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# Multiple Bifurcations of Sample Dynamical Systems

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#### x x2 4p1

of 39 /39
1 1 2 Expandx y 5 x 5 5x 4 y 10x 3 y 2 10x 2 y 3 5xy 4 y 5 Abs 4 4 Sin 0 Cos 1 Log 1 Log10, 100 2 comment Plot[f[x], {x, xmin, xmax}]; Solve[eqn,x]; D[f[x],x] PlotSinx, x, 10, 10 10 5 5 10 1.0 0.5 0.5 1.0 数学表式的一格式x 2 3x y x w 和二格式的快捷方式 x 2 : ctrl x 2 :ctrl ^ x : ctrl 2thenx x 2 :ctrl _ 2 100 1267650600228229401496703205376 12345 5555 2469 1111
Transcript

1 1

2

Expandx y5x5 5 x4 y 10 x3 y2 10 x2 y3 5 x y4 y5

Abs44

Sin0

Cos1

Log1

Log10, 1002

comment Plot[f[x], {x, xmin, xmax}]; Solve[eqn,x]; D[f[x],x]

PlotSinx, x, 10, 10

10 5 5 10

1.0

0.5

0.5

1.0

2:

ctrl x2 : ctrl ^ x : ctrl 2 then x x2 : ctrl _

2100

1 267 650 600 228 229 401 496 703 205 376

12 345 55552469

1111

0.239998

0.239998

0.12 10^11

1.2 1010

2 1

4 0.5

2.75

2 1 4 0.52.75

3 0.7 i

3 0.7 i

N[x] 将x转换成实数Rationalize[x] 给出x的有理数近似值

NPi, 113.1415926536

Rationalize, 0.000001355

113

Pi

E

Degree

°

i

i

Infinity

Infinity

2 1st.nb

NGoldenRatio1.61803

NumberForm[expr, n]以n位精度的实数形式输出实数exprScientificForm[expr]以科学记数法输出实数exprEngineeringForm[expr]以工程记数法输出实数expr

NPi^30, 308.21289330402749581586503585434 1014

NumberFormNPi^308.21289 1014

ScientificFormNPi^308.21289 1014

EngineeringFormNPi^30, 7821.2893 1012

x 3

3

x^2 2

11u, v, w 1, 2, 31, 2, 32 u 3 v w

11

u .

2 u 3 v w

9 2 u

x .

f x 2 11

x

2

f . x 1

3

2

1st.nb 3

f . x 2

2

f .

f x y x y^2 . x 3, y 1 a4 a 2 a2f .

fx_ x Sinx x^2x2 x Sinxf39 3 Sin3Plotft, t, 0, 2

0.5 1.0 1.5 2.0

1

2

3

4

5

ClearfPlotft, t, 0, 2

0.5 1.0 1.5 2.0

0.2

0.4

0.6

0.8

1.0

Removef

4 1st.nb

Plotft, t, 0, 2

0.5 1.0 1.5 2.0

0.2

0.4

0.6

0.8

1.0

fx_, y_ x y y Cosxx y y Cosxf2, 36 3 Cos2f .

fx_, y_ : x y y Cosxf2, 36 3 Cos2f .

fx_ : x 1 ; x 0

fx_ : x^2 ; x 1 && x 0fx_ : Sinx ; x 1

Plotfx, x, 2, 2

2 1 1 2

1.0

0.5

0.5

1.0

1st.nb 5

Ifx 0, x 1, Ifx 1, Sinx, x^2;Plotfx, x, 2, 2

2 1 1 2

1.0

0.5

0.5

1.0

1, 2, 31, 2, 31 x x^1 2 x, 1 2 x x2, 1 3 x x3D, x2, 2 2 x, 3 3 x2 . x 12, 4, 6Tablex i, i, 2, 62 x, 3 x, 4 x, 5 x, 6 xTablex^2, 4x2, x2, x2, x2Range101, 2, 3, 4, 5, 6, 7, 8, 9, 10Range8, 20, 28, 10, 12, 14, 16, 18, 20t Table2 i j, i, 1, 3, j, 3, 55, 6, 7, 7, 8, 9, 9, 10, 11TableFormt5 6 77 8 99 10 11

6 1st.nb

t27, 8, 9Expandx y^4 x y^2x5 4 x4 y 6 x3 y2 x4 y2 4 x2 y3 4 x3 y3 x y4 6 x2 y4 4 x y5 y6

Factorx y4 x y2ShortExpand1 x^301 30 x 435 x2 4060 x3 27 405 x4 142 506 x5 593 775 x6

17 593 775 x24 142 506 x25 27405 x26 4060 x27 435 x28 30 x29 x30

Short, 31 30 x 435 x2 4060 x3 27 405 x4 142 506 x5 593 775 x6 2035 800 x7 5852 925 x8 14307 150 x9 30045 015 x10 54627 300 x11 86493 225 x12 5

86493 225 x18 54627 300 x19 30045 015 x20 14307 150 x21 5852 925 x22

2035 800 x23 593 775 x24 142 506 x25 27405 x26 4060 x27 435 x28 30 x29 x30

x 2; y 9;

x y

False

3^2 y 1

True

LogicalExpand3 xx^2 yy 1 && 3^2 yyyy 9 && 3 xx2 1 yy

！ 非

&& 并

|| 或

Xor 异或If 条件

x .

SimplifyExpand2 x^4 1 x^4 3 x^33 x3 2 3 x x24p1 a^2 3 a 2; p2 a 1;

p1 p2

3 4 a a2

p1 p2

1 2 a a2

1st.nb 7

p1 p21 a 2 3 a a2p1 p22 3 a a2

1 a

Cancelp1 p22 a

PolynomialQuotientx^2 2 x 2, x 1, x1 x

PolynomialRemainderx^2 2 x 2, x 1, x1

Rootsx^2 3 x 2 0, xx 1 x 2

Solvex 1, x 2FindRoot3 Cosx Logx, x, 1x 1.44726FindRoot3 Cosx Logx, x, 5x 5.30199Plot3 Cosx, Logx, x, 0, 10

2 4 6 8 10

8

6

4

2

2

8 1st.nb

Solvex^3 5 x 3 0, xx 5

2

3 27 2229 13

1

227 2229 13

323 ,x

1 3 1

227 2229 132 323

5 1 3 223 3 27 2229 13 ,

x 1 3 1

227 2229 132 323

5 1 3 223 3 27 2229 13

Nx 0.5641, x 0.28205 2.28881 , x 0.28205 2.28881 x .; y .; NSolve2 x y 0, x 3 y 3 0, x, yx 0.6, y 1.2Solvea x^2 b x c 0, xx

b b2 4 a c

2 a, x

b b2 4 a c

2 a

Reducea x^2 b x c 0, xa 0 && x

b b2 4 a c

2 a x

b b2 4 a c

2 a

a 0 && b 0 && x c

b c 0 && b 0 && a 0

Solve, Roots只给出方程的一般解，而Reduce函数数可以给出方程的全部可能解

Sc x^2 y

x2 y

1st.nb 9

Solvex^4 b x^2 c 0, Sc, x, yy

1

2b b2 4 c , x

b b2 4 c

2, y

1

2b b2 4 c , x

b b2 4 c

2,

y 1

2b b2 4 c , x

b

21

2b2 4 c ,

y 1

2b b2 4 c , x

b

21

2b2 4 c

Sc .

Sc Sinx^2 Cosx^2 1

Cosx2 Sinx2 1

SolveCosx 2 Sinx 1, Sc, Sinx, CosxSinx 0, Cosx 1, Sinx 4

5, Cosx

3

5

Sumi, i, 1, 9, 225

Sum2 i 1, i, 1, 525

Sumi j, i, 1, 5, j, 1, 5225

Producti j, i, 1, 5, j, 1, 5619 173 642 240 000 000 000

NSum1 i^2, i, 1, Infinity1.64493

NSum1 i^2, i, 1, Infinity, 21.2337

NProduct1 i^2, i, 1, Infinity, 20.

10 1st.nb

gx_ Sinx^2 1 xPlotgx, x, 0, 2 PiSinx21 x

1 2 3 4 5 6

0.3

0.2

0.1

0.1

0.2

0.3

0.4

Plotgx, x, 0, 2 Pi, AspectRatio 1 2

1 2 3 4 5 6

0.3

0.2

0.1

0.1

0.2

0.3

0.4

Plotgx, x, 0, 2 Pi, Ticks none

1st.nb 11

Plotgx, x, 0, 2 Pi, AxesLabel "time", "height"

1 2 3 4 5 6time

0.3

0.2

0.1

0.1

0.2

0.3

0.4

height

Plotgx, x, 0, 2 Pi, AxesOrigin 3, 0, PlotLabel "Decay Waves"

0 1 2 4 5 6

0.3

0.2

0.1

0.1

0.2

0.3

0.4

Decay Waves

Plotgx, x, 0, 2 Pi, Ticks 0, Pi 2, 3 Pi 2, 2 Pi, Automatic

2

3

22

0.3

0.2

0.1

0.1

0.2

0.3

0.4

12 1st.nb

Plotgx, x, 0, 2 Pi, PlotRange 0.6, 0.6

1 2 3 4 5 6

0.6

0.4

0.2

0.2

0.4

0.6

g1 Plotgx, x, 0, 2 Pi;g2 Plotx Cosx 12, x, 0, 2 Pi;Showg1, g2

1 2 3 4 5 6

0.3

0.2

0.1

0.1

0.2

0.3

0.4

ListPlot[{y1, y2, … ..}] 绘出在x的值为1，2…时y1, y2, …的图形

ListPlot[{{x1, y1}, {x2, y2}, … ..}] 绘出离散点 （xi, yi）

ListPlot[List, PlotJoined -> True] 把离散点连成曲线

ParametricPlot[{fx,fy},{t,tmin,tmax}] 绘出参数图

ParametricPlot[{{fx,fy},{gx,gy},….},{t,tmin,tmax}] 绘出一组参数图

ParametricPlot[{fx,fy},{t,tmin,tmax},AspectRatio->Automatic] 设法保持曲线的形

1st.nb 13

ParametricPlotSin3 t Cost, Sin3 t Sint, t, 0, 2 Pi

0.5 0.5

1.0

0.5

0.5

ParametricPlotSin3 t Cost, Sin3 t Sin2 t, Sint, Cost,t, 0, 2 Pi, AspectRatio Automatic

1.0 0.5 0.5 1.0

1.0

0.5

0.5

1.0

List1 Tablei^3 i, i, 102, 10, 30, 68, 130, 222, 350, 520, 738, 1010

14 1st.nb

ListPlotList1

2 4 6 8 10

200

400

600

800

1000

ListPlotList1, PlotJoined True

2 4 6 8 10

200

400

600

800

1000

g1 GraphicsText"left", 1, 0, Text"right", 1, 0, Text"above", 0, 1,Text"below", 0, 1, PointSize0.4, Point0, 0, PlotRange All

left right

above

below

1st.nb 15

LineTablen, 1^n, n, 6Line1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1Graphics

ShowGraphics, Axes True

2 3 4 5 6

1.0

0.5

0.5

1.0

St : TableRectanglex, 0, x 0.08, Sinx, x, 0, 2 Pi, 0.15ShowGraphicsSt, Axes True

1 2 3 4 5 6

1.0

0.5

0.5

1.0

16 1st.nb

GraphicsCircle0, 0, 1, Axes True

1.0 0.5 0.5 1.0

1.0

0.5

0.5

1.0

ShowGraphicsCircle0, 0, 5, 3, Axes True

4 2 2 4

3

2

1

1

2

3

1st.nb 17

GraphicsCircle0, 0, 1, 0, Pi 2, Axes True

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

ShowGraphicsCircle0, 0, 5, 3, Pi 2, 3 Pi 2, Axes True, AspectRatio Automatic

5 4 3 2 1

3

2

1

1

2

3

18 1st.nb

GraphicsDisk0, 0, 1, Axes True

1.0 0.5 0.5 1.0

1.0

0.5

0.5

1.0

GraphicsRaster0, 0, 1, 0, 1, 0, 1, 0, 0

1st.nb 19

PlotSinx, Sin2 x, Sin3 x, x, 0, 2 Pi,PlotStyle RGBColor0.9, 0, 0, RGBColor0, 0.9, 0, RGBColor0, 0, 0.9

1 2 3 4 5 6

1.0

0.5

0.5

1.0

v1 1, 0, 0, 1, 1, 0, 0, 11, 0, 0, 1, 1, 0, 0, 1ShowGraphicsHue0.2, Polygon3 v1, Hue0.4, Polygon2 v1, Hue0.9, Polygonv1,AspectRatio Automatic

TablePointn^2, Primen, n, 5;

20 1st.nb

ShowGraphicsPointSize0.1, , PlotRange All

TableGraphicsAbsolutePointSized, Point0, 0, d, 0.5, 2, 7, 15

,

1st.nb 21

,

,

22 1st.nb

ShowGraphics

TableAbsoluteThicknessd, Line0, 0, 1, d, d, 5,Line0, 5, 1, 0

1st.nb 23

PlotSinx^2, x, Pi, Pi

3 2 1 1 2 3

1.0

0.5

0.5

1.0

Show, PlotRange 1, 2, Frame True

3 2 1 0 1 2 31.0

0.5

0.0

0.5

1.0

1.5

2.0

f1 Plotx Sin2 x Pi, x, 0, 4 Pi;f2 Plotx Cos2 x, x, 0, 4 Pi;Showf1, f2

2 4 6 8 10 12

10

5

5

10

24 1st.nb

ShowGraphicsArray, f1, , f22 4 6 8 10 12

10

5

5

10

2 4 6 8 10 12

10

5

5

10

2 4 6 8 10 12

10

5

5

10

2 4 6 8 10 12

10

5

5

10

t1 Plot3DSinx y Cosx y, x, 0, 4, y, 0, 4

1st.nb 25

Show, PlotRange 0, 0.5

Showt1, AxesLabel "time", "depth", "Value", FaceGrids All

26 1st.nb

Showt1, Axes False, Boxed False

Showt1, Mesh None

1st.nb 27

Plot3DSinx y Cosx y, x, 0, 4, y, 0, 4, Mesh None

Plot3DSinx y Cosx y, x, 0, 4, y, 0, 4, Shading FalsePlot3D::optx : Unknown option Shading in Plot3DSinx y Cosx y, x, 0, 4, y, 0, 4, Shading False. Plot3DSinx y Cosx y, x, 0, 4, y, 0, 4, Shading FalsePlot3DSinx y Cosx y, x, 0, 4, y, 0, 4, Lighting None

28 1st.nb

MyTable :TableSinx y RandomReal, 0.15, 0.15, x, 0, 3 Pi 2, Pi 15, y, 0, 3 Pi 2, Pi 15

ListPlot3DMyTable

ParametricPlot3D3 Cos4 t 1, Cos2 t 3, 4 Cos2 t 5, t, 0, Pi2

0

2

1.00.5

0.00.5

1.0

4

2

0

2

4

1st.nb 29

ParametricPlot3Dr, Expr^2 Cos4 r^2 Cost, Expr^2 Cos4 r^2 Sint, r, 1, 1, t, 0, 2 Pi

LimitSqrtx^2 2 3 x 6, x Infinity1

3

LimitSinx^2 x^2, x 01

LimitLogx x, x 0, Direction 1

DExpx Sinx, xx Cosx x SinxDExpx Sinx, x, 22 x CosxDSina x, xa Cosa x

30 1st.nb

DSina x, x, NonConstants aCosa x a x Da, x, NonConstants afx_, y_ x^2 y y^2

x2 y y2

Dfx, y, x2 x y

Dfx, y, yx2 2 y

Dfx, y, x, 22 y

Dfx, y, y, 22

Dfx, y, x, y2 x

Dx f3x, xf3x x f3xDf3f4x, xf3f4x f4xDExpx Sinx, x . x 2

2 Cos2 2 Sin2Dtx^2 y^2, x2 x 2 y Dty, xDfx^2 y^2Dfx2 y2Dtx^2 xy^3 yz, Constants z2 x Dtx, Constants z 3 xy2 Dtxy, Constants z Dtyz, Constants zDtx^2 xyx yx z2 x Dtx Dtz yx Dtx xyx z Dtx yx u 1 u2

2 11 u2u

1

12111 1 u2 3 11 ArcTanh1

311 1 u2

1st.nb 31

SinSinx SinxIntegrate::ivar : Sinx is not a valid variable. SinSinx Sinx SinSinx x SinSinx x a x2 b x c xc x

b x2

2a x3

3

4

6

x2 eax x

280 eax

3

1

1

x4x

1

3

1

1

xpx

IfRep 1,1

1 p, Integratexp, x, 1, , Assumptions Rep 1

NIntegrateSinSinx, x, 0, Pi1.78649

NIntegrate1 SqrtAbsx, x, 1, 0, 14.

NIntegrateExpx^2, x, 0, Infinity0.886227

DSinx y^2, x, x, y2 x y5 Cosx y2 4 y3 Sinx y2DSinx y^2, x, 2, y2 x y5 Cosx y2 4 y3 Sinx y2Dx^2 y^2, x, NonConstants y2 x 2 y Dy, x, NonConstants y

32 1st.nb

Dtx2 y32 x y3 Dtx 3 x2 y2 Dtyz x3 y x2 y2 3 x y2;

Dtz3 x2 y Dtx 3 y2 Dtx 2 x y2 Dtx x3 Dty 6 x y Dty 2 x2 y DtyCollectDtz, Dtx, Dty3 x2 y 3 y2 2 x y2 Dtx x3 6 x y 2 x2 y Dty . Dtx dx, Dty dydy x3 6 x y 2 x2 y dx 3 x2 y 3 y2 2 x y2Dtz, x3 x2 y 3 y2 2 x y2 x3 Dty, x 6 x y Dty, x 2 x2 y Dty, x . Dty, x 0

3 x2 y 3 y2 2 x y2

Dt5 y^2 Siny x^2, x10 y Dty, x Cosy Dty, x 2 x

Solve, Dty, xDty, x 2 x

10 y Cosy Dtx^2 y^2 z^2, x, Constants z2 x 2 y Dty, x, Constants x3 y 3 x y2 x2 y2Dtz, x, y3 x2 6 y 4 x y 6 x y Dtx, y 2 y2 Dtx, y 6 x Dty, x 2 x2 Dty, x 3 x2 Dtx, y Dty, x 6 y Dtx, y Dty, x 4 x y Dtx, y Dty, x

0

a0

bx^2 y^2 x y1

3a b a2 b2

NIntegrateSqrtx y, x, 0, 2, y, 0, Sqrtx 24.65557

NIntegrateSqrtx^2 z^2, x, 2, 2, y, x^2, 4, z, Sqrty x^2, Sqrty x^226.8083

y .; DSolvey'x 2 yx, yx, xyx 2 x C1

1st.nb 33

yx y0 y'x .

2 x C1 y0 yx解y[x]仅适合其本身，并不适合于y[x]的其它形式，如y’[x]，y[0]等，也就是说y[x]不是函数

DSolvey'x 2 yx, y, xy Functionx, 2 x C1yx y0 y'x . C1 3 2 x C1

y .; z .; DSolveyx z'x, zx y'x, y, z, xz Functionx, 1

2x 1 2 x C1 1

2x 1 2 x C2,

y Functionx, 1

2x 1 2 x C1 1

2x 1 2 x C2

y .; z .; DSolveyx z'x, zx y'x, yx, zx, xzx 1

2x 1 2 x C1 1

2x 1 2 x C2,

yx 1

2x 1 2 x C1 1

2x 1 2 x C2

DSolvey'x yx, y0 5, yx, xyx 5 xs1 NDSolvey'x 1 2 yx, y0.01 0.1, y, x, 0.01, 1y InterpolatingFunction0.01, 1., PlotEvaluateyx . s1, x, 0.01, 1, AxesOrigin 0, 0

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

t 10

10

34 1st.nb

Modulet, t 8; Printt8

t

10

fv_ : Modulet, t 1 v^2; Expandt; fa1 2 a a2

t

10

gu_ : Modulet u, t t t 1 u;ga ba b

a b

1 a b

x^2 1

1 x2

Blockx a 1, 1 1 a2x

x

m i^2

i2

Blocki a, i ma a2

Modulei a, i ma i2

Removeggx_ : 1 ; x 0

gx_ : 1 ; x 0

? g

Global`g

gx_ : 1 ; x 0

gx_ : 1 ; x 0

Removeh; hx_ : Whichx 0, 1, x 0, 0, x 0. 1

1st.nb 35

h1, h0, h3h1, h0, h3qx_ : SwitchModx, 3, 0, a, 1, b, 2, cq17c

If[x==y,a,b,c]可以给If加上第三个条件结果,这允许你测试的结果既不是真也不是假的情况下使用它除非表达式能得出真,否则都被假设为假

Ife f, a, b, cc

TrueQe fFalse

e f

False

Mathematica处理逻辑表达式的方法允许你组合一系列的测试条件,且只有当前面条件满足时才处理后面的条件

DoPrinti i^2, i, 1, 42

6

12

20

DoPrinti, j, i, 4, j, i 12, 13, 13, 24, 14, 24, 3t 67;DoPrintt; t Floort 2, 3

67

33

16

n 25;Whilen Floorn 3 0, Printn

36 1st.nb

8

2

Fori 1, i 5, i, Printi1

2

3

4

x .; Fori 1; t x, i^2 10, i, t t^2 i; Printt1 x2

2 1 x223 2 1 x222

Nestf, x, 5fffffxNestFunctiont, 1 Sqrt1 t^2, x, 2

1

1 1

1x2

FixedPointFunctiont, Printt; Floort 3, 6767

22

7

2

0

0

t 1;Dot k; Printt; Ift 20, Break, k, 10

1

2

6

24

t 1;Dot k; Printt; Ift 3, Continue; t 2, k, 5

1st.nb 37

1

2

6

32

170

Removeffx_ : Ifx 5, Returnbig; t x^3; Returnt 7f3big

f5118

hx_ : Ifx 0, Throwerror, xCatchh36

(不产生error，且出示Catch的结果无效)

Catchh3error

Residuefz z^5, z, 00

Residue1 Sinz^5, z, 03

8

SeriesExpx, x, 0, 101 x

x2

2x3

6x4

24

x5

120

x6

720

x7

5040

x8

40 320

x9

362 880

x10

3 628 800 Ox11

Seriesx^x, x, 0, 41 Logx x 1

2Logx2 x2 1

6Logx3 x3 1

24Logx4 x4 Ox5

Normal1 x Logx 1

2x2 Logx2 1

6x3 Logx3 1

24x4 Logx4

38 1st.nb

Sum1 2 n 1 2 n 1, n, 1, Infinity1

2

SumLogn 1 n, n, 1, InfinitySum::div : Sum does not converge. n1

Log 1 nn

1st.nb 39

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