Chapter (dynamical system) a n+1 = 2a n ; a 0 = 1. a n = 2 n f(x) = 2x a n+1 = f(a n ) a 1 = f(a 0 ), a 2 = f(f(a 0 )) a 3 = f(f(f(a

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1 Chapter (dynamical system) a n+1 = 2a n ; a 0 = 1. a n = 2 n f(x) = 2x a n+1 = f(a n ) a 1 = f(a 0 ), a 2 = f(f(a 0 )) a 3 = f(f(f(a 0 ))) f a 0 1

2 *1 {a n } R 0, ±1, ±2, x 1 f(x) = 2x R f (dynamical system) *2 x 0 R x 0 f 2x 0 x 0 f f(x 0 ) f 2 2 x 0 f f(f(x 0 )) f f x 0 (orbit) {a n } a 0 = 1 f f f f f n f n {f n (x 0 )} *3 f f(x) 2 f 10 (x) x 1024 Mathematica 14.2 p y = f(x) p f(p) f(f(p)) f(f(f(p)))... y = f(x) xy (p, p) (f(p), f(p)) *1 3 *2 (discrete dynamical system) *3 f n (x) f f f f(x) f(x) n {f(x)} n

3 Step 0 y = f(x) y = x Step 1 (p, p) y = f(x) (p, f(p)) Step 2 (p, f(p)) y = x (f(p), f(p)) Step 1 2 y = x p 14.1 (graphical analysis) web diagram y y = x f(p) p y = f(x) O p f(p) x 14.1 f(x) (p, p) (f 2 (p).f 2 (p)) f(x) = 2x x 0 = ±1/ Mathematica ListLinePlot Show [1] Step 0 y = f(x) = 2x y = x In[1]:= f[x_] := 2 x; gr = Plot[{f[x], x}, {x, -2, 8}, AspectRatio -> Automatic]

4 f(x) = 2x x 0 = 1/4 x 0 = 1/ Out[1]= [2] Step 1 Step 2 (p, p) (p, f(p)) (f(p), f(p)) In[2]:= tateyoko[p_] := {{p, f[p]}, {f[p], f[p]}}; [3] p n In[3]:= weblist[p_, n_] := (w = {{p, p}}; x = p; Do[(w = Join[w, tateyoko[x]]; x = f[x]), {i, 1, n}]; w); [4] In[4]:= weblist[1/2, 3] Out[4]= 1 2, 1 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 4 2

5 [5] In[5]:= webdiag[p_, n_] := ListLinePlot[weblist[p, n], PlotStyle -> Thick, AspectRatio -> Automatic, PlotRange -> All]; PlotRange -> All [6] In[6]:= webdiag[1/2, 3] 4 3 Out[6]= [7] Show In[7]:= Show[gr, webdiag[1/2, 3]] Out[7]= [8] Manipulate p n In[8]:= Manipulate[Show[gr, webdiag[p, n]], {{p, 1}, -1, 4}, {{n, 3}, 0, 10, 1}]

6 6 14 p n 15 Out[8]= n 14.1 ( ) 2 g a (x) = ax(1 x) 0 a 4 [0, 1] [0, 1] Manipulate a [1] [8] a In[ ]:= g[a_, x_] := a x (1 - x); gr2[a_] := Plot[{g[a, x], x}, {x, 0, 1}, AspectRatio -> Automatic] tateyoko2[a_, p_] := {{p, g[a, p]}, {g[a, p], g[a, p]}}; weblist2[a_, p_, n_] := (w = {{p, p}}; x = p; Do[(w = Join[w, tateyoko2[a, x]]; x = g[a, x]), {i, 1, n}]; w); webdiag2[a_, p_, n_] := ListLinePlot[weblist2[a, p, n], PlotStyle -> Thick, PlotRange -> All] Manipulate[Show[gr2[a], webdiag2[a, p, n]], {{a, 3}, 0, 4}, {{p, 0.2}, 0, 1}, {{n, 5}, 0, 10, 1}]

7 a p n ( ) 1 (1) (2) y = x ListLinePlot Graphics Graphics In[ ]:= tateyoko3[p_, q_] := (pp = {p, p}; pq = {p, q}; qq = {q, q}; {Line[{pp, pq, qq}], {Red, PointSize[Medium], Point[{pp, qq}]}, {Blue, PointSize[Medium], Point[pq]} }); q g[a, p] tateyoko3 n In[ ]:= weblist3[a_, p_, n_] := ( w = {{Red, PointSize[Large], Point[{p, p}]}}; x = p; Do[(w = Join[w, tateyoko3[x, g[a, x] ]]; x = g[a, x]), {i, 1, n}]; w);

8 8 14 Manipulate In[ ]:= Manipulate[ Show[gr2[a], Graphics[weblist3[a, p, n]]], {{a, 3}, 0, 4}, {{p, 0.25}, 0, 1}, {{n, 5}, 1, 10, 1}] a p n f(x) f(x) = 0 (Newton s method) (1) y = f(x) (2) x 0 (x 0, f(x 0 )) (3) x (x 1, 0)

9 x 0 f(x) = 0 α x 1 α *4 (x 1, f(x 1 )) x 1 = x 0 f(x 0 )/f (x 0 ) N f (x) := x f(x) f (x) N f x 0 N f x1 = N f (x 0 ) Nf x 2 = Nf 2 f (x 0 ) α N f f 14.3 [9] f(x) = x f In[9]:= f[x_] = x^2-2; df[x_] = D[f[x], x]; newton[x_] = x - f[x]/df[x] *4 f C 2

10 10 14 Out[9]= x 2 x2 2 x N f (x) = x x [10] NestList x 0 = 1 x 0, x 1,..., x 5 In[10]:= app = NestList[newton, 1, 5] Out[10]= 1, 3 2, 17 12, ,, [11] 20 In[11]:= N[app, 20] //TableForm Out[11]//TableForm= TableForm [12] Sqrt[2] *5 In[12]:= N[{app, app - Sqrt[2]}, 20] //Transpose//TableForm Out[12]//TableForm= *5 Transpose[N[{app, app - Sqrt[2]}, 20]] //TableForm TableForm[N[{app, app - Sqrt[2]}, 20], TableDirections -> Row]

11 [13] In[13]:= seg[p_] := {{p, f[p]}, {newton[p], 0}}; seglist[p_, n_] := (w = {{p, 0}}; x = p; Do[ (w = Join[w, seg[x]]; x = newton[x]), {i, 1, n}]; w); seggr[p_, n_] := ListLinePlot[seglist[p, n], PlotRange -> All, PlotStyle -> Thick] gr = Plot[f[x], {x, -5, 5}]; Manipulate[ Show[gr, seggr[p, n]], {{p, 5}, -5, 5}, {{n, 3}, 1, 10, 1}] p n 20 Out[13]= p 2 p 2 * ( ) 3 g a (x) = x 3 3x + a (a > 0) Manipulate a a g a (x) = 0 *6 p 0 f

12 12 14 a p n C f(z) 1 (complex dynamics) f(z) f c (z) = z 2 + c (c C) 2 f c c C f c z fc n (z) (n ) *7 B c := {z C fc n (z) (n )} c f c (basin at infinity) B c K c := C B c *7 z 2 + c f c(z) 2 z f(z) = z 2 + c z z c (2 + c ) z c 2 z + c ( z 1) 2 z + c (1 + c ) 2 z. f n (z) 2 n z (n ) z max{2, c } f n (z) (n )

13 (filled Julia set) B c K c f c (Julia set) J c J c Mathematica * 8 c 2 z C fc k (z) 2 k fc k+n (z) n B c (k) := { z C f k c (z) 2 } B c (1) B c (2) B c = k 1 B c(k) k B c (k) B c k = 50 [14] c = i B c K c f c In[14]:= c = I; f[z_] := z^2 + c; [15] B c In[15]:= col[z_] := (p = z; k = 0; While[(Abs[p] < 2.0) && (k < 50), k) (p = f[p]; k = k + 1)]; While f k c (z) < 2 k < 50 (...) p = f[p]; k = k + 1 k *8 Mathematica C Java Mathematica C

14 14 14 fc k (z) 2 k 49 k = 50 fc k (z) < 2 col z B c 50 z K c [16] {x + yi 2 x 2, 2 y 2} d = 0.01 Table col In[16]:= d = 0.01; tab = Table[col[x + y I], {x, -2, 2, d}, {y, -2, 2, d}]; * 9 [17] tab ArrayPlot complexap (complex Array Plot ) 14.4 In[17]:= complexap[t_] := ArrayPlot[Reverse[Transpose[t]]]; * = col col 50 d

14.4 15 14.4 complexap d col tab Transpose Reverse ArrayPlot [18] tab complexap In[18]:= complexap[tab] Out[18]= col B c K c 14.

15 complexap d col tab Transpose Reverse ArrayPlot [18] tab complexap In[18]:= complexap[tab] Out[18]= col B c K c 14.4 ( ) B c ArrayPlot ColorFunction complexap In[ ]:= complexap2[t_] := ArrayPlot[Reverse[Transpose[t]], ColorFunction -> "LightTemperatureMap"]; [16] tab complexap2[tab] 14.5 c f c z = 0 f c(z) = 0 z = 0 f c z = 0 0 B c c H := {c C fc n (0) (n )}

M := C H (the Mandelbrot set) * 10 14.5 ( ) complexap2 M In[ ]:= colm[c_] := (p = 0.

16 ColorFunction "LightTemperatureMap" "MintColors", "WatermelonColors" "RedBlueTones". M := C H (the Mandelbrot set) * ( ) complexap2 M In[ ]:= colm[c_] := (p = 0.0; k = 0; While[(Abs[p] < 2.0) && (k < 100), (p = p^2 + c; k = k + 1)]; k) d = 0.01 In[ ]:= tabm = Table[colM[a + b I], {a, -2, 0.6, d}, {b, -1.3, 1.3, d}]; *10 K c c

17 c = 1 c = i, i c = complexap2[tabm]

18 14. f(z) = z 3 1 = 0 z 0 N f (z) 2 1 3 N f (z) f f, df, newton d = 0.

18 f(z) = z 3 1 = 0 z 0 N f (z) N f (z) f f, df, newton d = 0.01; f[x_] = x^3-1; df[x_] = D[f[x], x] newton[x_] := x - f[x]/df[x]; coln[z_] := (p = z; k = 0; While[(Abs[p^3-1] > 0.1) && (k < 50), (p = newton[p]; k = k + 1)]; k) tabn = Table[colN[x + y I], {x, -2, 2, d}, {y, -2, 2, d}]; complexap2[tabn]

19 (1) R. Devaney 2 (2) 2

Chapter 3 Mathematica Mathematica e a n = ( ) n b n = n 1! + 1 2! n! b n a n e 3/n b n e 2/n! b n a n b n M athematica Ma

Mathematica Workbook Workbook Mathematica Mathematica A4 12 A5 9 14 4 2 Chapter 3 Mathematica Mathematica e a n = ( 1 + 1 ) n b n = 1 + 1 n 1! + 1 2! + + 1 n! b n a n e 3/n b n e 2/n! b n a n b n M athematica