 ) = k k f (0) f ( z ) 5 L k (0,z ) L k 1 (0,z ) >ε, k=k+1, 3 6 k, c 7 2~6, D, f M 6(a) z z 2 +c M, 6(b)

6 75 6(c) 6(d) α=5 α=14, M,,,,, 6(a), ;, DRM,DRM 6 [18] M,, α>1, M DRM M, M,, DRM, [19,20] (a) α=2 (b) Zoomed image for α=2 (c) α=5 (d) α=14 (a) α=2 (b) α=2 (c) α=5 (d) α=14 Fig6 DRM for z z α +c 6 z z α +c M α<0, DRM, DRM, 7 α= 2 DRM DRM 1 DRM, 2, (a) (b) (c) (d) Fig7 Distance ratio fractal image for complex mapping z z 2 +c 7 α= 2 DRM z z 2 +c,, n, n=1+ α n, n,,,

7 76 Journal of Software Vol17, Supplement, November α= 2 M, 7(a), DRM M DRM, 9,, M M, DRM M,, α<0, M, Fig8 Mandelbrot set for α= 2 Fig9 DRM and M-set for α= 2 8 α= 2 M 9 α= 2 M DRM α<0,drm α +1,, ; α +1, ; α +1,, ;DRM M, 4, M-J,,,, M-J M-J α>1, M-J, M-J ; α<0, M M, M,, Julia,, References [1] Mandelbrot BB The Fractal Geometry of Nature WH Freeman, [2] Spehar B, Clifford C, Newell B, Taylor R Universal aesthetic of fractals Computers & Graphics, 2003,27(5) [3] Liaw SA Find the mandelbrot-like sets in any mapping Fractals, 2002,10(2) [4] Buchanan W, Gomatam J, Steves B Generalized Mandelbrot sets for meromorphic complex and quaternionic maps Int l Journal of Bifurcation and Chaos, 2002,12(8) [5] Leys J Sphere inversion fractals Computers & Graphics, 2005,29(3) [6] Wang XY Researches on some internal structures of the general Mandelbrot sets for negative real index number Journal of Computer-Aided Design & Computer Graphics, 2001,13(10) (in Chinese with English abstract) [7] Wang XY, Liu XD, Zhu WY, Gu SS Analysis of c-plane fractal images from z z α +c for (α<0) Fractals, 2000,8(3) [8] Wang XY, Shi QJ The generalized M-J sets of complex exponent mapping Progress in Natural Science, 2004,14(8) (in Chinese with English abstract) [9] Chen N, Zhu WYM-Fractal image from complex mapping z z w +c (w=α+iβ) in the complex C-plane Journal of Computer Research and Development, 1997,34(12) (in Chinese with English abstract)

8 77 [10] Chen N, Zhu WY J fractal image constructed from complex mapping z z w +c(w=α+iβ) Journal of Computer Research and Development, 1998,35(11) (in Chinese with English abstract) π [11] Chen N, Zhu WY General Mandelbrot set and Julia set constructed by complex mapping i e 2 ( z m ) + c Journal of Computer Research and Development, 1997,34(5) (in Chinese with English abstract) [12] Hooper KJ A note on some internal structures of the Mandelbrot set Computers & Graphics, 1991,15(2) [13] Pickover CA Chaos and Fractals A Computer Graphical Journey Elsevier Science, [14] Wang XY Fractal structures of the non-boundary region of the generalized Mandelbrot set Progress in Natural Science, 2001, 11(9) (in Chinese with English abstract) [15] Wang XY, Chang PJ Using lyapunov exponent and period points to analysis generalized M-J sets Progress in Natural Science, 2005,15(4) (in Chinese with English abstract) [16] Dhurandhar SV Analysis of z-plane fractal images from z z α +c for α<0 Computers & Graphics, 1993,17(1)89 94 [17] Gujar UG, Bhavsar VC, Vangala N Fractal images from z z α +c in the complex z-plane Computers & Graphics, 1992,16(1) [18] Gujar UG, Bhavsar VC Fractals from z z α +c in the complex c-plane Computers & Graphics, 1991,15(3) [19] Sasmor JC Fractals for functions with rational exponent Computers & Graphics, 2004,28(4) [20] Huang YN Global analysis of the Mandelbrot set and the general Mandelbrot set China Science (Series A), 1991,(8) (in Chinese with English abstract) [6] M,2001,13(10) [8], M-J,2004,14(8) [9], z z w +c (w=α+iβ) M,1997,34(12) [10], z z w +c (w=α+iβ) J,1998,35(11) [11], π i e 2 ( z m ) + c Mandelbrot Julia,1997,34(5) [14] C M,2001,11(9) [15], Lyapunov M-J,2005,15(4) [20] Mandelbrot (A ),1991,(8) (1978 ),,,, (1945 ),,,,,, (1979 ),,,,,

% GIS / / Fig. 1 Characteristics of flood disaster variation in suburbs of Shang

% GIS / / Fig. 1 Characteristics of flood disaster variation in suburbs of Shang 20 6 2011 12 JOURNAL OF NATURAL DISASTERS Vol. 20 No. 6 Dec. 2011 1004-4574 2011 06-0094 - 05 200062 1949-1990 1949 1977 0. 8 0. 03345 0. 01243 30 100 P426. 616 A Risk analysis of flood disaster in Shanghai