Presentation is loading. Please wait.

Presentation is loading. Please wait.

Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))

Similar presentations


Presentation on theme: "Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))"— Presentation transcript:

1 Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z)) 3. Rational maps (z n + /z n )

2 We’ll investigate chaotic behavior in the dynamical plane (the Julia sets) z 2 + c exp(z) z 2 + /z 2

3 As well as the structure of the parameter planes. z 2 + c exp(z)z 3 + /z 3 (the Mandelbrot set)

4 A couple of subthemes: 1.Some “crazy” mathematics 2.Great undergrad research topics

5 The Fractal Geometry of the Mandelbrot Set

6 How to count The Fractal Geometry of the Mandelbrot Set

7 The Fractal Geometry of the Mandelbrot Set How to add How to count

8 Many people know the pretty pictures...

9 but few know the even prettier mathematics.

10

11

12

13

14

15

16

17

18

19

20

21

22

23 Oh, that's nothing but the 3/4 bulb....

24 ...hanging off the period 16 M-set.....

25 ...lying in the 1/7 antenna...

26 ...attached to the 1/3 bulb...

27 ...hanging off the 3/7 bulb...

28 ...on the northwest side of the main cardioid.

29 Oh, that's nothing but the 3/4 bulb, hanging off the period 16 M-set, lying in the 1/7 antenna of the 1/3 bulb attached to the 3/7 bulb on the northwest side of the main cardioid.

30 Start with a function: x + constant 2

31 Start with a function: x + constant 2 and a seed: x 0

32 Then iterate: x = x + constant 10 2

33 Then iterate: x = x + constant

34 Then iterate: x = x + constant

35 Then iterate: x = x + constant

36 Then iterate: x = x + constant Orbit of x 0 etc. Goal: understand the fate of orbits.

37 Example: x + 1 Seed 0 2 x = 0 0 x =

38 Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x =

39 Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x =

40 Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 4 5 6

41 Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 26 4 x = 5 6

42 Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 26 4 x = big 5 x = 6

43 Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 26 4 x = big 5 x = BIGGER 6

44 Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 26 4 x = big 5 x = BIGGER 6 “Orbit tends to infinity”

45 Example: x + 0 Seed 0 2 x = 0 0 x =

46 Example: x + 0 Seed 0 2 x = x =

47 Example: x + 0 Seed 0 2 x = x =

48 Example: x + 0 Seed 0 2 x = x = 4 5 6

49 Example: x + 0 Seed 0 2 x = “A fixed point”

50 Example: x - 1 Seed 0 2 x = 0 0 x =

51 Example: x - 1 Seed 0 2 x = 0 0 x = -1 1 x =

52 Example: x - 1 Seed 0 2 x = 0 0 x = -1 1 x = 0 2 x =

53 Example: x - 1 Seed 0 2 x = 0 0 x = -1 1 x = 0 2 x = -1 3 x = 4 5 6

54 Example: x - 1 Seed 0 2 x = 0 0 x = -1 1 x = 0 2 x = -1 3 x = 0 4 x = 5 6

55 Example: x - 1 Seed 0 2 x = 0 0 x = -1 1 x = 0 2 x = -1 3 x = 0 4 x = -1 5 x = 0 6 “A two- cycle”

56 Example: x Seed 0 2 x = 0 0 x =

57 Example: x Seed 0 2 x = 0 0 x = x =

58 Example: x Seed 0 2 x = 0 0 x = x = x =

59 Example: x Seed 0 2 x = 0 0 x = x = x = time for the computer!

60 Observation: For some real values of c, the orbit of 0 goes to infinity, but for other values, the orbit of 0 does not escape.

61 Complex Iteration Iterate z + c 2 complex numbers

62 Example: z + i Seed 0 2 z = 0 0 z =

63 Example: z + i Seed 0 2 z = 0 0 z = i 1 z =

64 Example: z + i Seed 0 2 z = 0 0 z = i 1 z = -1 + i 2 z =

65 Example: z + i Seed 0 2 z = 0 0 z = i 1 z = -1 + i 2 z = -i 3 z = 4 5 6

66 Example: z + i Seed 0 2 z = 0 0 z = i 1 z = -1 + i 2 z = -i 3 z = -1 + i 4 z = 5 6

67 Example: z + i Seed 0 2 z = 0 0 z = i 1 z = -1 + i 2 z = -i 3 z = -1 + i 4 z = -i 5 z = 6

68 Example: z + i Seed 0 2 z = 0 0 z = i 1 z = -1 + i 2 z = -i 3 z = -1 + i 4 z = -i 5 z = -1 + i 6 2-cycle

69 Example: z + i Seed i -i

70 Example: z + i Seed i -i

71 Example: z + i Seed i -i

72 Example: z + i Seed 0 2 -i 1 i

73 Example: z + i Seed i -i

74 Example: z + i Seed 0 2 -i 1 i

75 Example: z + i Seed i -i

76 Example: z + i Seed 0 2 -i 1 i

77 Example: z + 2i Seed 0 2 z = 0 0 z =

78 Example: z + 2i Seed 0 2 z = 0 0 z = 2i 1 z = i 2 z = i 3 z = i 4 z = big 5 z = BIGGER 6 Off to infinity

79 Same observation Sometimes orbit of 0 goes to infinity, other times it does not.

80 The Mandelbrot Set: All c-values for which orbit of 0 does NOT go to infinity. Why do we care about the orbit of 0?

81 The Mandelbrot Set: All c-values for which orbit of 0 does NOT go to infinity. As we shall see, the orbit of the critical point determines just about everything for z 2 + c. 0 is the critical point of z 2 + c.

82 Algorithm for computing M Start with a grid of complex numbers

83 Algorithm for computing M Each grid point is a complex c-value.

84 Algorithm for computing M Compute the orbit of 0 for each c. If the orbit of 0 escapes, color that grid point. red = fastest escape

85 Algorithm for computing M Compute the orbit of 0 for each c. If the orbit of 0 escapes, color that grid point. orange = slower

86 Algorithm for computing M Compute the orbit of 0 for each c. If the orbit of 0 escapes, color that grid point. yellow green blue violet

87 Algorithm for computing M Compute the orbit of 0 for each c. If the orbit of 0 does not escape, leave that grid point black.

88 Algorithm for computing M Compute the orbit of 0 for each c. If the orbit of 0 does not escape, leave that grid point black.

89 The eventual orbit of 0

90

91 3-cycle

92 The eventual orbit of 0 3-cycle

93 The eventual orbit of 0 3-cycle

94 The eventual orbit of 0 3-cycle

95 The eventual orbit of 0 3-cycle

96 The eventual orbit of 0 3-cycle

97 The eventual orbit of 0 3-cycle

98 The eventual orbit of 0 3-cycle

99 The eventual orbit of 0 3-cycle

100 The eventual orbit of 0

101

102 4-cycle

103 The eventual orbit of 0 4-cycle

104 The eventual orbit of 0 4-cycle

105 The eventual orbit of 0 4-cycle

106 The eventual orbit of 0 4-cycle

107 The eventual orbit of 0 4-cycle

108 The eventual orbit of 0 4-cycle

109 The eventual orbit of 0 4-cycle

110 The eventual orbit of 0

111

112 5-cycle

113 The eventual orbit of 0 5-cycle

114 The eventual orbit of 0 5-cycle

115 The eventual orbit of 0 5-cycle

116 The eventual orbit of 0 5-cycle

117 The eventual orbit of 0 5-cycle

118 The eventual orbit of 0 5-cycle

119 The eventual orbit of 0 5-cycle

120 The eventual orbit of 0 5-cycle

121 The eventual orbit of 0 5-cycle

122 The eventual orbit of 0 5-cycle

123 The eventual orbit of 0 2-cycle

124 The eventual orbit of 0 2-cycle

125 The eventual orbit of 0 2-cycle

126 The eventual orbit of 0 2-cycle

127 The eventual orbit of 0 2-cycle

128 The eventual orbit of 0 fixed point

129 The eventual orbit of 0 fixed point

130 The eventual orbit of 0 fixed point

131 The eventual orbit of 0 fixed point

132 The eventual orbit of 0 fixed point

133 The eventual orbit of 0 fixed point

134 The eventual orbit of 0 fixed point

135 The eventual orbit of 0 fixed point

136 The eventual orbit of 0 goes to infinity

137 The eventual orbit of 0 goes to infinity

138 The eventual orbit of 0 goes to infinity

139 The eventual orbit of 0 goes to infinity

140 The eventual orbit of 0 goes to infinity

141 The eventual orbit of 0 goes to infinity

142 The eventual orbit of 0 goes to infinity

143 The eventual orbit of 0 goes to infinity

144 The eventual orbit of 0 goes to infinity

145 The eventual orbit of 0 goes to infinity

146 The eventual orbit of 0 goes to infinity

147 The eventual orbit of 0 gone to infinity

148 One reason for the importance of the critical orbit: If there is an attracting cycle for z 2 + c, then the orbit of 0 must tend to it.

149 How understand the of the bulbs? periods

150 How understand the of the bulbs? periods

151 junction point three spokes attached

152 Period 3 bulb junction point three spokes attached

153

154

155 Period 4 bulb

156

157

158 Period 5 bulb

159

160

161 Period 7 bulb

162

163

164

165 Period 13 bulb

166 Filled Julia Set:

167 Fix a c-value. The filled Julia set is all of the complex seeds whose orbits do NOT go to infinity.

168 Example: z 2 Seed: 0 In filled Julia set?

169 Example: z 2 Seed: 0Yes In filled Julia set?

170 Example: z 2 Seed: 0Yes 1 In filled Julia set?

171 Example: z 2 Seed: 0Yes 1 In filled Julia set?

172 Example: z 2 Seed: 0Yes 1 In filled Julia set?

173 Example: z 2 Seed: 0Yes 1 Yes In filled Julia set?

174 Example: z 2 Seed: 0Yes 1 Yes i In filled Julia set?

175 Example: z 2 Seed: 0Yes 1 Yes i In filled Julia set?

176 Example: z 2 Seed: 0Yes 1 Yes i 2i In filled Julia set?

177 Example: z 2 Seed: 0Yes 1 Yes i 2i No In filled Julia set?

178 Example: z 2 Seed: 0Yes 1 Yes i 2i No 5 In filled Julia set?

179 Example: z 2 Seed: 0Yes 1 Yes i 2i No 5No way In filled Julia set?

180 Filled Julia Set for z 2 All seeds on and inside the unit circle. i 1

181 The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic”

182 The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

183 The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

184 The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

185 The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

186 The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

187 The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

188 The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

189 The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

190 The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

191 The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

192 Other filled Julia sets

193 c = 0

194 Other filled Julia sets c = -1

195 Other filled Julia sets c = -1

196 Other filled Julia sets c = -1

197 Other filled Julia sets c = -1

198 Other filled Julia sets c = -1

199 Other filled Julia sets c = -1

200 Other filled Julia sets c = -1

201 Other filled Julia sets c = -1

202 Other filled Julia sets c = i

203 Other filled Julia sets c = i

204 Other filled Julia sets c = i

205 Other filled Julia sets c = i

206 Other filled Julia sets c = i

207 Other filled Julia sets c = i

208 If c is in the Mandelbrot set, then the filled Julia set is always a connected set.

209 Other filled Julia sets But if c is not in the Mandelbrot set, then the filled Julia set is totally disconnected.

210 Other filled Julia sets c =.3

211 Other filled Julia sets c =.3

212 Other filled Julia sets c =.3

213 Other filled Julia sets c =.3

214 Other filled Julia sets c =.3

215 Other filled Julia sets c = i

216 Another reason why we use the orbit of the critical point to plot the M-set: Theorem: (Fatou & Julia) For z 2 + c:

217 Another reason why we use the orbit of the critical point to plot the M-set: Theorem: (Fatou & Julia) For z 2 + c: If the orbit of 0 goes to infinity, the Julia set is a Cantor set (totally disconnected, “fractal dust,” a scatter of uncountably many points.

218 Another reason why we use the orbit of the critical point to plot the M-set: Theorem: (Fatou & Julia) For z 2 + c: But if the orbit of 0 does not go to infinity, the Julia set is connected (just one piece). If the orbit of 0 goes to infinity, the Julia set is a Cantor set (totally disconnected, “fractal dust,” a scatter of uncountably many points.

219 Animations: In and out of M arrangement of the bulbs Saddle node Period doubling Period 4 bifurcation

220 How do we understand the arrangement of the bulbs?

221 How do we understand the arrangement of the bulbs? Assign a fraction p/q to each bulb hanging off the main cardioid. q = period of the bulb

222 Where is the smallest spoke in relation to the “principal spoke”? p/3 bulb principal spoke

223 1/3 bulb principal spoke The smallest spoke is located 1/3 of a turn in the counterclockwise direction from the principal spoke.

224 1/3 bulb 1/3

225 1/3 bulb 1/3

226 1/3 bulb 1/3

227 1/3 bulb 1/3

228 1/3 bulb 1/3

229 1/3 bulb 1/3

230 1/3 bulb 1/3

231 1/3 bulb 1/3

232 1/3 bulb 1/3

233 1/3 bulb 1/3

234 ??? bulb 1/3

235 1/4 bulb 1/3

236 1/4 bulb 1/3 1/4

237 1/4 bulb 1/3 1/4

238 1/4 bulb 1/3 1/4

239 1/4 bulb 1/3 1/4

240 1/4 bulb 1/3 1/4

241 1/4 bulb 1/3 1/4

242 1/4 bulb 1/3 1/4

243 1/4 bulb 1/3 1/4

244 1/4 bulb 1/3 1/4

245 ??? bulb 1/3 1/4

246 2/5 bulb 1/3 1/4

247 2/5 bulb 1/3 1/4 2/5

248 2/5 bulb 1/3 1/4 2/5

249 2/5 bulb 1/3 1/4 2/5

250 2/5 bulb 1/3 1/4 2/5

251 2/5 bulb 1/3 1/4 2/5

252 ??? bulb 1/3 1/4 2/5

253 3/7 bulb 1/3 1/4 2/5

254 3/7 bulb 1/3 1/4 2/5 3/7

255 3/7 bulb 1/3 1/4 2/5 3/7

256 3/7 bulb 1/3 1/4 2/5 3/7

257 3/7 bulb 1/3 1/4 2/5 3/7

258 3/7 bulb 1/3 1/4 2/5 3/7

259 3/7 bulb 1/3 1/4 2/5 3/7

260 3/7 bulb 1/3 1/4 3/7 2/5

261 ??? bulb 1/3 1/4 3/7 2/5

262 1/2 bulb 1/3 1/4 3/7 1/2 2/5

263 1/2 bulb 1/3 1/4 3/7 1/2 2/5

264 1/2 bulb 1/3 1/4 3/7 1/2 2/5

265 1/2 bulb 1/3 1/4 3/7 1/2 2/5

266 ??? bulb 1/3 1/4 3/7 1/2 2/5

267 2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5

268 2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5

269 2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5

270 2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5

271 2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5

272 2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5

273 How to count

274 1/4 How to count

275 1/3 1/4 How to count

276 1/3 1/4 2/5 How to count

277 1/3 1/4 2/5 3/7 How to count

278 1/3 1/4 2/5 3/7 1/2 How to count

279 1/3 1/4 2/5 3/7 1/2 2/3 How to count

280 1/3 1/4 2/5 3/7 1/2 2/3 The bulbs are arranged in the exact order of the rational numbers. How to count

281 1/3 1/4 2/5 3/7 1/2 2/3 The bulbs are arranged in the exact order of the rational numbers. 1/101 32,123/96,787 How to count

282 Animations: Mandelbulbs Spiralling fingers

283 How to add

284 1/2

285 How to add 1/2 1/3

286 How to add 1/2 1/3 2/5

287 How to add 1/2 1/3 2/5 3/7

288 += 1/2 + 1/3 = 2/5

289 += 1/2 + 2/5 = 3/7

290 22 1/2 0/1 Here’s an interesting sequence:

291 22 1/2 0/1 Watch the denominators 1/3

292 22 1/2 0/1 Watch the denominators 1/3 2/5

293 22 1/2 0/1 Watch the denominators 1/3 2/5 3/8

294 22 1/2 0/1 Watch the denominators 1/3 2/5 3/8 5/13

295 22 1/2 0/1 What’s next? 1/3 2/5 3/8 5/13

296 22 1/2 0/1 What’s next? 1/3 2/5 3/8 5/13 8/21

297 22 1/2 0/1 The Fibonacci sequence 1/3 2/5 3/8 5/13 8/21 13/34

298 The Farey Tree

299 How get the fraction in between with the smallest denominator?

300 The Farey Tree How get the fraction in between with the smallest denominator? Farey addition

301 The Farey Tree

302

303 .... essentially the golden number

304 Another sequence (denominators only) 1 2

305 Another sequence (denominators only) 1 2 3

306 Another sequence (denominators only)

307 Another sequence (denominators only)

308 Another sequence (denominators only)

309 Another sequence (denominators only)

310 sequence Devaney

311 The Dynamical Systems and Technology Project at Boston University website: math.bu.edu/DYSYS: Have fun! Mandelbrot set explorer; Applets for investigating M-set; Applets for other complex functions; Chaos games, orbit diagrams, etc.

312 Farey.qt Farey tree D-sequence Continued fraction expansion Far from rationals Other topics Website

313 Continued fraction expansion Let’s rewrite the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,..... as a continued fraction:

314 Continued fraction expansion 1212 = 1212 the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

315 Continued fraction expansion 1313 = the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

316 Continued fraction expansion 2525 = the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

317 Continued fraction expansion 3838 = the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

318 Continued fraction expansion = the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

319 Continued fraction expansion = the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

320 Continued fraction expansion = the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

321 Continued fraction expansion = essentially the 1/golden number the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

322 We understand what happens for = 1a1a + 1b1b + 1c1c 1d1d + 1e1e + 1f1f + 1g1g + where all entries in the sequence a, b, c, d,.... are bounded above. But if that sequence grows too quickly, we’re in trouble!!! etc.

323 The real way to prove all this: Need to measure: the size of bulbs the length of spokes the size of the “ears.”

324 There is an external Riemann map : C - D C - M taking the exterior of the unit disk to the exterior of the Mandelbrot set.

325 takes straight rays in C - D to the “external rays” in C - M 01/2 1/3 2/3 external ray of angle 1/3

326 Suppose p/q is periodic of period k under doubling mod 1: period 2 period 3 period 4

327 Suppose p/q is periodic of period k under doubling mod 1: period 2 period 3 period 4 Then the external ray of angle p/q lands at the “root point” of a period k bulb in the Mandelbrot set.

328 0 1/3 2/3 0 is fixed under angle doubling, so lands at the cusp of the main cardioid.

329 0 1/3 2/3 1/3 and 2/3 have period 2 under doubling, so and land at the root of the period 2 bulb. 2

330 0 1/3 2/3 And if lies between 1/3 and 2/3, then lies between and. 2

331 0 1/3 2/3 So the size of the period 2 bulb is, by definition, the length of the set of rays between the root point rays, i.e., 2/3-1/3=1/3. 2

332 0 1/3 2/3 1/15 and 2/15 have period 4, and are smaller than 1/ /7 2/7 3/7 4/7 5/7 6/ /15 2/15

333 0 1/3 2/3 1/15 and 2/15 have period 4, and are smaller than 1/ /7 2/7 3/7 4/7 5/7 6/ /15 2/15

334 0 1/3 2/3 1/7 2/7 3/7 4/7 5/7 6/ /15 2/15 3/15 and 4/15 have period 4, and are between 1/7 and 2/7....

335 0 1/3 2/3 3/15 and 4/15 have period 4, and are between 1/7 and 2/ /7 2/7 3/7 4/7 5/7 6/ /15 2/15

336 1/72/7 3/15 and 4/15 have period 4, and are between 1/7 and 2/7....

337 1/72/7 3/15 and 4/15 have period 4, and are between 1/7 and 2/ /154/15

338 So what do we know about M? All rational external rays land at a single point in M.

339 So what do we know about M? All rational external rays land at a single point in M. Rays that are periodic under doubling land at root points of a bulb. Non-periodic rational rays land at Misiurewicz points (how we measure length of antennas).

340 So what do we know about M? “Highly irrational” rays also land at unique points, and we understand what goes on here. “Highly irrational" = “far” from rationals, i.e.,

341 So what do we NOT know about M? But we don't know if irrationals that are “close” to rationals land. So we won't understand quadratic functions until we figure this out.

342 MLC Conjecture: The boundary of the M-set is “locally connected” --- if so, all rays land and we are in heaven!. But if not......

343 The Dynamical Systems and Technology Project at Boston University website: math.bu.edu/DYSYS Have fun!

344 A number is far from the rationals if:

345

346 This happens if the “continued fraction expansion” of has only bounded terms.


Download ppt "Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))"

Similar presentations


Ads by Google