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</o:shapelayout></xml><![endif]--></head><body lang=FR link=blue vlink=purple><div class=WordSection1><p class=MsoNormal><span style='font-size:10.5pt;font-family:Monospaced;color:#B01813'>Hello,<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:10.5pt;font-family:Monospaced;color:#B01813'>This is what I meant when I suggested to remove the singularity, and seems to be close to what you want; and it runs smoothly with Scilab 5.3.2 for a slightly larger domain of parameters (limited by deltaPsB >~0.943, for gsR=10) than without singularity removal.<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:10.5pt;font-family:Monospaced;color:#B01813'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:10.5pt;font-family:Monospaced;color:#B01813'> function</span><span style='font-size:10.5pt;font-family:Monospaced'> <span style='color:#4A55DB'>[</span><b><span style='color:#834310'>Xprime</span></b><span style='color:#4A55DB'>]</span><span style='color:#5C5C5C'>=</span><u><span style='color:black'>VanDerPol</span></u><span style='color:#4A55DB'>(</span><b><span style='color:#834310'>t</span></b><span style='color:black'>, </span><b><span style='color:#834310'>X</span></b><span style='color:#4A55DB'>)</span><o:p></o:p></span></p><p class=MsoNormal><span style='font-size:10.5pt;font-family:Monospaced'> if <b><span style='color:#834310'>X</span></b><span style='color:#4A55DB'>(</span><span style='color:rosybrown'>1</span><span style='color:#4A55DB'>)==</span><span style='color:orchid'> %pi</span><span style='color:#5C5C5C'>/</span><span style='color:rosybrown'>2</span><span style='color:#5C5C5C'> then singularity=0; else singularity=</span><span style='color:#4A55DB'>(</span><span style='color:rosybrown'>1</span><span style='color:#5C5C5C'>-</span><span style='color:#32B9B9'>sin</span><span style='color:#4A55DB'>(</span><b><span style='color:#834310'>X</span></b><span style='color:#4A55DB'>(</span><span style='color:rosybrown'>1</span><span style='color:#4A55DB'>)))</span><span style='color:#5C5C5C'>/</span><span style='color:#4A55DB'>(</span><span style='color:orchid'>%pi</span><span style='color:#5C5C5C'>/</span><span style='color:rosybrown'>2</span><span style='color:#5C5C5C'>-</span><b><span style='color:#834310'>X</span></b><span style='color:#4A55DB'>(</span><span style='color:rosybrown'>1</span><span style='color:#4A55DB'>)); end;</span><o:p></o:p></span></p><p class=MsoNormal><span style='font-size:10.5pt;font-family:Monospaced'> <b><span style='color:#834310'>Xprime</span></b><span style='color:#4A55DB'>(</span><span style='color:rosybrown'>1</span><span style='color:#4A55DB'>)</span> <span style='color:#5C5C5C'>=</span> <b><span style='color:#834310'>X</span></b><span style='color:#4A55DB'>(</span><span style='color:rosybrown'>2</span><span style='color:#4A55DB'>)</span><o:p></o:p></span></p><p class=MsoNormal><span style='font-size:10.5pt;font-family:Monospaced'> <b><span style='color:#834310'>Xprime</span></b><span style='color:#4A55DB'>(</span><span style='color:rosybrown'>2</span><span style='color:#4A55DB'>)</span> <span style='color:#5C5C5C'>=</span> <span style='color:black'>gsR</span><span style='color:#5C5C5C'>*singularity -</span><span style='color:black'>deltaPsB</span><span style='color:#5C5C5C'>*</span><b><span style='color:#834310'>X</span></b><span style='color:#4A55DB'>(</span><span style='color:rosybrown'>2</span><span style='color:#4A55DB'>)^2</span><span style='color:#5C5C5C'>/</span><span style='color:#4A55DB'>(</span><span style='color:orchid'>%pi</span><span style='color:#5C5C5C'>/</span><span style='color:rosybrown'>2</span><span style='color:#5C5C5C'>-</span><b><span style='color:#834310'>X</span></b><span style='color:#4A55DB'>(</span><span style='color:rosybrown'>1</span><span style='color:#4A55DB'>))</span><o:p></o:p></span></p><p class=MsoNormal><span style='font-size:10.5pt;font-family:Monospaced'> <o:p></o:p></span></p><p class=MsoNormal><span style='font-size:10.5pt;font-family:Monospaced;color:#B01813'>endfunction</span><span style='font-size:10.5pt;font-family:Monospaced'><o:p></o:p></span></p><p class=MsoNormal><span style='font-size:10.5pt;font-family:Monospaced;color:black'>gsR</span><span style='font-size:10.5pt;font-family:Monospaced;color:#5C5C5C'>=</span><span style='font-size:10.5pt;font-family:Monospaced;color:rosybrown'>10</span><span style='font-size:10.5pt;font-family:Monospaced;color:black'>;</span><span style='font-size:10.5pt;font-family:Monospaced'><o:p></o:p></span></p><p class=MsoNormal><span style='font-size:10.5pt;font-family:Monospaced;color:black'>deltaPsB</span><span style='font-size:10.5pt;font-family:Monospaced;color:#5C5C5C'>=</span><span style='font-size:10.5pt;font-family:Monospaced;color:rosybrown'>1</span><span style='font-size:10.5pt;font-family:Monospaced;color:black'>;</span><span style='font-size:10.5pt;font-family:Monospaced'><o:p></o:p></span></p><p class=MsoNormal><span style='font-size:10.5pt;font-family:Monospaced'> <span style='color:black'>t</span><span style='color:#5C5C5C'>=</span><u><span style='color:#AE5CB0'>linspace</span></u><span style='color:#4A55DB'>(</span><span style='color:rosybrown'>0</span><span style='color:black'>,</span><span style='color:rosybrown'>10</span><span style='color:black'>,</span><span style='color:rosybrown'>10000</span><span style='color:#4A55DB'>)</span><span style='color:black'>;</span> <span style='color:black'>X0</span><span style='color:#5C5C5C'>=</span><span style='color:#4A55DB'>[</span><span style='color:rosybrown'>85</span><span style='color:#5C5C5C'>*</span><span style='color:orchid'>%pi</span><span style='color:#5C5C5C'>/</span><span style='color:rosybrown'>180</span><span style='color:black'>;</span><span style='color:rosybrown'>0</span><span style='color:#4A55DB'>]</span><span style='color:black'>;</span><o:p></o:p></span></p><p class=MsoNormal><span style='font-size:10.5pt;font-family:Monospaced'> <span style='color:#4A55DB'>[</span><span style='color:black'>u</span><span style='color:#4A55DB'>]</span><span style='color:#5C5C5C'>=</span><span style='color:#32B9B9'>ode</span><span style='color:#4A55DB'>(</span><span style='color:black'>X0,</span><span style='color:rosybrown'>0</span><span style='color:black'>,t,<u>VanDerPol</u></span><span style='color:#4A55DB'>)</span><span style='color:black'>;</span><o:p></o:p></span></p><p class=MsoNormal><span style='font-size:10.5pt;font-family:Monospaced;color:#32B9B9'>xset</span><span style='font-size:10.5pt;font-family:Monospaced;color:#4A55DB'>(</span><span style='font-size:10.5pt;font-family:Monospaced;color:rosybrown'>'window'</span><span style='font-size:10.5pt;font-family:Monospaced;color:black'>,</span><span style='font-size:10.5pt;font-family:Monospaced;color:rosybrown'>0</span><span style='font-size:10.5pt;font-family:Monospaced;color:#4A55DB'>)</span><span style='font-size:10.5pt;font-family:Monospaced;color:black'>;</span><span style='font-size:10.5pt;font-family:Monospaced'><o:p></o:p></span></p><p class=MsoNormal><u><span style='font-size:10.5pt;font-family:Monospaced;color:#AE5CB0'>plot</span></u><span style='font-size:10.5pt;font-family:Monospaced;color:#4A55DB'>(</span><span style='font-size:10.5pt;font-family:Monospaced;color:black'>t,u</span><span style='font-size:10.5pt;font-family:Monospaced;color:#4A55DB'>(</span><span style='font-size:10.5pt;font-family:Monospaced;color:rosybrown'>1</span><span style='font-size:10.5pt;font-family:Monospaced;color:black'>,</span><span style='font-size:10.5pt;font-family:Monospaced;color:#FFAA00'>:</span><span style='font-size:10.5pt;font-family:Monospaced;color:#4A55DB'>)</span><span style='font-size:10.5pt;font-family:Monospaced;color:#5C5C5C'>*</span><span style='font-size:10.5pt;font-family:Monospaced;color:rosybrown'>180</span><span style='font-size:10.5pt;font-family:Monospaced;color:#5C5C5C'>/</span><span style='font-size:10.5pt;font-family:Monospaced;color:orchid'>%pi</span><span style='font-size:10.5pt;font-family:Monospaced;color:black'>,</span><span style='font-size:10.5pt;font-family:Monospaced;color:rosybrown'>'r'</span><span style='font-size:10.5pt;font-family:Monospaced;color:#4A55DB'>)</span><span style='font-size:10.5pt;font-family:Monospaced'><o:p></o:p></span></p><p class=MsoNormal><span style='font-size:10.5pt;font-family:Monospaced;color:#32B9B9'>xset</span><span style='font-size:10.5pt;font-family:Monospaced;color:#4A55DB'>(</span><span style='font-size:10.5pt;font-family:Monospaced;color:rosybrown'>'window'</span><span style='font-size:10.5pt;font-family:Monospaced;color:black'>,</span><span style='font-size:10.5pt;font-family:Monospaced;color:rosybrown'>2</span><span style='font-size:10.5pt;font-family:Monospaced;color:#4A55DB'>)</span><span style='font-size:10.5pt;font-family:Monospaced;color:black'>;</span><span style='font-size:10.5pt;font-family:Monospaced'><o:p></o:p></span></p><p class=MsoNormal><i><span style='font-size:10.5pt;font-family:Monospaced;color:#01A801'>// plot(t,R*u(2,:))</span></i><span style='font-size:10.5pt;font-family:Monospaced'><o:p></o:p></span></p><p class=MsoNormal><span style='font-size:10.5pt;font-family:Monospaced'> <u><span style='color:#AE5CB0'>plot</span></u><span style='color:#4A55DB'>(</span><span style='color:black'>t,u</span><span style='color:#4A55DB'>(</span><span style='color:rosybrown'>2</span><span style='color:black'>,</span><span style='color:#FFAA00'>:</span><span style='color:#4A55DB'>))</span><o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:10.5pt;font-family:Monospaced;color:#B01813'>This suggests that the 2nd singularity (last term scaled with deltaP) is the reason for the problem. Furthermore, I don't understand the physical meaning of this term W'²/W…<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:10.5pt;font-family:Monospaced;color:#B01813'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D'>In addition, it behaves as one expects: as the motion is damped by the second term, is goes slowly (if deltaPsB is large) towards X=%pi/2. I doubt that with such an equation you can find an oscillatory behaviour (which is what the Van der Pol equation should describe, shouldn't it ?-)<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D'>Denis <o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D'><o:p> </o:p></span></p><div><div style='border:none;border-top:solid #B5C4DF 1.0pt;padding:3.0pt 0cm 0cm 0cm'><p class=MsoNormal><b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'>De :</span></b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'> arnaud dujeu [mailto:fangorn813@hotmail.com] <br><b>Envoyé :</b> jeudi 21 juillet 2011 16:20<br><b>À :</b> users@lists.scilab.org<br><b>Objet :</b> RE: [scilab-Users] problem with scilab<o:p></o:p></span></p></div></div><p class=MsoNormal><o:p> </o:p></p><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'>In fact I have try several methods and no one is working, do you know if it's possible with scilab to plot alpha(t) and alpha'(t) when the differential equation is like : <o:p></o:p></span></p><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'>alpha''(t) = A*(1-sin(alpha(t)))/(pi/2-alpha(t))-B*(alpha'(t))^2/(pi/2-alpah(t))<o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'><o:p> </o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'>What would be the function to use ?<o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'><o:p> </o:p></span></p></div><div><p class=MsoNormal style='margin-bottom:12.0pt'><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'><o:p> </o:p></span></p><div><div class=MsoNormal align=center style='text-align:center'><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'><hr size=2 width="100%" align=center id=stopSpelling></span></div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'>Date: Thu, 21 Jul 2011 14:06:16 +0200<br>From: Serge.Steer@inria.fr<br>To: users@lists.scilab.org<br>Subject: Re: [scilab-Users] problem with scilab<br><br><br>What do you mean by "<span style='color:#4A55DB'>it doesn't work.</span>" error message?, unexpected results?<br><br>I was not able to run it because g R DeltaP and B are undefined.<br><br><br>Serge Steer<br>INRIA<br><br>Le 21/07/2011 11:25, arnaud dujeu a écrit : <o:p></o:p></span></p><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'>Hi, <o:p></o:p></span></p><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'><o:p> </o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'>I've got a problem to solve a non linear differential equation, which is :<o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'><o:p> </o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'>y1'=A*(1-sin(y2))/(pi/2-y2)-B*y1^2<o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'>y2'=y1<o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'><o:p> </o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'><o:p> </o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'>I'm using ode : <o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'><o:p> </o:p></span></p></div><div><pre><span style='font-size:10.5pt;font-family:Monospaced;color:#B01813'>function</span><span style='font-size:10.5pt;font-family:Monospaced'> <span style='color:#4A55DB'>[</span><b><span style='color:#834310'>Xprime</span></b><span style='color:#4A55DB'>]</span><span style='color:#5C5C5C'>=</span><u><span style='color:black'>VanDerPol</span></u><span style='color:#4A55DB'>(</span><b><span style='color:#834310'>t</span></b><span style='color:black'>, </span><b><span style='color:#834310'>X</span></b><span style='color:#4A55DB'>)</span><o:p></o:p></span></pre><pre><span style='font-size:10.5pt;font-family:Monospaced'> <o:p></o:p></span></pre><pre><span style='font-size:10.5pt;font-family:Monospaced'> <b><span style='color:#834310'>Xprime</span></b><span style='color:#4A55DB'>(</span><span style='color:rosybrown'>1</span><span style='color:#4A55DB'>)</span> <span style='color:#5C5C5C'>=</span> <b><span style='color:#834310'>X</span></b><span style='color:#4A55DB'>(</span><span style='color:rosybrown'>2</span><span style='color:#4A55DB'>)</span><o:p></o:p></span></pre><pre><span style='font-size:10.5pt;font-family:Monospaced'> <b><span style='color:#834310'>Xprime</span></b><span style='color:#4A55DB'>(</span><span style='color:rosybrown'>2</span><span style='color:#4A55DB'>)</span> <span style='color:#5C5C5C'>=</span> <span style='color:black'>g</span><span style='color:#5C5C5C'>/</span><span style='color:black'>R</span><span style='color:#5C5C5C'>*</span><span style='color:#4A55DB'>(</span><span style='color:rosybrown'>1</span><span style='color:#5C5C5C'>-</span><span style='color:#32B9B9'>sin</span><span style='color:#4A55DB'>(</span><b><span style='color:#834310'>X</span></b><span style='color:#4A55DB'>(</span><span style='color:rosybrown'>1</span><span style='color:#4A55DB'>)))</span>/<span style='color:#4A55DB'>(</span><span style='color:orchid'>%pi</span><span style='color:#5C5C5C'>/</span><span style='color:rosybrown'>2</span><span style='color:#5C5C5C'>-</span><b><span style='color:#834310'>X</span></b><span style='color:#4A55DB'>(</span><span style='color:rosybrown'>1</span><span style='color:#4A55DB'>))</span><span style='color:#5C5C5C'>-</span><span style='color:black'>deltaP</span><span style='color:#5C5C5C'>/</span><span style='color:#4A55DB'>(</span><span style='color:black'>B</span><span style='color:#5C5C5C'>*</span><span style='color:#4A55DB'>(</span><span style='color:orchid'>%pi</span><span style='color:#5C5C5C'>/</span>2<span style='color:#5C5C5C'>-</span><b><span style='color:#834310'>X</span></b><span style='color:#4A55DB'>(</span><span style='color:rosybrown'>1</span><span style='color:#4A55DB'>)))</span><o:p></o:p></span></pre><pre><span style='font-size:10.5pt;font-family:Monospaced'> <o:p></o:p></span></pre><pre><span style='font-size:10.5pt;font-family:Monospaced;color:#B01813'>endfunction</span><span style='font-size:10.5pt;font-family:Monospaced'><o:p></o:p></span></pre><pre><span style='font-size:10.5pt;font-family:Monospaced'><o:p> </o:p></span></pre><pre><span style='font-size:10.5pt;font-family:Monospaced'> <span style='color:black'>t</span><span style='color:#5C5C5C'>=</span><u><span style='color:#AE5CB0'>linspace</span></u><span style='color:#4A55DB'>(</span><span style='color:rosybrown'>0</span><span style='color:black'>,</span><span style='color:rosybrown'>4</span><span style='color:black'>,</span><span style='color:rosybrown'>3000</span><span style='color:#4A55DB'>)</span><span style='color:black'>;</span> <span style='color:black'>X0</span><span style='color:#5C5C5C'>=</span><span style='color:#4A55DB'>[</span><span style='color:rosybrown'>85</span><span style='color:#5C5C5C'>*</span><span style='color:orchid'>%pi</span><span style='color:#5C5C5C'>/</span><span style='color:rosybrown'>180</span><span style='color:black'>;</span><span style='color:rosybrown'>0</span><span style='color:#4A55DB'>]</span><span style='color:black'>;</span><o:p></o:p></span></pre><pre><span style='font-size:10.5pt;font-family:Monospaced'> <span style='color:#4A55DB'>[</span><span style='color:black'>u</span><span style='color:#4A55DB'>]</span><span style='color:#5C5C5C'>=</span><span style='color:#32B9B9'>ode</span><span style='color:#4A55DB'>(</span><span style='color:black'>X0,</span><span style='color:rosybrown'>0</span><span style='color:black'>,t,<u>VanDerPol</u></span><span style='color:#4A55DB'>)</span><span style='color:black'>;</span><o:p></o:p></span></pre><pre><span style='font-size:10.5pt;font-family:Monospaced'><o:p> </o:p></span></pre><pre><u><span style='font-size:10.5pt;font-family:Monospaced;color:#AE5CB0'>plot</span></u><span style='font-size:10.5pt;font-family:Monospaced;color:#4A55DB'>(</span><span style='font-size:10.5pt;font-family:Monospaced;color:black'>t,u</span><span style='font-size:10.5pt;font-family:Monospaced;color:#4A55DB'>(</span><span style='font-size:10.5pt;font-family:Monospaced;color:rosybrown'>1</span><span style='font-size:10.5pt;font-family:Monospaced;color:black'>,</span><span style='font-size:10.5pt;font-family:Monospaced;color:#FFAA00'>:</span><span style='font-size:10.5pt;font-family:Monospaced;color:#4A55DB'>)</span><span style='font-size:10.5pt;font-family:Monospaced;color:#5C5C5C'>*</span><span style='font-size:10.5pt;font-family:Monospaced;color:rosybrown'>180</span><span style='font-size:10.5pt;font-family:Monospaced;color:#5C5C5C'>/</span><span style='font-size:10.5pt;font-family:Monospaced;color:orchid'>%pi</span><span style='font-size:10.5pt;font-family:Monospaced;color:black'>,</span><span style='font-size:10.5pt;font-family:Monospaced;color:rosybrown'>'r'</span><span style='font-size:10.5pt;font-family:Monospaced;color:#4A55DB'>)</span><span style='font-size:10.5pt;font-family:Monospaced'><o:p></o:p></span></pre><pre><span style='font-size:10.5pt;font-family:Monospaced'> <u><span style='color:#AE5CB0'>plot</span></u><span style='color:#4A55DB'>(</span><span style='color:black'>t,R</span><span style='color:#5C5C5C'>*</span><span style='color:black'>u</span><span style='color:#4A55DB'>(</span><span style='color:rosybrown'>2</span><span style='color:black'>,</span><span style='color:#FFAA00'>:</span><span style='color:#4A55DB'>))</span><o:p></o:p></span></pre><pre><span style='font-size:10.5pt;font-family:Monospaced;color:#4A55DB'><o:p> </o:p></span></pre><pre><span style='font-size:10.5pt;font-family:Monospaced;color:#4A55DB'><o:p> </o:p></span></pre><pre><span style='font-size:10.5pt;font-family:Monospaced;color:#4A55DB'>but it doesn't work.</span><span style='font-size:10.5pt;font-family:Monospaced'><o:p></o:p></span></pre><pre><span style='font-size:10.5pt;font-family:Monospaced;color:#4A55DB'><o:p> </o:p></span></pre><pre><span style='font-size:10.5pt;font-family:Monospaced;color:#4A55DB'>Do you know why?</span><span style='font-size:10.5pt;font-family:Monospaced'><o:p></o:p></span></pre><pre><span style='font-size:10.5pt;font-family:Monospaced;color:#4A55DB'><o:p> </o:p></span></pre><pre><span style='font-size:10.5pt;font-family:Monospaced;color:#4A55DB'>Best regards.</span><span style='font-size:10.5pt;font-family:Monospaced'><o:p></o:p></span></pre></div></div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'><o:p> </o:p></span></p></div></div></div></div></body></html>