Multiple Bifurcations of Sample Dynamical Systems

1. 11
2
Expandx  y5 
x5  5 x4 y  10 x3 y2  10 x2 y3  5 x y4  y5
Abs 4
4
Sin
0
Cos
1
Log
1
Log10, 100
2
comment Plot[f[x], {x, xmin, xmax}]; Solve[eqn,x]; D[f[x],x]
PlotSinx, x,  10, 10
1.0
0.5
10 5 5 10
0.5
1.0
x  2  3 x  y  x  w
x
:
ctrl  
2
x2 : ctrl  ^ x : ctrl  2 then x x2 : ctrl  _
2100
1 267 650 600 228 229 401 496 703 205 376
12 345  5555
2469
1111

2. 2 1st.nb
0.239998
0.239998
0.12  10 ^ 11
1.2  1010
1
2  0.5
4
2.75
2  1  4  0.5
2.75
3  0.7 i
3  0.7 i
N[x] x
Rationalize[x] x
NPi, 11
3.1415926536
Rationalize, 0.000001
355
113
Pi

E

Degree
°
i
i
Infinity

 Infinity


3. 1st.nb 3
NGoldenRatio
1.61803
NumberForm[expr, n] n expr
ScientificForm[expr] expr
EngineeringForm[expr] expr
NPi ^ 30, 30
8.21289330402749581586503585434  1014
NumberFormNPi ^ 30
8.21289  1014
ScientificFormNPi ^ 30
8.21289  1014
EngineeringFormNPi ^ 30, 7
821.2893  1012
x3
3
x^2  2
11
u, v, w  1, 2, 3
1, 2, 3
2u3vw
11
u .
2u3vw
92u
x .
f  x21
x
1
2
f . x  1
3
2

4. 4 1st.nb
f . x  2
2
f .
f  x  y x  y ^ 2 . x  3, y  1  a
4  a 2  a2
f .
fx_  x  Sinx  x ^ 2
x2  x Sinx
f3
9  3 Sin3
Plotft, t, 0, 2
5
4
3
2
1
0.5 1.0 1.5 2.0
Clearf
Plotft, t, 0, 2
1.0
0.8
0.6
0.4
0.2
0.5 1.0 1.5 2.0
Removef

5. 1st.nb 5
Plotft, t, 0, 2
1.0
0.8
0.6
0.4
0.2
0.5 1.0 1.5 2.0
fx_, y_  x  y  y  Cosx
x y  y Cosx
f2, 3
6  3 Cos2
f .
fx_, y_ : x  y  y  Cosx
f2, 3
6  3 Cos2
f .
fx_ : x  1 ; x  0
fx_ : x ^ 2 ; x   1 && x  0
fx_ : Sinx ; x   1
Plotfx, x,  2, 2
1.0
0.5
2 1 1 2
0.5
1.0

6. 6 1st.nb
Ifx  0, x  1, Ifx   1, Sinx, x ^ 2;
Plotfx, x,  2, 2
1.0
0.5
2 1 1 2
0.5
1.0
1, 2, 3
1, 2, 3
1   x  x^
1  2 x, 1  2 x  x2 , 1  3 x  x3 
D, x
2, 2  2 x, 3  3 x2 
 . x  1
2, 4, 6
Tablex  i, i, 2, 6
2 x, 3 x, 4 x, 5 x, 6 x
Tablex ^ 2, 4
x2 , x2 , x2 , x2 
Range10
1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Range8, 20, 2
8, 10, 12, 14, 16, 18, 20
t  Table2  i  j, i, 1, 3, j, 3, 5
5, 6, 7, 7, 8, 9, 9, 10, 11
TableFormt
5 6 7
7 8 9
9 10 11

7. 1st.nb 7
t2
7, 8, 9
Expandx  y ^ 4 x  y ^ 2
x5  4 x4 y  6 x3 y2  x4 y2  4 x2 y3  4 x3 y3  x y4  6 x2 y4  4 x y5  y6
Factor
x  y4 x  y2 
ShortExpand1  x ^ 30
1  30 x  435 x2  4060 x3  27 405 x4  142 506 x5  593 775 x6 
17  593 775 x24  142 506 x25  27 405 x26  4060 x27  435 x28  30 x29  x30
Short, 3
1  30 x  435 x2  4060 x3  27 405 x4  142 506 x5  593 775 x6  2 035 800 x7 
5 852 925 x8  14 307 150 x9  30 045 015 x10  54 627 300 x11  86 493 225 x12  5 
86 493 225 x18  54 627 300 x19  30 045 015 x20  14 307 150 x21  5 852 925 x22 
2 035 800 x23  593 775 x24  142 506 x25  27 405 x26  4060 x27  435 x28  30 x29  x30
x  2; y  9;
xy
False
3^2  y  1
True
LogicalExpand3 xx ^ 2  yy  1 && 3 ^ 2  yy
yy  9 && 3 xx2  1  yy
&&
||
Xor
If
x .
SimplifyExpand2  x ^ 4 1  x ^ 4 3  x ^ 3
3  x3 2  3 x  x2 
4
p1  a ^ 2  3 a  2; p2  a  1;
p1  p2
3  4 a  a2
p1  p2
1  2 a  a2

8. 8 1st.nb
p1  p2
1  a 2  3 a  a2 
p1  p2
2  3 a  a2
1a
Cancelp1  p2
2a
PolynomialQuotientx ^ 2  2 x  2, x  1, x
1x
PolynomialRemainderx ^ 2  2 x  2, x  1, x
1
Rootsx ^ 2  3 x  2  0, x
x  1  x  2
Solve
x  1, x  2
FindRoot3  Cosx  Logx, x, 1
x  1.44726
FindRoot3  Cosx  Logx, x, 5
x  5.30199
Plot3  Cosx, Logx, x, 0, 10
2
2 4 6 8 10
2
4
6
8

9. 1st.nb 9
Solvex ^ 3  5 x  3  0, x
  27  2229 
13
x   5 ,
13 1
2 2
3  27  2229 

323
1   3   27  2229  5 1   3 
13
x   ,
1
2
223 3  27  2229 

2 323 13
1   3   27  2229  5 1   3 
13
x   
1
2
3  27  2229 

2 323 223
13
N
x   0.5641, x  0.28205  2.28881 , x  0.28205  2.28881 
x .; y .; NSolve2 x  y  0, x  3 y  3  0, x, y
x   0.6, y  1.2
Solvea  x ^ 2  b  x  c  0, x
x  , x  
b  b2  4 a c b  b2  4 a c
2a 2a
Reducea  x ^ 2  b  x  c  0, x
 x  
b  b2  4 a c b  b2  4 a c
a  0 && x 
2a 2a
 c  0 && b  0 && a  0
c
a  0 && b  0 && x  
b
Solve, Roots Reduce
Sc  x ^ 2  y
x2  y

10. 10 1st.nb
Solvex ^ 4  b  x ^ 2  c  0, Sc, x, y
y  , y  ,
1 b  b2  4 c 1 b  b2  4 c
b  b2  4 c ,x b  b2  4 c ,x
2 2 2 2
y  b2  4 c ,
1 b 1
b  b2  4 c ,x  
2 2 2
y  b2  4 c 
1 b 1
b  b2  4 c ,x  
2 2 2
Sc .
Sc  Sinx ^ 2  Cosx ^ 2  1
Cosx2  Sinx2  1
SolveCosx  2 Sinx  1, Sc, Sinx, Cosx
Sinx  0, Cosx  1, Sinx  
4 3
, Cosx  
5 5
Sumi, i, 1, 9, 2
25
Sum2 i  1, i, 1, 5
25
Sumi  j, i, 1, 5, j, 1, 5
225
Producti  j, i, 1, 5, j, 1, 5
619 173 642 240 000 000 000
NSum1  i ^ 2, i, 1, Infinity
1.64493
NSum1  i ^ 2, i, 1, Infinity, 2
1.2337
NProduct1  i ^ 2, i, 1, Infinity, 2
0.

11. 1st.nb 11
gx_  Sinx ^ 2  1  x
Plotgx, x, 0, 2 Pi
Sinx2 
1x
0.4
0.3
0.2
0.1
1 2 3 4 5 6
0.1
0.2
0.3
Plotgx, x, 0, 2 Pi, AspectRatio  1  2
0.4
0.3
0.2
0.1
1 2 3 4 5 6
0.1
0.2
0.3
Plotgx, x, 0, 2 Pi, Ticks  none

12. 12 1st.nb
Plotgx, x, 0, 2 Pi, AxesLabel  "time", "height"
height
0.4
0.3
0.2
0.1
time
1 2 3 4 5 6
0.1
0.2
0.3
Plotgx, x, 0, 2 Pi, AxesOrigin  3, 0, PlotLabel  "Decay Waves"
Decay Waves
0.4
0.3
0.2
0.1
0 1 2 4 5 6
0.1
0.2
0.3
Plotgx, x, 0, 2 Pi, Ticks  0, Pi  2, 3 Pi  2, 2 Pi, Automatic
0.4
0.3
0.2
0.1
 3
2
2 2
0.1
0.2
0.3

13. 1st.nb 13
Plotgx, x, 0, 2 Pi, PlotRange   0.6, 0.6
0.6
0.4
0.2
1 2 3 4 5 6
0.2
0.4
0.6
g1  Plotgx, x, 0, 2 Pi;
g2  Plotx  Cosx  12, x, 0, 2 Pi;
Showg1, g2
0.4
0.3
0.2
0.1
1 2 3 4 5 6
0.1
0.2
0.3
ListPlot[{y1, y2, … ..}] x 1 2… y1, y2, …
ListPlot[{{x1, y1}, {x2, y2}, … ..}] xi, yi
ListPlot[List, PlotJoined -> True]
ParametricPlot[{fx,fy},{t,tmin,tmax}]
ParametricPlot[{{fx,fy},{gx,gy},….},{t,tmin,tmax}]
ParametricPlot[{fx,fy},{t,tmin,tmax},AspectRatio->Automatic]

14. 14 1st.nb
ParametricPlotSin3 t Cost, Sin3 t Sint, t, 0, 2 Pi
0.5
0.5 0.5
0.5
1.0
ParametricPlotSin3 t Cost, Sin3 t Sin2 t, Sint, Cost,
t, 0, 2 Pi, AspectRatio  Automatic
1.0
0.5
1.0 0.5 0.5 1.0
0.5
1.0
List1  Tablei ^ 3  i, i, 10
2, 10, 30, 68, 130, 222, 350, 520, 738, 1010

15. 1st.nb 15
ListPlotList1
1000
800
600
400
200
2 4 6 8 10
ListPlotList1, PlotJoined  True
1000
800
600
400
200
2 4 6 8 10
g1  GraphicsText"left",  1, 0, Text"right", 1, 0, Text"above", 0, 1,
Text"below", 0,  1, PointSize0.4, Point0, 0, PlotRange  All
above
left right
below

16. 16 1st.nb
LineTablen,  1 ^ n, n, 6
Line1,  1, 2, 1, 3,  1, 4, 1, 5,  1, 6, 1
Graphics
ShowGraphics, Axes  True
1.0
0.5
2 3 4 5 6
0.5
1.0
St : TableRectanglex, 0, x  0.08, Sinx, x, 0, 2 Pi, 0.15
ShowGraphicsSt, Axes  True
1.0
0.5
1 2 3 4 5 6
0.5
1.0

17. 1st.nb 17
GraphicsCircle0, 0, 1, Axes  True
1.0
0.5
1.0 0.5 0.5 1.0
0.5
1.0
ShowGraphicsCircle0, 0, 5, 3, Axes  True
3
2
1
4 2 2 4
1
2
3

18. 18 1st.nb
GraphicsCircle0, 0, 1, 0, Pi  2, Axes  True
1.0
0.8
0.6
0.4
0.2
0.2 0.4 0.6 0.8 1.0
ShowGraphicsCircle0, 0, 5, 3, Pi  2, 3  Pi  2, Axes  True, AspectRatio  Automatic
3
2
1
5 4 3 2 1
1
2
3

19. 1st.nb 19
GraphicsDisk0, 0, 1, Axes  True
1.0
0.5
1.0 0.5 0.5 1.0
0.5
1.0
GraphicsRaster0, 0, 1, 0, 1, 0, 1, 0, 0

20. 20 1st.nb
PlotSinx, Sin2 x, Sin3 x, x, 0, 2 Pi,
PlotStyle  RGBColor0.9, 0, 0, RGBColor0, 0.9, 0, RGBColor0, 0, 0.9
1.0
0.5
1 2 3 4 5 6
0.5
1.0
v1   1, 0, 0, 1, 1, 0, 0,  1
 1, 0, 0, 1, 1, 0, 0,  1
ShowGraphicsHue0.2, Polygon3  v1, Hue0.4, Polygon2  v1, Hue0.9, Polygonv1,
AspectRatio  Automatic
TablePointn ^ 2, Primen, n, 5;

21. 1st.nb 21
ShowGraphicsPointSize0.1, , PlotRange  All
TableGraphicsAbsolutePointSized, Point0, 0, d, 0.5, 2, 7, 15
 ,

22. 22 1st.nb
,
,


23. 1st.nb 23

Show

Graphics
AbsoluteThicknessd, Line0, 0, 1, d, d, 5,
Table
Line0, 5, 1, 0




24. 24 1st.nb
PlotSinx ^ 2, x,  Pi, Pi
1.0
0.5
3 2 1 1 2 3
0.5
1.0
Show, PlotRange   1, 2, Frame  True
2.0
1.5
1.0
0.5
0.0
0.5
1.0
3 2 1 0 1 2 3
f1  Plotx  Sin2 x  Pi, x, 0, 4 Pi;
f2  Plotx  Cos2 x, x, 0, 4 Pi;
Showf1, f2
10
5
2 4 6 8 10 12
5
10

25. 1st.nb 25
ShowGraphicsArray, f1, , f2
10 10
5 5
2 4 6 8 10 12 2 4 6 8 10 12
5 5
10 10
10 10
5 5
2 4 6 8 10 12 2 4 6 8 10 12
5 5
10 10
t1  Plot3DSinx  y  Cosx  y, x, 0, 4, y, 0, 4

26. 26 1st.nb
Show, PlotRange   0, 0.5
Showt1, AxesLabel  "time", "depth", "Value", FaceGrids  All

27. 1st.nb 27
Showt1, Axes  False, Boxed  False
Showt1, Mesh  None

28. 28 1st.nb
Plot3DSinx  y  Cosx  y, x, 0, 4, y, 0, 4, Mesh  None
mesh plot3D shading lighting false
Plot3DSinx  y  Cosx  y, x, 0, 4, y, 0, 4, Shading  False
Plot3D::optx : Unknown option Shading in Plot3DSinx  y Cosx  y, x, 0, 4, y, 0, 4, Shading  False. 
Plot3DSinx  y Cosx  y, x, 0, 4, y, 0, 4, Shading  False
Plot3DSinx  y  Cosx  y, x, 0, 4, y, 0, 4, Lighting  None

29. 1st.nb 29
TableSinx  y  RandomReal,  0.15, 0.15, x, 0, 3 Pi  2, Pi  15, y, 0, 3 Pi  2, Pi  15
MyTable :
ListPlot3DMyTable
ParametricPlot3D3 Cos4 t  1, Cos2 t  3, 4 Cos2 t  5, t, 0, Pi
1.0
0.5 2
0.0
0.5 0
1.0
2
4
2
0
2
4

30. 30 1st.nb
r, Exp r ^ 2 Cos4 r ^ 2  Cost, Exp r ^ 2 Cos4 r ^ 2  Sint, r,  1, 1, t, 0, 2 Pi
ParametricPlot3D
LimitSqrtx ^ 2  2  3 x  6, x  Infinity
1
3
LimitSinx ^ 2  x ^ 2, x  0
1
LimitLogx  x, x  0, Direction   1

DExpx  Sinx, x
x Cosx  x Sinx
DExpx  Sinx, x, 2
2 x Cosx
DSina  x, x
a Cosa x

31. 1st.nb 31
DSina  x, x, NonConstants  a
Cosa x a  x Da, x, NonConstants  a
fx_, y_  x ^ 2  y  y ^ 2
x2 y  y2
Dfx, y, x
2xy
Dfx, y, y
x2  2 y
Dfx, y, x, 2
2y
Dfx, y, y, 2
2
Dfx, y, x, y
2x
Dx  f3x, x
f3x  x f3 x
Df3f4x, x
f3 f4x f4 x
DExpx  Sinx, x . x  2
2 Cos2  2 Sin2
Dtx ^ 2  y ^ 2, x
2 x  2 y Dty, x
Dfx ^ 2  y ^ 2
Dfx2  y2 
Dtx ^ 2  xy ^ 3  yz, Constants  z
2 x Dtx, Constants  z  3 xy2 Dtxy, Constants  z  Dtyz, Constants  z
Dtx ^ 2  xyx  yx z
2 x Dtx  Dtz yx  Dtx xy x  z Dtx y x

u 1  u2
u
2  11 u2
1  u2 
1 1
11 1  u2  3 11 ArcTanh 11
121 3

32. 32 1st.nb
 SinSinx  Sinx
Integrate::ivar : Sinx is not a valid variable. 
 SinSinx  Sinx
 SinSinx  x
 SinSinx  x
 a  x  b  x  c  x
2
b x2 a x3
cx 
2 3
 x e x
6
2 ax
4
280 eax
3

 1
x
1 x4
1
3

 1
x
1 xp
, Integratexp , x, 1, , Assumptions  Rep  1
1
IfRep  1,
1  p
NIntegrateSinSinx, x, 0, Pi
1.78649
NIntegrate1  SqrtAbsx, x,  1, 0, 1
4.
NIntegrateExp x ^ 2, x, 0, Infinity
0.886227
DSinx  y ^ 2, x, x, y
 2 x y5 Cosx y2   4 y3 Sinx y2 
DSinx  y ^ 2, x, 2, y
 2 x y5 Cosx y2   4 y3 Sinx y2 
Dx ^ 2  y ^ 2, x, NonConstants  y
2 x  2 y Dy, x, NonConstants  y

33. 1st.nb 33
Dtx2 y3 
2 x y3 Dtx  3 x2 y2 Dty
z  x3 y  x2 y2  3 x  y2 ;
Dtz
3 x2 y Dtx  3 y2 Dtx  2 x y2 Dtx  x3 Dty  6 x y Dty  2 x2 y Dty
CollectDtz, Dtx, Dty
3 x2 y  3 y2  2 x y2  Dtx  x3  6 x y  2 x2 y Dty
 . Dtx  dx, Dty  dy
dy x3  6 x y  2 x2 y  dx 3 x2 y  3 y2  2 x y2 
Dtz, x
3 x2 y  3 y2  2 x y2  x3 Dty, x  6 x y Dty, x  2 x2 y Dty, x
 . Dty, x  0
3 x2 y  3 y2  2 x y2
Dt5  y ^ 2  Siny  x ^ 2, x
10 y Dty, x  Cosy Dty, x  2 x
Solve, Dty, x
Dty, x  
2x
10 y  Cosy
Dtx ^ 2  y ^ 2  z ^ 2, x, Constants  z
2 x  2 y Dty, x, Constants  x3 y  3 x y2  x2 y2 
Dtz, x, y
3 x2  6 y  4 x y  6 x y Dtx, y  2 y2 Dtx, y  6 x Dty, x  2 x2 Dty, x 
3 x2 Dtx, y Dty, x  6 y Dtx, y Dty, x  4 x y Dtx, y Dty, x
  x ^ 2  y ^ 2  x  y
a b
0 0
a b a2  b2 
1
3
NIntegrateSqrtx  y, x, 0, 2, y, 0, Sqrtx  2
4.65557
NIntegrateSqrtx ^ 2  z ^ 2, x,  2, 2, y, x ^ 2, 4, z,  Sqrty  x ^ 2, Sqrty  x ^ 2
26.8083
y .; DSolvey 'x  2 yx, yx, x
yx  2 x C1

34. 34 1st.nb
yx  y0  y 'x . 
2 x C1  y0  y x
y[x] y[x] y’[x] y[0] y[x]
DSolvey 'x  2 yx, y, x
y  Functionx, 2 x C1
yx  y0  y 'x . 
C1  3 2 x C1
y y
y .; z .; DSolveyx   z 'x, zx   y 'x, y, z, x
z  Functionx, x 1  2 x  C1  x  1  2 x  C2,
1 1
2 2
x  1  2 x  C1  x 1  2 x  C2
1 1
y  Functionx, 
2 2
y .; z .; DSolveyx   z 'x, zx   y 'x, yx, zx, x
zx  x 1  2 x  C1  x  1  2 x  C2,
1 1
2 2
x  1  2 x  C1  x 1  2 x  C2
1 1
yx  
2 2
DSolvey 'x  yx, y0  5, yx, x
yx  5 x 
s1  NDSolvey 'x  1  2  yx, y0.01  0.1, y, x, 0.01, 1
y  InterpolatingFunction0.01, 1., 
PlotEvaluateyx . s1, x, 0.01, 1, AxesOrigin  0, 0
1.0
0.8
0.6
0.4
0.2
0.2 0.4 0.6 0.8 1.0
t  10
10

35. 1st.nb 35
Modulet, t  8; Printt
8
t
10
fv_ : Modulet, t  1  v ^ 2; Expandt; fa
1  2 a  a2
t
10
gu_ : Modulet  u, t  t  t  1  u;
ga  b
ab
ab
1ab
x^2  1
1  x2
Blockx  a  1, 
1  1  a2
x
x
m  i^2
i2
Blocki  a, i  m
a  a2
Modulei  a, i  m
a  i2
Removeg
gx_ : 1 ; x  0
gx_ :  1 ; x  0
?g
Global`g
gx_ : 1 ; x  0
gx_ :  1 ; x  0
Removeh; hx_ : Whichx  0, 1, x  0, 0, x  0.  1

36. 36 1st.nb
h 1, h0, h3
h 1, h0, h3
qx_ : SwitchModx, 3, 0, a, 1, b, 2, c
q17
c
If[x==y,a,b,c] If ,
,
Ife  f, a, b, c
c
TrueQe  f
False
e  f
False
Mathematica ,
DoPrinti  i ^ 2, i, 1, 4
2
6
12
20
DoPrinti, j, i, 4, j, i  1
2, 1
3, 1
3, 2
4, 1
4, 2
4, 3
t  67;
DoPrintt; t  Floort  2, 3
67
33
16
n  25;
Whilen  Floorn  3  0, Printn

37. 1st.nb 37
8
2
Fori  1, i  5, i , Printi
1
2
3
4
x .; Fori  1; t  x, i ^ 2  10, i , t  t ^ 2  i; Printt
1  x2
2  1  x2 
2
3  2  1  x2  
2 2
Nestf, x, 5
fffffx
NestFunctiont, 1  Sqrt1  t ^ 2, x, 2
1
1
1
1x2
FixedPointFunctiont, Printt; Floort  3, 67
67
22
7
2
0
0
t  1;
Dot  k; Printt; Ift  20, Break, k, 10
1
2
6
24
t  1;
Dot  k; Printt; Ift  3, Continue; t  2, k, 5

38. 38 1st.nb
1
2
6
32
170
Removef
fx_ : Ifx  5, Returnbig; t  x ^ 3; Returnt  7
f3
big
f5
118
hx_ : Ifx  0, Throwerror, x 
Catchh3
6
( error Catch )
Catchh 3
error
Residuefz  z ^ 5, z, 0
0
Residue1  Sinz ^ 5, z, 0
3
8
SeriesExpx, x, 0, 10
x2 x3 x4 x5 x6 x7 x8 x9 x10
1x          Ox11
2 6 24 120 720 5040 40 320 362 880 3 628 800
Seriesx ^ x, x, 0, 4
1 1 1
1  Logx x  Logx2 x2  Logx3 x3  Logx4 x4  Ox5
2 6 24
Normal
1 1 1
1  x Logx  x2 Logx2  x3 Logx3  x4 Logx4
2 6 24

39. 1st.nb 39
Sum1  2 n  1  2 n  1, n, 1, Infinity
1
2
SumLogn  1  n, n, 1, Infinity
Sum::div : Sum does not converge. 
 Log 
 1n
n1 n