one man

Alexey Ivanov

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@C_R 
I don't know, such expressions make me doubt the reality of the model. The only thing, it made me remember the way of implementing Draghilev's method (I already mentioned  here), when we do not calculate the minors of the Jacobian matrix in symbolic form. I have some Pascal (Delphi) texts that I translated from FORTRAN at one time. One numerical method for solving ODEs (the Hamming method) is implemented there. The point, of course, is not in the method of solving ODE. But there, precisely because of the limited symbolic capabilities of more lower-level languages, only the Jacobi matrix was used (calculated manually). All left-hand sides of the ODE are obtained numerically: at each step, a system of linear equations is solved.
The calculations are quite fast. And, most importantly, we are much less dependent on the dimensionality of the original system of equations and on the cumbersome form of partial derivatives, because we do not perform symbolic transformations.
It is not yet clear to me how to adequately translate this approach to Maple.. So far, the only thing in my head is to rewrite the text from Pascal to Maple. I'm thinking about it.

@C_R 
I sometimes use the op and lhs functions. I just can't execute the program and apply them, for example, to eqm2. My computer freezes right at the start.
So far, from what you've said, I can understand that we have 3 equations and 4 variables, and very cumbersome expressions, right? Of course, the Method is quite applicable for such dimensions, but we need an appropriate computer at least to see the equations themselves. Apparently, your computer is suitable for this. Then we can use the Method, avoiding working with large formulas, stopping at the matrix of partial derivatives, but without calculating the determinant in symbolic form.

@C_R 
Just saw your request. I can't understand from the text what form these equations are.
Right, these equations eqm, eqm2, eqTPO? I don't see their content. Can you show me their contents separately?

@C_R 
Yes, the second question touches upon the strongest side of the Method. For example, in that very Schatz mechanism, one equation was discarded from the system of equations describing geometric connections, and we obtained a motion with small deformations. To generalize, we can find the infinite subset of solutions. After all, the set of solutions can break down into disconnected subsets (branches), and we need to find each branch. But it also happens that the entire set of solutions is connected, and in this case we solve the system at once, that is, in this case there is only one "pink" line. But we always need one starting point on the pink line, because this pink line is the solution to the Cauchy problem.
As for the first question (finding point solutions of the NxN system), the Method cannot guarantee a complete solution head-on, there are just successful examples. It is necessary to try to work with the system of equations in parts, as if working with a part of its equations. But even applying the Method head-on to NxN systems has undeniable advantages, the only thing, as it seems to me, is that it may be inferior to modern optimization methods for individual solutions. Therefore, optimization methods or Gröbner bases for polynomial systems of small dimension are very convenient for finding the starting point for pink lines.

@dharr  Yes, I also think that the successful solution of the original transcendental system  using polynomial theory is a fortunate coincidence of many circumstances.
Thanks again.

@dharr With your permission, I moved your comment to the answers, and +.
Thank you, there is something to learn about working with Maple.

@vv Yes, I used the Groebner package and figured out plex, but these were very simple and small examples. My PC got to your nops(G) in a reasonable time, but I'm a bit scared to try to go further.
Agree that it is possible to bring the work to the solution of a polynomial system only if there is a very serious interest. You have satisfied my curiosity completely.
If you are interested, at one time (back in the era of large computers) a group of several people was created for this task. True, I don’t know anything about the results.

@vv  Very well, +. Now we know that the polynomial interpretation also has solutions. (It is clear that a finite set of solutions of a polynomial system corresponds to a countable set of solutions of the original system.) Wonder how long it will take to find these solutions...

@C_R 
In my opinion, waiting so long is already a lot of work. Yes, I have long had suspicions about the real possibilities of solving polynomial systems on a theoretical basis. 
By the way, Draghilev's method allows us to draw conclusions about the solution of the original system itself, we can even say, to obtain its complete solution: for example, you can solve 4 of 5 equations in turn, tracking the change in the sign of the remaining equation.
In any case, I'm sure this is a very good example for Maple.

@janhardo 
maybe this is a call for everyone to use the search function on the forum more actively.

There was one activist here (Markiyan Hirnyk), so in order to protect myself from his nastiness, I turned to the community for help. But now he visits other forums.
(For some time now, I have also formally been a moderator, and, just in case, I will say right away that I have nothing to do with deleting non-advertising posts, that is, I have never deleted your posts.)

@Carl Love   I read both your messages in the thread with great pleasure. It seems to me that the online translator conveys the level of your mastery of the word.

@vv  No, it's not compact, it's professional☺.

@vv  

Thank you for your participation. Yes, it can be proved both geometrically and analytically.
This example was made specifically because of the animation. (I really like moving pictures.)

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