Python

2017 10 25

1

x(t) = (x1(t), . . . , xn(t)) x(t0) = (x1(t0), . . . , xn(t0)) M

dx1dt

= v1(x, t), x1(t0) = x10

...

dxndt

= vn(x, t), xn(t0) = xn0

dx

dt= v(x, t), x(t0) = x0

x t v Rn v : M Rn

(vector filed)

t x

dx

dt= lim

t0

x(t+t) x(t)t

t

x(t+t) x(t)t

f(x, t)

t t x(t)

t = t0 x(t0) {xn}(n = 0, 1, 2, . . . )x0 = x(t0) t

R3 Lorentz

dx

dt= x+ y

dy

dt= rx y + xz

dz

dt= bz + xy

Lorenz (1963) = 10, r = 24.74, b = 8/3

1.1 Euler

Euler x0,x1, . . . ,xn, . . .

xn+1 = xn +tf(xn, tn), tn = t0 + nt.

1

http://journals.ametsoc.org/doi/pdf/10.1175/1520-0469(1963)020%3C0130%3ADNF%3E2.0.CO%3B2

Python euler_orbit(x0, T, dt, f) Euler

vectorfield v m [v1, . . . , vm]

t0 m2 width x0 = [x01, . . . , x0m]

t0 t0 + T dt N = T/dt m [x0,x1, . . . ,xN ] 3 x0 x while

orbit.append(list(x)) orbit

1 def euler_orbit(x0, T, dt, vectorfield):

2 width = len(x0)

3 x = x0

4 t = 0

5 orbit = []

6 orbit.append(list(x))

7 while t > def func(x, y, *args):

>> print a, b, args

>> print func(a, b, c, d, e, f)

(a, b, (c, d, e, f))

>> print func(a, b, c, d)

(a, b, (c, d))

>> print func(a, b, c)

2

(a, b, (c,))

>> print func(a, b)

(a, b, ())

1.3 Euler Lorenz

Lorenz (25), (26), (27) Euler diff_euler.py

vector = [v1, v2, v3] Lorenz x0 = [0.1, 0.1, 0.1]

1 0.01 Euler100 orbit

diff euler.py

1 #!/usr/bin/env python

2 # -*- coding: utf-8 -*-

3

4 # vector fields v1, v2, v3 of Lorenz eq as functions of t, x, y, z.

5 p, r, b = 10.0, 28.0, 8.0/3.0

6 def v1(t, x, y, z):

7 return(-p * x + p * y)

8 def v2(t, x, y, z):

9 return(-x * z + r * x - y)

10 def v3(t, x, y, z):

11 return(x * y - b * z)

12

13 def euler_orbit(x0, T, dt, vectorfield):

14 width = len(x0)

15 x = x0

16 t = 0

17 orbit = []

18 while t

numpy matplotlibdiff euler.py

1 # -*- coding: utf-8 -*-

2 import numpy as np

3 import matplotlib.pyplot as plt

4 from mpl_toolkits.mplot3d.axes3d import Axes3D# for 3dim plot

5 ....

6 ....

7 orbit = euler_orbit(x0, T, dt, vector)

8

9 xlist = []

10 ylist = []

11 zlist = []

12 for x, y, z in orbit:

13 xlist.append(x)

14 ylist.append(y)

15 zlist.append(z)

16

17 # plot in 3-din space

18 fig = plt.figure()

19 ax = fig.gca(projection=3d)

20 ax.plot(xorbit, yorbit, zorbit)

21 ax.set_xlabel(x(t))

22 ax.set_ylabel(y(t))

23 ax.set_zlabel(z(t))

24 plt.show()

1.2 diff_euler.py T= 50, dt = 0.01

1.3.1 Numpy

9-15 matplotlib

Euler 3 x, y, z ( N

N 3N 3 xorbit, yorbit zorbit N 3 NumPy NumPy ( import numpy as np

np.ndarray(object)

object np.ndarray

np.array()

Python Shell a 4 3Python a a[i][j]

3 a[1][2] 2 35a[:]

a[:][2] 2 3

1 >>> import numpy as np

2 >>> a = [[1,2,3], [4,5,6], [7,8,9], [10,11,12]]

3 >>> a[1][2]

4 6

5 >>> a[:][2]

6 [7, 8, 9]

7 >>> na = np.array(a)

8 >>> na[:, 2]

9 array([ 3, 6, 9, 12])

10 >>> na[1, :]

4

11 array([4, 5, 6])

7 na = np.array(a) ndarray8 310 2

arrayndarry A A[j,

k]A[j][k]

NumPy orbit 3 nporbit = np.array(orbit) N 3 ndarray xorbit, yorbit, zorbit16

matplotlibdiff euler.py

1 ....

2 orbit = euler_orbit(x0, T, dt, vector)

3 nporbit = np.array(orbit)

4

5 #xorbit = []

6 #yorbit = []

7 #zorbit = []

8 #for x, y, z in orbit:

9 # xorbit.append(x)

10 # yorbit.append(y)

11 # zorbit.append(z)

12

13 # plot in 3-din space

14 fig = plt.figure()

15 ax = fig.gca(projection=3d)

16 ax.plot(nporbit[:, 0], nporbit[:, 1], nporbit[:, 2])

17 .....

1.3 EulerNumPy ndarray diff_euler.py

T= 50, dt = 0.01

1.4 Runge-Kutta

Euler

Runge-Kutta dt

Python runge_kutta_orbit(x0, T, dt, vectorfield) Runge-Kutta

vectorfield v m [v1, . . . , vm]

t0 m 2 width x0 =

[x01, . . . , x0m] t0 t0+T dtN = T/dt m [x0,x1, . . . ,xN ] 3 x0 x while

orbit.append(list(x)) orbit

Euler while tn = nt xn tn+1 = tn + dt

xn+1 22,23 dt/2 k1, k2, k3, k4

4

1 def runge_kutta(x0, T, dt, vectorfield):

2 width = len(x0)

3 x = x0

4 t = 0.0

5 orbit = []

6 while t

10 k1 = list(map(lambda v: v(t, *x1), vectorfield))

11 x2 = x

12 for i in range(width):

13 x2[i] += dt / 2.0 * k1[i]

14 k2 = list(map(lambda v: v(t + dt / 2.0, *x2), vectorfield))

15 x3 = x

16 for i in range(width):

17 x3[i] += dt / 2 * k2[i]

18 k3 = list(map(lambda v: v(t + dt / 2.0, *x3), vectorfield))

19 x4 = x

20 for i in range(width):

21 x4[i] += dt * k3[i]

22 k4 = list(map(lambda v: v(t + dt, *x4), vectorfield))

23 for i in range(width):

24 x[i] += dt / 6.0 * (k1[i] + 2.0 * k2[i] + 2.0 * k3[i] + k4[i])

25 t += dt

26 return(orbit)

1.4 diff_euler.pyRunge-Kutta Lorenz diff_runge.py

T= 50, dt = 0.01 3

LorenzEuler

1 # lorenz system

2 p, r, b = 10.0, 28.0, 8.0 / 3.0

3 def v1(t, x, y, z):

4 return(-p * x + p * y)

5 def v2(t, x, y, z):

6 return(-x * z + r * x - y)

7 def v3(t, x, y, z):

8 return(x * y - b * z)

9 ....

10 ....

11 vector = [v1, v2, v3]

12 x0 = [0.1, 0.1, 0.1]

13 dt = 0.01

14 T = 20

15 orbit = runge_kutta(x0, T, dt, vector)

2 Scipy

Python Scipy Numpy Matplotlib

Python matplotlib 3 1 Runge-

Kett lorenz 3d.py dt = 0.005 T = 20

5matplotlib.pyplot4mpl toolkits.mplot3d Axes3D

Axes3D 74,75 ax75

80(79) 3

lorenz 3d.py1 #!/usr/bin/env python

2 # -*- coding: utf-8 -*-

3

4 from mpl_toolkits.mplot3d import Axes3D

5 import matplotlib.pyplot as plt

6

7 # vector fields f1, f2, f3 of Lorenz eq as functions of t, x, y, z.

8 p, r, b = 10.0, 28.0, 8.0/3.0

6

1 lorenz 3d.py x0 = (0.1, 0.1, 0.1), dt = 0.005 T = 20.

viewpoint

9 def f1(t, x, y, z):

10 return -p * x + p * y

11 def f2(t, x, y, z):

12 return -x * z + r * x - y

13 def f3(t, x, y, z):

14 return x * y - b * z

15

16 def euler(x0, T, dt, f):

17 width = len(x0)

18 x = x0

19 t = 0

20 orbit = []

21 orbit.append(list(x))

22 while t

52 x[i] += dt/6 * (f1[i] + 2 * f2[i] + 2 * f3[i] + f4[i])

53 t += dt

54 return orbit

55

56

57 func = [f1, f2, f3]

58 x0 = [0.1, 0.1, 0.1]

59 dt = 0.005

60 T = 20

61

62 #### orbit = [[x0,y0,z0], [x1,y1,z1],[x2,y2,z2], ...]

63 #orbit = euler(x0, T, dt, func)

64 orbit = runge_kutta(x0, T, dt, func)

65

66 xorbit = []

67 yorbit = []

68 zorbit = []

69 for i in range(len(orbit)):

70 xorbit.append(orbit[i][0])

71 yorbit.append(orbit[i][1])

72 zorbit.append(orbit[i][2])

73

74 fig = plt.figure()

75 ax = fig.gca(projection=3d)

76 ax.set_xlabel(X)

77 ax.set_ylabel(Y)

78 ax.set_zlabel(Z)

79 #ax.scatter(xorbit, yorbit, zorbit, zdir=y)

80 ax.plot(xorbit, yorbit, zorbit, zdir=y)

81 plt.show()

3

t t 0 span class=red/span

t

t f(x, t) t

x(t+t) x(t) Coulomb

strong/strong

Runge-Kutta

Lorenz

Smale(1998)Mathematical Problems for the Next CenturyMathematical Intelligencer 20 (2): 7

15 14

W. Tucker(2002) A Rigorous ODE Solver and Smales 14th

Problem

4 Lorenz

Edward N. Lorenz Deterministic Nonperiodic Flow J.

8

http://www6.cityu.edu.hk/ma/doc/people/smales/pap104.pdfhttp://www2.math.uu.se/~warwick/main/rodes.htmlhttp://www2.math.uu.se/~warwick/main/rodes.htmlhttp://www.math.sci.hokudai.ac.jp/~zin/papers/kokyuroku.pdfhttp://www.math.sci.hokudai.ac.jp/~zin/papers/kokyuroku.pdfhttp://journals.ametsoc.org/doi/pdf/10.1175/1520-0469%281963%29020%3C0130%3ADNF%3E2.0.CO%3B2

Atmos. Sci., 20(11963), 130 141)

sensitive dependence

on initial conditionschaos

p = 10.0 , r = 28.04, b = 8.0/3

p = 10, b = 8/3 r

4.1 Poincare

1 < r (0, 0, 0) 2 C

C = (b(r 1),

b(r 1), r 1)

z 2 r 1 (x(t), y(t), z(t)) z = r 1 xy- z=r1 transverse z=r1 xy-/p p Lorenz 3

x = (x, y, z) [x0, x1, . . . , xN ] z- [z0, z1, . . . , zN ]

z z=r1

[z0 (r 1), z1 (r 1), . . . , zN (r 1)]=[s0, s1, . . . , sN ], sj = zj (r 1)

k xk z=r1 k + 1 zk+1

zk (r 1) < 0 and zk+1 (r 1) < 0 sk > 0 and sksk+1 < 0

z- xk z=r1

z- xk (x, y) z=r1

P Lorenz 2 C

z=r1

x x

P :

x P P (x) = x

Poincaren

n 1 Poincare lorenz section.py T = 100, dt = 0.05 Runge-Kutta Lorenz

x, y, z- xorbit, yorbit, zorbit z- z=r1

x, y- xsection, ysectionxy-

2

1 label=lorenz\_section.py]

2 import matplotlib.pyplot as plt

3 ...

4 ...

5 xorbit = []

6 yorbit = []

7 zorbit = []

8 for i in range(len(orbit)):

9 xorbit.append(orbit[i][0])

10 yorbit.append(orbit[i][1])

11 zorbit.append(orbit[i][2])

9

12

13 xsection = []

14 ysection = []

15 for i in range(len(zorbit)-1):

16 if zorbit[i]-(r-1) > 0 and zorbit[i+1] -(r-1) < 0:

17 xsection.append(xorbit[i])

18 ysection.append(yorbit[i])

19

20 plt.xlabel(x)

21 plt.ylabel(y)

22 plt.plot(xsection, ysection,rx)

23 plt.show()

2 Lorenz Poincare : T = 100 dt = 0.05 (0.1, 0.1, 0.1)

SciPy z=r1

2Lorenz Lorenz Lorenz

Lorenz

4.1 lorenz section.py z=r1

Poincare

4.2 Lorenz

E. N. Lorenz Deterministic Nonperiodic Flow J. Atmos. Sci., 20(11963), 130 141)

span class=bold blueLorenz/span

z = r 1 2 C C+ C

x(t) = (x1(t), x2(t), . . . , xn(t)

xi- xi(t) tk tk t (tk, tk+) x(t) < x(tk) (ti)iN = t1, t2, . . . , tk, . . .xi(t) tk

xk

x1, x2, . . . , x

k, . . .

(x1, x2), (x

2, x

3), . . . , (x

k1, x

k), (x

k, x

k+1), . . . )

Lorenz

Lorenz3 x = (x, y, z) [x0, x1, . . . , xN ] z-

10

http://journals.ametsoc.org/doi/pdf/10.1175/1520-0469%281963%29020%3C0130%3ADNF%3E2.0.CO%3B2

z (z0, z1, . . . , zN ) dzi = zi+1 zi

z1 z0, z2 z1, . . . , zi+1 zi, . . .

zi1 < zi and zi > zi+1

zi zmaxi Lorenz z-

(zi) (0, 1] 2

T (x) =

{2x, x (0, 1/2)2 2x, x [1/2, 1)

lorenz plot.py T = 100, dt = 0.05 Runge-Kutta Lorenz

x, y, z- xorbit, yorbit, zorbit z zorbit Lorenz

zorbit = [z0, z1, . . . , zN ] zdifi = zi+1 zi zi1 < zi and zi > zi+1 zi zextremum Lorenz

transit = 2000

lorenz plot.py

1 # -*- coding: utf-8 -*-

2 # numerila solution fo Lorenz system using scipy.integrate

3 # Lorenz section

4

5 import numpy as np

6 import matplotlib.pyplot as plt

7 from scipy.integrate import odeint

8

9 # lorenz system

10 p, r, b = 10.0, 28.0, 8.0 / 3.0

11 def lorenz_system(x, t):

12 vxt = -p * x[0] + p * x[1]

13 vyt = -x[0] * x[2] + r * x[0] - x[1]

14 vzt = x[0] * x[1] - b * x[2]

15 return([vxt, vyt, vzt])

16

17 x0 = [0.1, 0.1, 0.1]# initial points

18 t0 = 0

19 T = 100

20 dt = 0.01

21

22 # numerila solution using scipy.integrate

23 times = np.arange(t0, T, dt)

24 orbit = odeint(lorenz_system, x0, times)

25 #print(orbit)

26 # Poincare surface of section

27 # z = r-1 x-y (x,y) plot28 xsectionlist = []

29 ysectionlist = []

30

31 zhight = [0, 0, r-1]

32 for k in range(len(orbit)-1):

33 if (orbit[k] - zhight)[2] < 0 and (orbit[k] - zhight)[2] * (orbit[k+1] - zhight)[2] < 0:

34 # if (orbit[k] - zhight)[2] * (orbit[k+1] - zhight)[2] < 0:

35 xsectionlist.append(orbit[k,0])

11

36 ysectionlist.append(orbit[k,1])

37

38

39 #plot using matplotlib

40 ax = plt.axes()

41 plt.xlabel(x)

42 plt.ylabel(y)

43 plt.plot(xsectionlist, ysectionlist, "x", markersize=1)

44 plt.show()

lorenz plot.py21-23 z-

zextremum = [zmax,1, . . . , zmax,k1, zmax,k, zmax,k+1, . . . , zmax,Nk ] 26 29

zpair = [. . . , (zmax,k1, zmax,k), (zmax,k, zmax,k+1), . . . ]

2 {(zmax,k, zmax,i+k)}matplotlib x- y- zpairx, zpairy

Lorenz 3

3 Lorenz: T = 100, dt = 0.005, (x0, y0, z0) = (0.1, 0.1, 0.1)

3 1 I I L

L : I I

z0 I zn+1 = Lzn = Lnz0 L z0, z1, z2, . . . Lorenz z

Lorenz

1 I L

4.2 lorenz plot.pyLorenz

12

1 1.1 Euler1.2 1.3 EulerLorenz1.4 Runge-Kutta

2 Scipy3 4 Lorenz4.1 Poincare4.2 Lorenz