In[214]:=

omegax = Pi omegay = 2 Pi RowBox[{Plot, [, RowBox[{{Cos[omegax t] , .5 Cos[omegay t]}, ,, {t, ... RowBox[{{, RowBox[{RowBox[{Thickness, [, 0.005, ]}], ,, RGBColor[0, 0, 1]}], }}]}], }}]}]}], ]}]

Out[214]= 

Pi

Out[215]= 

2 Pi

[Graphics:HTMLFiles/index_2.gif]

Out[216]= 

-Graphics-

In[217]:=

RowBox[{, RowBox[{omegax = Pi, , RowBox[{omegay, , =, , RowBox[{2.3, , Pi}] ... ness, [, 0.005, ]}], ,, RGBColor[0, 0, 1]}], }}]}], }}]}]}], ]}], , , }]}]

Out[217]= 

Pi

Out[218]= 

7.22566

[Graphics:HTMLFiles/index_4.gif]

Out[219]= 

-Graphics-

In[220]:=

RowBox[{, RowBox[{x[t_] = Cos[Pi t], , dx[t_] = D[x[t], t], ,  ... ParametricPlot[{x[t], y[t]}, {t, 1, 5500}, PlotStyle {{RGBColor[1, 0, 0]}}], }]}]

Out[220]= 

Cos[Pi t]

Out[221]= 

-(Pi Sin[Pi t])

[Graphics:HTMLFiles/index_6.gif]

Out[222]= 

-Graphics-

Out[223]= 

Cos[6.59734 t]

Out[224]= 

-6.59734 Sin[6.59734 t]

[Graphics:HTMLFiles/index_7.gif]

Out[225]= 

-Graphics-

[Graphics:HTMLFiles/index_8.gif]

Out[226]= 

-Graphics-

[Graphics:HTMLFiles/index_9.gif]

Out[227]= 

-Graphics-

Out[228]= 

Cos[Pi t]

Out[229]= 

-(Pi Sin[Pi t])

[Graphics:HTMLFiles/index_10.gif]

Out[230]= 

-Graphics-

Out[231]= 

Cos[Sqrt[2] Pi t]

Out[232]= 

-(Sqrt[2] Pi Sin[Sqrt[2] Pi t])

[Graphics:HTMLFiles/index_11.gif]

Out[233]= 

-Graphics-

In[162]:=

x[t_] = Cos[Pi t] Exp[-.1 t] dx[t_] = D[x[t], t]  ParametricPlot[{x[t], dx[t]} , {t, 1, 10}, PlotStyle {{RGBColor[1, 0, 0]}}]

Out[162]= 

Cos[Pi t]
---------
  0.1 t
 E

Out[163]= 

-0.1 Cos[Pi t]   Pi Sin[Pi t]
-------------- - ------------
     0.1 t           0.1 t
    E               E

[Graphics:HTMLFiles/index_13.gif]

Out[164]= 

-Graphics-

RowBox[{, , (*THE AXION*), (*Frank Wilczek -- Today ' s Nobel Prize ; also S . ... le {{RGBColor[0, 0, 1]}}, ,, AxesLabel-> {"t", "s"}}], ]}], ;}]}]}]

Out[209]= 

{{a -> InterpolatingFunction[{{0., 15.}}, <>]}}

[Graphics:HTMLFiles/index_15.gif]

In[211]:=

sol1 = NDSolve[{a''[t] + 2/t a '[t] + Pi a[t] 0, a[1] 1, a '[1] == 0}, a, {t, ... ,, RGBColor[0, 1, 0]}], }}], }}]}], ,, AxesLabel-> {"t", "s"}}], ]}], ;}]

Out[211]= 

{{a -> InterpolatingFunction[{{1., 50.}}, <>]}}

[Graphics:HTMLFiles/index_17.gif]

[Graphics:HTMLFiles/index_18.gif]

(*CHAOS*)

In[693]:=

omega = .7 c = .01 F = .02 Clear[x] sol1 = NDSolve[{x''[t] + c  x '[t] + Sin[x[t]] - ... ] ParametricPlot[{x[t], x '[t]}/.sol1 , {t, 1400, 1500}, PlotStyle {{RGBColor[1, 0, 0]}}]

Out[693]= 

0.7

Out[694]= 

0.01

Out[695]= 

0.02

Out[697]= 

{{x -> InterpolatingFunction[{{0., 1500.}}, <>]}}

[Graphics:HTMLFiles/index_21.gif]

[Graphics:HTMLFiles/index_22.gif]

Out[699]= 

-Graphics-

[Graphics:HTMLFiles/index_23.gif]

Out[700]= 

-Graphics-

In[701]:=

c = .03 F = .2 Clear[x] sol1 = NDSolve[{x''[t] + c  x '[t] + Sin[x[t]] - F Cos[omega ... ] ParametricPlot[{x[t], x '[t]}/.sol1 , {t, 1400, 1500}, PlotStyle {{RGBColor[1, 0, 0]}}]

Out[701]= 

0.03

Out[702]= 

0.2

Out[704]= 

{{x -> InterpolatingFunction[{{0., 1500.}}, <>]}}

[Graphics:HTMLFiles/index_25.gif]

[Graphics:HTMLFiles/index_26.gif]

Out[706]= 

-Graphics-

[Graphics:HTMLFiles/index_27.gif]

Out[707]= 

-Graphics-

In[708]:=

c = .01 F = .6 Clear[x] sol1 = NDSolve[{x''[t] + c  x '[t] + Sin[x[t]] - F Cos[omega ... ] ParametricPlot[{x[t], x '[t]}/.sol1 , {t, 1400, 1500}, PlotStyle {{RGBColor[1, 0, 0]}}]

Out[708]= 

0.01

Out[709]= 

0.6

Out[711]= 

{{x -> InterpolatingFunction[{{0., 1500.}}, <>]}}

[Graphics:HTMLFiles/index_29.gif]

[Graphics:HTMLFiles/index_30.gif]

Out[713]= 

-Graphics-

[Graphics:HTMLFiles/index_31.gif]

Out[714]= 

-Graphics-

In[722]:=

c = .03 F = .6 Clear[x] sol1 = NDSolve[{x''[t] + c  x '[t] + Sin[x[t]] - F Cos[omega ... ParametricPlot[{x[t], x '[t]}/.sol1 , {t, 10000, 15000}, PlotStyle {{RGBColor[1, 0, 0]}}]

Out[722]= 

0.03

Out[723]= 

0.6

Out[725]= 

{{x -> InterpolatingFunction[{{0., 15000.}}, <>]}}

[Graphics:HTMLFiles/index_33.gif]

[Graphics:HTMLFiles/index_34.gif]

Out[727]= 

-Graphics-

[Graphics:HTMLFiles/index_35.gif]

Out[728]= 

-Graphics-

RowBox[{(*Here we very the initial conditions .   The transition to Chaos is Clearly ... ,, {t, 0, 1500}, ,, , MaxSteps1000000}], ]}]}], ;}], , x[1500]/.sol1}]}]

Out[1181]= 

{0.0303327}

Out[1184]= 

{0.0303277}

Out[1190]= 

{-56.5483}

Out[1193]= 

{-56.4883}

Out[1197]= 

{13.8196}

Out[1200]= 

{-47.4458}

In[753]:=

omega = .7 ; c = .03 ; F = .6 ; Clear[x] ; sol1 = NDSolve[{x''[t] + c  x '[t] + Sin[ ... [0] == 0}], }}], ,, x, ,, {t, 0, 1500}, ,, , MaxSteps1000000}], ]}]}], ;}] x[1500]/.sol1

Out[758]= 

{-101.73}

Out[761]= 

{84.7761}

Created by Mathematica  (October 5, 2004)