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Offline Razonar  
#41 Posted : 08 November 2018 23:37:16(UTC)
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Originally Posted by: Ber7 Go to Quoted Post
The parameterization by Draghilev method.The starting point is taken near the bifurcation point (0,0).


Just a very little modifications: using norme(J) and the SMath ability for handling undefined parameters.

Clipboard02.gif
ContourTifoleum.sm (17kb) downloaded 14 time(s).

Best regards.
Alvaro.
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Offline Ber7  
#42 Posted : 09 November 2018 11:29:40(UTC)
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Thank you, Alvaro. The norme function simplifies the code and reduces the calculation time.
Example of using norme when solving ODE

LorenzPointsAreEquidistantc.sm (7kb) downloaded 14 time(s).
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Offline Razonar  
#43 Posted : 09 November 2018 19:08:46(UTC)
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Hi Ber. I don't remember to read nothing in the literature about equally spaced points in the numerical solution of the ode, except for the opposite: adaptive steps, but referring for the time variable, not the X,Y,Z solution points. About how you apparently get the same distance between solution points, i.e. sqrt(X^2+Y^2+Z^2), I guess that the background theory must to be in the Draghilev method and how the solution (X,Y,Z) is obtained from the differential equation. You have a very interesting point for investigate and publish about it.

I try to investigate the relationship between the symbolic ode solution and the paramatrization, but the symbolic solutions are quite complicated, and I don't have simple examples.

Apparently the distance between the points is 1 (guess can be easily proved because you divide by the norme the system), and this seems to provide more stable numerical solutions for the system (can be applied here Lyapunov's theorems?)

Clipboard08.gif

Unfortunately in this example I introduce the factor 1/1000 for avoid numerical over max limit error for the case without norme.

Best regards.
Alvaro.
Offline Jean Giraud  
#44 Posted : 11 November 2018 05:39:23(UTC)
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Originally Posted by: Ber7 Go to Quoted Post
The parameterization by Draghilev method.The starting point is taken near the bifurcation point (0,0).

Thanks Ber7, gorgeous
For this particular Trifolium, once created it is easy to collect
as many as desired to any scale corresponding to Draghilev 'a'
The great tool here, is the bidirectional fmesh(f(x),x0,x1,mesh)

Cheers ... Jean

2D Parametric Plot [Create Trifolium].sm (32kb) downloaded 12 time(s).
Offline Ber7  
#45 Posted : 11 November 2018 08:48:41(UTC)
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Thank you Jean, I suggest a small change in the animation.
TrifoliumAnim.sm (25kb) downloaded 12 time(s).

Edited by user 11 November 2018 12:44:29(UTC)  | Reason: Not specified

Offline Jean Giraud  
#46 Posted : 11 November 2018 17:48:44(UTC)
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Originally Posted by: Ber7 Go to Quoted Post
I suggest a small change in the animation.

Thanks Ber7 ... even more compacted.
Trifolium:=stack(b1,b2,b3,b4) from inversing f4(x) <= f3(x).

0Anim Trifolium [Windmill Ber7].sm (26kb) downloaded 12 time(s).

EolNew.gif

Offline Ber7  
#47 Posted : 24 November 2018 22:03:02(UTC)
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Graph of implicit function with bifurcation point (Problem on
the calculation of the arch)

https://en.smath.info/forum/yaf_...thod-in-Engineering.aspx

The graph consists of three curves that occur at the bifurcation point.
1. Find the coordinates of the bifurcation point
2.The starting point for each of the three graphs is taken near the bifurcation point
3. Build graphics by Draghilev method

Point Bifurcation.sm (39kb) downloaded 13 time(s).
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Offline Jean Giraud  
#48 Posted : 26 November 2018 04:29:54(UTC)
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Originally Posted by: Ber7 Go to Quoted Post
The graph consists of three curves that occur at the bifurcation point.
1. Find the coordinates of the bifurcation point
2.The starting point for each of the three graphs is taken near the bifurcation point
3. Build graphics by Draghilev method


Thanks Ber7.
This version works fine compared to the previous "arca" that never stopped pedaling .
By same token, I'm puzzled by the Lagrange points. Where those contours come from ?

LagrangePoints.PNG
Offline frapuano  
#49 Posted : 26 November 2018 10:56:38(UTC)
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Joan

from Wikipedia it is stated that they are involved in Astronomy and that for 2 large bodies there are 5 of these points so I guess that your pictures/worksheet refers on how to calculate them all(their positions).

https://en.wikipedia.org/wiki/Lagrangian_point

Best regards

Franco
Offline Jean Giraud  
#50 Posted : 28 November 2018 16:30:11(UTC)
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Originally Posted by: Ber7 Go to Quoted Post
Thank you, Alvaro. The norme function simplifies the code and reduces the calculation time.
Example of using norme when solving ODE


dn_GearsBDF is nearly ½ time. I think its Lorentz

LorentzAlvaro.PNG
Offline алексей_алексей  
#51 Posted : 05 January 2019 16:22:29(UTC)
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Maple, Draghilev's method. The inverse problem of kinematics. For those who want to try hard and to do better in SMath.
Offline Jean Giraud  
#52 Posted : 05 January 2019 18:10:14(UTC)
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Originally Posted by: алексей_алексей Go to Quoted Post
hose who want to try hard and to do better in SMath.

Thanks for the suggestion. My head is not oblate like Extraterrestrials !
Offline Ber7  
#53 Posted : 06 January 2019 11:18:45(UTC)
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Originally Posted by: Razonar Go to Quoted Post
. I don't remember to read nothing in the literature about equally spaced points in the numerical solution of the ode, except for the opposite: adaptive steps, but referring for the time variable, not the X,Y,Z solution points. About how you apparently get the same distance between solution points, i.e. sqrt(X^2+Y^2+Z^2), I guess that the background theory must to be in the Draghilev method and how the solution (X,Y,Z) is obtained from the differential equation. You have a very interesting point for investigate and publish about it.
Apparently the distance between the points is 1 (guess can be easily proved because you divide by the norme the system), and this seems to provide more stable numerical solutions for the system (can be applied here Lyapunov's theorems?)

Best regards.
Alvaro.

An article about the effectiveness of the solution of the system diff. equations for parameterization integral curve through arc length
Russiy.pdf (608kb) downloaded 23 time(s).

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Offline Razonar  
#54 Posted : 09 January 2019 00:02:47(UTC)
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Originally Posted by: Ber7 Go to Quoted Post
Originally Posted by: Razonar Go to Quoted Post
. I don't remember ...

An article about the effectiveness of the solution of the system diff. equations for parameterization integral curve through arc length
Russiy.pdf (608kb) downloaded 23 time(s).



Thanks for the paper. Now you found the keywords for this point, which seems to be "Arc Length Method". This give only 79 results at google search:
Clipboard04.gif

Results are related with mechanical engineering for finite elements analysis. From the first result, having this attached file: https://scholar.harvard....sios/files/ArcLength.pdf
Clipboard02.gif

Clipboard03.gif

But actually I don't find any appointment nor observation that solution points are equally spaced. Notice that It could be some "obvious" point for, given f(t,x,x' ) = 0, plot for x(t) it's equally spaced if one transform it to f(s,x,x' ) with s as the arc length. But it's immediate for me that the plot for the state space (x,x' ) seems to be equally spaced too, as in some smath examples here in this topic.

Also, for the observation that can apply Lyapunov theorems about stability of solutions, there are some references in the 79 google's search results. The application here of Lyapunov is related about the stability of the found solutions, because authors using the parametrization along the arc length for ill conditioned systems.

Originally Posted by: Razonar Go to Quoted Post
. Apparently the distance between the points is 1 (guess can be easily proved because you divide by the norme the system), and this seems to provide more stable numerical solutions for the system (can be applied here Lyapunov's theorems?)


Best regards.
Alvaro.

Edited by user 09 January 2019 00:15:12(UTC)  | Reason: Not specified

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Offline Ber7  
#55 Posted : 26 February 2019 20:32:50(UTC)
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Finding Minimum Distanceof a Point from Curve
(with Draghilev method)



MinDistDragilev.sm (24kb) downloaded 13 time(s).

Edited by user 27 February 2019 11:55:44(UTC)  | Reason: Not specified

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Offline Jean Giraud  
#56 Posted : 26 February 2019 21:07:47(UTC)
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Originally Posted by: Ber7 Go to Quoted Post
Finding Minimum Distance of a Point from Curve

Thanks Ber7 ... works fine.

Offline Ber7  
#57 Posted : 13 March 2019 20:47:37(UTC)
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Distance from point to implicit curve(with al_nleqsolve)
A thick red line is the normal to the curve at point a.


FindDist.pdf (244kb) downloaded 10 time(s).
FindDist.sm (73kb) downloaded 12 time(s).
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