Re: Re: ORDINARY DIFFERENTIAL EQUATION

• To: mathgroup at smc.vnet.net
• Subject: [mg32271] Re: [mg32229] Re: [mg32170] ORDINARY DIFFERENTIAL EQUATION
• From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
• Date: Wed, 9 Jan 2002 23:37:30 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```I suggest that the best thing to do is to obtain the book
"VisualDSolve" by Dan Schwalbe and Stan Wagon together with the
accompanying package. It is of course possible to study such questions
with Mathematica "from scratch" but Visual DSolve makes it much easier
and the pictures you get draw faster and are more revealing that what I
could produce in the short time I can devote to this.

Andrzej

Andrzej Kozlowski
Toyama International University
JAPAN
http://platon.c.u-tokyo.ac.jp/andrzej/

On Wednesday, January 9, 2002, at 05:17  PM, Khaled Saad wrote:

>
> Dear Dr. Andrzej Kozlowski
> I would like to thank you for send me the answer my question .
> I have another question can the mathematica plot the bifurcation of
> these
> system.if it can could you give me the code of the plot (if you do not
> mind).
> With best regard
> Thanks
>    Khaled
>
>
>> From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
To: mathgroup at smc.vnet.net
>> CC: mathgroup at smc.vnet.net
>> Subject: [mg32271] [mg32229] Re: [mg32170] ORDINARY DIFFERENTIAL EQUATION
>> Date: Wed, 2 Jan 2002 22:55:41 +0900
>>
>> A few minutes after posting the answer below I noticed the obvious fact
>> that in your particular case yo do not need even to solve the
>> differential equations to see the flow field. In fact the easiest way
>> to
>> do so is simply load the package:
>>
>> <<Graphics`PlotField`
>>
>> define the function:
>>
>>
>> h[x_,y_]:={(1-x-0.5*y)*x,(1-y-0.8*x)*y}
>>
>> And then plot the flow vector field:
>>
>>
>> PlotVectorField[h[x,y],{x,0,1,0.1},{y,0,1,0.1}]
>>

>>
>>
>> On Wednesday, January 2, 2002, at 09:25  PM, Andrzej Kozlowski wrote:
>>
>>> You should specify the region over which you wish to plot the flow
>>> lines, which means both the "starting points" and the duration of
>>> flows. Since you do not I shall choose it myself.
>>>
>>>
>>>
>>> In[1]:=
>>> g[a_, b_] := Module[{x, y, t},
>>>    {x, y} /. Flatten[NDSolve[{Derivative[1][x][t] == (1 - x[t] -
>>> 0.5*y[t])*x[t],
>>>        Derivative[1][y][t] == (1 - y[t] - 0.8*x[t])*y[t], x[0] == a,
>>> y[0] == b}, {x, y},
>>>       {t, 0, 10}]]]
>>>
>>> In[2]:=
>>> pl[x_, y_] := ParametricPlot[Evaluate[Through[g[x, y][t]]], {t, 0,
>>> 10},
>>>    DisplayFunction -> Identity]
>>>
>>> In[3]:=
>>> flowLines = Table[pl[x, y], {x, 0, 1, 0.1}, {y, 0, 1, 0.1}];
>>>
>>> In[4]:=
>>> Show[flowLines, DisplayFunction -> \$DisplayFunction];
>>>
>>> You might also wish to see some arrows to make the directions of the
>>> flows more clear. Too many arrows tend to make a picture messy (unless
>>> one takes a lot of care to get their proportions right, for which I do
>>> not have time), so here are just a few:
>>>
>>> In[5]:=
>>> << "Graphics`Arrow`"
>>>
>>> In[6]:=
>>> vectors[x_, y_] := Graphics[Table[Arrow[Through[g[x, y][t]],
>>> Through[g[x, y][t]] +
>>>       (D[Through[g[x, y][s]], s] /. s -> t)/3], {t, 0, 10}]]
>>>
>>> In[7]:=
>>> vectorField = Table[vectors[x, y], {x, 0, 1, 0.2}, {y, 0, 1, 0.2}];
>>>
>>> In[8]:=
>>> Show[vectorField, DisplayFunction -> \$DisplayFunction];
>>>
>>
>>>
>>> On Wednesday, January 2, 2002, at 05:28  PM, Khaled Saad wrote:
>>>
>>>> Dear mathgroup
>>>> I am trayin to plot the directin of vector field of system of
>>>> ordinary
>>>> differential equation but i could not
>>>>
>>>> the system is
>>>> x'[t]=(1-x[t]-0.5 y[t])x[t]
>>>> y'[t]=(1-y[t]-0.8 x[t])y[t]
>>>>
>>>> With best regards
>>>> Khaled
>>>>
>>>> _________________________________________________________________
>>>> Chat with friends online, try MSN Messenger: http://messenger.msn.com
>>>>
>>>>
>>>>
>>>>
>>>
>>
>
>
> _________________________________________________________________