MaplePrimes - comments on Post, Pushing dsolve to its limits

MaplePrimes - comments on Post, Pushing dsolve to its limits http://www.mapleprimes.com/posts/124914-Pushing-Dsolve-To-Its-Limits en-us 2026 Maplesoft, A Division of Waterloo Maple Inc. Maplesoft Document System Tue, 09 Jun 2026 05:11:26 GMT Tue, 09 Jun 2026 05:11:26 GMT The latest comments added to the Post, Pushing dsolve to its limits http://www.mapleprimes.com/images/mapleprimeswhite.jpg MaplePrimes - comments on Post, Pushing dsolve to its limits http://www.mapleprimes.com/posts/124914-Pushing-Dsolve-To-Its-Limits animated gif of the simulations http://www.mapleprimes.com/posts/124914-Pushing-Dsolve-To-Its-Limits?ref=Feed:MaplePrimes:Pushing dsolve to its limits:Comments#comment124916 <img src="http://patrick.toche.free.fr/stuff/animation.gif" alt="animated gif" width="512" height="512">   The animated gif of my simulations is too large to be hosted with mapleprimes, so I uploaded it elsewhere and placed a link to it above. The preview is pretty bad, I don't know if this is caused by my browser or what. What you can see is a sharp change in the slope of q (relative to u as well as relative to time t). The red line and some of the lines right next to it are simulations that didn't converge. Any suggestions welcome! Thanks. The latest comments added to the Post, Pushing dsolve to its limits 124916 Sat, 20 Aug 2011 03:42:53 Z PatrickT PatrickT Warnings http://www.mapleprimes.com/posts/124914-Pushing-Dsolve-To-Its-Limits?ref=Feed:MaplePrimes:Pushing dsolve to its limits:Comments#comment124919 Are you satisfied that all the Warnings issued in your worksheet from dsolve/numeric are acceptable and may be safely ignored? The halting on events might be less suspicious (you did specify halting events, after all). But what about the warnings about stopping prematurely on account of the maxfun limit being hit? The latest comments added to the Post, Pushing dsolve to its limits 124919 Sat, 20 Aug 2011 07:40:54 Z pagan pagan warnings http://www.mapleprimes.com/posts/124914-Pushing-Dsolve-To-Its-Limits?ref=Feed:MaplePrimes:Pushing dsolve to its limits:Comments#comment124925 <a href="http://www.mapleprimes.com/posts/124914-Pushing-Dsolve-To-Its-Limits#comment124919">@pagan</a>  Thanks pagan for your feedback. I should have said something about the warnings. Do you have any suggestions in this respect? The warnings related to the "events" are exactly what I expect. I set them up to prevent Maple from integrating beyond the region of interest. The non-converging trajectories (red and red-ish colours) have a different kind of warning. <ol> <li>If I comment out the relerr, abserr, initstep, maxstep and use rkf45 or rosenbrock, for a quick and dirty simulation, I get something like: Warning, cannot evaluate the solution further left of -14.780539, probably a singularity </li> <li>If I set relerr, abserr, initstep, maxstep to attemp a finer simulation, with initial points closer to the stationary points, then I sometimes get a complaint about maxfun being exceeded. Then if I raise maxfun, this message is usually eliminated, but instead I get the warning about "probably a singularity"</li> </ol> The maxfun warning can be eliminated by setting maxfun well in excess of the default value, but this doesn't help dsolve converge, in my experience. The latest comments added to the Post, Pushing dsolve to its limits 124925 Sat, 20 Aug 2011 13:28:01 Z PatrickT PatrickT dealing with the stiffness http://www.mapleprimes.com/posts/124914-Pushing-Dsolve-To-Its-Limits?ref=Feed:MaplePrimes:Pushing dsolve to its limits:Comments#comment124995 I have two problems really, the first problem is to find out whether the non-converging trajectories result from the true dynamics of the system (I have a local existence result, but no global existence, so maybe after all the divergence is what actually happens), and --- assuming that the divergence is caused by the lack of accuracy of the simulations (most likely my inability to use dsolve, digits, and all that) --- the second problem is to get those tricky trajectories to converge. To investigate the behaviour near the points, I simulated a bunch of trajectories (500 to be exact) in the neighborhood of the points of interest. This is what I get near a point in the blue region, where simulations converge: <img src="http://patrick.toche.free.fr/stuff/cluster2.gif" alt="animated gif"> This is what I get near a point in the red region, where simulations diverge: <img src="http://patrick.toche.free.fr/stuff/cluster1.gif" alt="animated gif"> Out of 500 initial conditions in a neighborhood of this "stiff" point, only a small number converged (or appear to have done).  I used the default dsolve method (rkf45) and default values for relerr, abserr, etc.. Perhaps I need a scheme to identify such initial conditions and take it from there. I'm still thinking... Suggestions always welcome. EDIT. I should add that in both cases trajectories from "below" converge well, it is trajectories from "above" that cause problems. These are the trajectories characterized by a sharp change in direction very near the stationary points. The sharp change in direction occurs closer to the stationary point, the deeper you go into the "red-colored" region, and that is precisely the region where dsolve doesn't get convergence. You can see this from the picture I posted earlier (http://www.mapleprimes.com/view.aspx?sf=124914/419404/urq50a.jpg). EDIT 2. Further analysis of the system revealed a condition on the parameters necessary for stability. I discuss it in a comment below. The latest comments added to the Post, Pushing dsolve to its limits 124995 Mon, 22 Aug 2011 13:08:00 Z PatrickT PatrickT Message to mapleprimes administrators http://www.mapleprimes.com/posts/124914-Pushing-Dsolve-To-Its-Limits?ref=Feed:MaplePrimes:Pushing dsolve to its limits:Comments#comment124996 I was unable to upload the animated gifs posted above (they were too large) and therefore hosted them on an outside website. Do feel free to make local copies of the images and direct my links to the local copies, so that the images will be accessible even if my website is down. Thanks. The latest comments added to the Post, Pushing dsolve to its limits 124996 Mon, 22 Aug 2011 13:18:52 Z PatrickT PatrickT The diverging trajectories do diverge http://www.mapleprimes.com/posts/124914-Pushing-Dsolve-To-Its-Limits?ref=Feed:MaplePrimes:Pushing dsolve to its limits:Comments#comment125075 Further analysis of the system revealed a condition on the parameters necessary for stability. Roughly speaking, this condition arises when one "plugs in" the center manifold reduction back into the original system. I hadn't seen it. All this is very new to me (and quite frankly I don't even know if I'm doing it correctly). Once this condition is imposed, all the simulated trajectories converge (i.e. the red-colored steady states are outside the stable range). I have thus resolved the problem I had with the simulations --- they did not converge because they do not converge. There remains the issue of the sharp turn that q(t) makes as it approaches the steady state. (as seen here http://www.mapleprimes.com/view.aspx?sf=124914/419404/q10a.jpg). Is this "kink" a non-removable singularity or a manifestation of the slow-fast nature of the system? This "kink" is present on one side of the steady-state points only. The bit of the trajectory where q(t) declines towards its steady-state value, (following a long flat segment) is, visually speaking, in "close" agreement with the center manifold approximation, as would be expected if my analysis is correct.The sharp turn from flat to downward-sloping takes very little time, this would be a manifestation of the "fast" in slow-fast... (I've read a few introductions to slow-fast systems, but I can't say that I understand it well at all) The latest comments added to the Post, Pushing dsolve to its limits 125075 Wed, 24 Aug 2011 00:18:21 Z PatrickT PatrickT