1167:
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1149:
1140:
1131:
1122:
1113:
1104:
28:
40:
20:
48:
880:
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just shifting the iteration sequence, and keeping the starting value 0.5. In practice, shifting this sequence leads to changes in the fractal, as some branches get covered by others. For instance, the
Lyapunov fractal for the iteration sequence AB (see top figure on the right) is not perfectly symmetric with respect to
285:
of the iterative function. The other (even complex valued) critical points of the iterative function during one entire round are those that pass through the value 0.5 in the first round. A convergent cycle must attract at least one critical point. Therefore, all convergent cycles can be obtained by
1180:
660:
2003:
274:. For larger values, the interval is no longer stable, and the sequence is likely to be attracted by infinity, although convergent cycles of finite values continue to exist for some parameters. For all iteration sequences, the diagonal
1183:
1188:
1186:
1182:
1181:
1187:
1204:
characters, e.g. "ABBBCA" for a 3D fractal, which can be visualized either as 3D object or as an animation showing a "slice" in the C direction for each animation frame, like the example given here.
875:{\displaystyle \lambda =\lim _{N\rightarrow \infty }{1 \over N}\sum _{n=1}^{N}\log \left|{dx_{n+1} \over dx_{n}}\right|=\lim _{N\rightarrow \infty }{1 \over N}\sum _{n=1}^{N}\log |r_{n}(1-2x_{n})|}
998:
1185:
651:
402:
198:
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573:
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537:
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471:
438:
1066:
921:
331:
272:
211:
2021:
1184:
1375:
Markus, Mario; Hess, Benno (1998). "Chapter 12. Lyapunov exponents of the logistic map with periodic forcing". In
Clifford A. Pickover (ed.).
1414:
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1961:
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219:
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1986:
51:
Generalized
Lyapunov logistic fractal with iteration sequence BBBBBBAAAAAA, in the growth parameter region (
2035:
1594:
1460:
215:
1820:
1512:
339:
1981:
1976:
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109:
fractal is constructed by mapping the regions of stability and chaotic behaviour (measured using the
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1716:
1651:
1599:
1584:
1517:
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Markus, Mario; Hess, Benno (1989). "Lyapunov exponents of the logistic map with periodic forcing".
223:
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151:
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1946:
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80:
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545:
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Lyapunov fractals can be calculated in more than two dimensions. The sequence string for a
1039:
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27:
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Generalized
Lyapunov logistic fractal with iteration sequence AABAB, in the region × .
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Standard
Lyapunov logistic fractal with iteration sequence AB, in the region × .
2016:
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Markus, Mario (1990). "Chaos in Maps with
Continuous and Discontinuous Maxima".
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47:
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formed by successive terms in the string, repeated as many times as necessary.
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Markus, Mario, "Die Kunst der
Mathematik", Verlag Zweitausendeins, Frankfurt
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1534:
303:
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Lyapunov fractals were discovered in the late 1980s by the
Germano-Chilean
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is always the same as for the standard one parameter logistic function.
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1673:
1476:
83:
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1744:
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Choose a string of As and Bs of any nontrivial length (e.g., AABAB).
35:
in the form of a swallow. Iteration sequence AB, in the region x .
1178:
46:
38:
26:
18:
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The sequence is usually started at the value 0.5, which is a
1445:
1200:-dimensional fractal has to be built from an alphabet with
1192:
Animation of a 3D Lyapunov fractal with the sequence ABBBCA
1312:
Dewdney, A.K. (1991). "Leaping into
Lyapunov Space".
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Repeat steps (3–7) for each point in the image plane.
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in which the degree of the growth of the population,
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Lyapunov fractals are generally drawn for values of
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992:
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567:
531:
498:
465:
432:
396:
325:
306:for computing Lyapunov fractals works as follows:
266:
192:
166:
124:
1379:Chaos and Fractals. A Computer Graphical Journey
775:
671:
993:{\displaystyle r_{0}(1-2x_{0})=r_{n}\cdot 0=0}
214:. They were introduced to a large public by a
1461:
1428:EFG's Fractals and Chaos – Lyapunov Exponents
903:is approximated by choosing a suitably large
8:
212:Max Planck Institute of Molecular Physiology
646:{\displaystyle x_{n+1}=r_{n}x_{n}(1-x_{n})}
94:, periodically switches between two values
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2022:List of fractals by Hausdorff dimension
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148:. In the images, yellow corresponds to
1290:
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140:plane for given periodic sequences of
174:(stability), and blue corresponds to
7:
1393:10.1016/B978-0-444-50002-1.X5000-0
1326:10.1038/scientificamerican0991-178
923:and dropping the first summand as
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681:
14:
2004:How Long Is the Coast of Britain?
86:derived from an extension of the
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1147:
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397:{\displaystyle (a,b)\in \times }
2028:The Fractal Geometry of Nature
1055:
1043:
962:
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842:
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782:
678:
656:Compute the Lyapunov exponent:
640:
621:
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261:
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1:
193:{\displaystyle \lambda >0}
167:{\displaystyle \lambda <0}
1347:10.1016/0097-8493(89)90019-8
2044:Chaos: Making a New Science
575:, and compute the iterates
2093:
1068:according to the value of
1026:{\displaystyle x_{0}=0.5}
568:{\displaystyle x_{0}=0.5}
1293:, pp. 481, 483 and
1081:{\displaystyle \lambda }
896:{\displaystyle \lambda }
220:recreational mathematics
125:{\displaystyle \lambda }
77:Markus–Lyapunov fractals
1439:The Chaos Hypertextbook
532:{\displaystyle S_{n}=B}
499:{\displaystyle r_{n}=b}
466:{\displaystyle S_{n}=A}
433:{\displaystyle r_{n}=a}
313:Construct the sequence
2036:The Beauty of Fractals
1335:Computers and Graphics
1295:Markus & Hess 1998
1235:Markus & Hess 1989
1222:Markus & Hess 1989
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216:science popularization
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1383:. Elsevier. pp.
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1061:{\displaystyle (a,b)}
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1982:Lewis Fry Richardson
1977:Hamid Naderi Yeganeh
1767:Burning Ship fractal
1699:Weierstrass function
1356:Computers in Physics
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407:Define the function
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1740:Space-filling curve
1717:Multifractal system
1600:Space-filling curve
1585:Sierpinski triangle
1314:Scientific American
224:Scientific American
59:) in × , known as
1967:Aleksandr Lyapunov
1947:Desmond Paul Henry
1911:Self-avoiding walk
1906:Percolation theory
1550:Iterated function
1491:Fractal dimensions
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2010:Coastline paradox
1987:Wacław Sierpiński
1972:Benoit Mandelbrot
1896:Fractal landscape
1804:Misiurewicz point
1709:Strange attractor
1590:Apollonian gasket
1580:Sierpinski carpet
1415:978-3-86150-767-3
1402:978-0-444-50002-1
1368:10.1063/1.4822940
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916:{\displaystyle N}
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326:{\displaystyle S}
111:Lyapunov exponent
73:Lyapunov fractals
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1927:Michael Barnsley
1794:Lyapunov fractal
1652:Sierpiński curve
1605:Blancmange curve
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1435:"Lyapunov Space"
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267:{\displaystyle }
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242:in the interval
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33:Lyapunov fractal
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1991:
1942:Felix Hausdorff
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1879:Brownian motion
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1694:Pythagoras tree
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1540:Self-similarity
1484:Characteristics
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1175:More dimensions
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1096:More Iterations
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75:(also known as
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16:Type of fractal
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1782:Newton fractal
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1679:Vicsek fractal
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1570:Koch snowflake
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1433:Elert, Glenn.
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1422:External links
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1341:(4): 553–558.
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1320:(3): 130–132.
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1280:, p. 486.
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31:Detail of the
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1315:
1310:
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1304:
1296:
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1266:
1260:
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1253:
1247:
1244:
1240:
1236:
1230:
1227:
1223:
1217:
1214:
1207:
1205:
1203:
1199:
1174:
1168:
1163:
1159:
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1141:
1136:
1132:
1127:
1123:
1118:
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1095:
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1075:
1052:
1049:
1046:
1035:
1020:
1017:
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987:
984:
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978:
973:
969:
965:
957:
953:
949:
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935:
931:
910:
890:
883:In practice,
859:
855:
851:
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837:
833:
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675:
667:
664:
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631:
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616:
612:
606:
602:
598:
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587:
583:
562:
559:
554:
550:
541:
526:
523:
518:
514:
493:
490:
485:
481:
460:
457:
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448:
427:
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419:
415:
406:
388:
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335:
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258:
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222:published in
221:
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187:
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161:
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155:
147:
143:
139:
135:
119:
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108:
103:
101:
97:
93:
89:
85:
82:
81:bifurcational
78:
74:
70:
62:
58:
54:
49:
41:
34:
29:
21:
2056:Chaos theory
2051:Kaleidoscope
2042:
2034:
2026:
1952:Gaston Julia
1932:Georg Cantor
1793:
1757:Escape-time
1689:Gosper curve
1637:Lévy C curve
1622:Dragon curve
1501:Box-counting
1438:
1378:
1359:
1355:
1338:
1334:
1317:
1313:
1285:
1272:
1259:
1252:Dewdney 1991
1246:
1229:
1216:
1201:
1197:
1195:
301:
291:
287:
280:
275:
239:
235:
233:
208:Mario Markus
202:
145:
141:
137:
133:
104:
99:
95:
91:
88:logistic map
76:
72:
66:
60:
56:
52:
32:
2047:(1987 book)
2039:(1986 book)
2031:(1982 book)
2017:Fractal art
1937:Bill Gosper
1901:Lévy flight
1647:Peano curve
1642:Moore curve
1528:Topological
1513:Correlation
1291:Markus 1990
1278:Markus 1990
1265:Markus 1990
1239:Markus 1990
218:article on
69:mathematics
61:Zircon Zity
1855:Orbit trap
1850:Buddhabrot
1843:techniques
1831:Mandelbulb
1632:Koch curve
1565:Cantor set
1362:(5): 481.
1305:References
230:Properties
1962:Paul Lévy
1841:Rendering
1826:Mandelbox
1772:Julia set
1684:Hexaflake
1615:Minkowski
1535:Recursion
1518:Hausdorff
1088:obtained.
1076:λ
979:⋅
947:−
891:λ
849:−
825:
802:∑
786:∞
783:→
721:
698:∑
682:∞
679:→
665:λ
628:−
377:×
359:∈
304:algorithm
298:Algorithm
226:in 1991.
210:from the
205:physicist
200:(chaos).
182:λ
156:λ
132:) in the
120:λ
2077:Fractals
2071:Category
1872:fractals
1759:fractals
1727:L-system
1669:T-square
1477:Fractals
107:Lyapunov
84:fractals
1821:Tricorn
1674:n-flake
1523:Packing
1506:Higuchi
1496:Assouad
136:−
1920:People
1870:Random
1777:Filled
1745:H tree
1664:String
1552:system
1413:
1399:
473:, and
79:) are
1996:Other
1387:-78.
1208:Notes
276:a = b
1411:ISBN
1397:ISBN
1289:See
1276:See
1263:See
1250:See
1237:and
1233:See
1220:See
1000:for
542:Let
302:The
290:and
238:and
185:>
159:<
144:and
98:and
1389:doi
1364:doi
1343:doi
1322:doi
1318:265
1021:0.5
822:log
776:lim
718:log
672:lim
563:0.5
506:if
440:if
67:In
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1339:13
1337:.
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105:A
102:.
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1324::
1297:.
1254:.
1241:.
1202:n
1198:n
1056:)
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1050:,
1047:a
1044:(
1033:.
1018:=
1013:0
1009:x
988:0
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982:0
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958:0
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911:N
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860:n
856:x
852:2
846:1
843:(
838:n
834:r
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817:N
812:1
809:=
806:n
796:N
793:1
780:N
772:=
768:|
760:n
756:x
752:d
745:1
742:+
739:n
735:x
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725:|
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705:=
702:n
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668:=
653:.
641:)
636:n
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539:.
527:B
524:=
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404:.
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380:[
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250:[
240:B
236:A
188:0
162:0
146:b
142:a
138:b
134:a
100:B
96:A
92:r
63:.
57:B
55:,
53:A
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