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Lyapunov Time (Read 4656 times)
APODman
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Lyapunov Time
11/21/07 at 04:58:37
 
Hy !
 
I'm testing some orbital integrators and recently I found Gravity. I really like of the visual aspect of the simulation in this soft ( and in the version beta ) for the easiness in obtaining the orbital data to be worked in Excel.
 
However I also used Solsyin that supplies robust results and allows very interesting graphic analyses directly of the program.
 
One of the possible graphs to obtain it is an analysis of the orbital uncertainty propagation being permitted estimates the Lyapunov Time of the orbital evolution of an object.
 
Does some form exist of obtaining numeric information of the orbital uncertainty propagation from the Gravity Simulator to draw a graph?
 
 
( I read in this forum about some results using a soft called Picard. Could someone indicate me where I can find it?? )  
 
 
Thanks !
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Re: Lyapunov Time
Reply #1 - 11/21/07 at 17:33:09
 
Hi, APODman.
There's no way to output Lyapunov Times.  As a workaround to the problem, you could probably set up some 2 body problems.  They have know analytical solutions.  The orbital elements, except for Mean Anomoly, should hold perfectly steady.  So anything other than a flatline graph will illustrate the error.  And Mean Anomoly should be a constant slope linear function.  You'll get different results for different time steps, with the slower timesteps approaching a true flatline.
 
Picard is an additional integrator written by forum member Frankuitaalst.  There is an unreleased beta version that uses this.  It's very fast, but is limited to only a small amount of objects (~20 if I remember correctly).  Like the routine in the Solsyin, it is variable step:  rather than the user controlling the timestep, it chooses the fastest timestep for the desired accuracy.
 
With a slow enough timestep, forces not modelled will become more significant than the integrator error.  There's a link in the "Asteroids" thread of a paper describing the difficulties of nailing down Apophis' future Earth encounters.  This describes how unmodeled forces, and lack of precision in the modeled forces prevent an accurate solution.  Keep an eye on the "Asteroids" thread, as soon I'll post the results of 2 simulations, one with, and one without additional perturbing asteroids in the Apophis simulation.
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Re: Lyapunov Time
Reply #2 - 11/22/07 at 09:41:33
 
Quote from Tony on 11/21/07 at 17:33:09:

Picard is an additional integrator written by forum member Frankuitaalst. There is an unreleased beta version that uses this. It's very fast, but is limited to only a small amount of objects (~20 if I remember correctly). Like the routine in the Solsyin, it is variable step: rather than the user controlling the timestep, it chooses the fastest timestep for the desired accuracy.
.

The Picard integrator is called after the Picard algoritm which uses a finite power series ( Taylor ) for describing the motion and velocity of an object . Advantage of this integrator is that the path of a body is described analytically for a large period of time . For the Earth-Moon system fi. about 5 days can be expressed by one formula . After that time a new formula is calculated ..and so on . In principle up to eternity .  
Another advantage is that one can give the cut-off accurancy of the motion , also the disired accurancy of computation . Normally I work with an accurancy of 10 meters after 5 days of simulation ! Thats pretty good . Picard performs well for a small number of bodies . Asteroid paths in the solar system , usually requiring 18 bodies are simulated quickly with a great accurancy . As the number of bodies increases the algorithm slows down . A hundred bodies can still be reasonable simulated . Increasing the number of bodies further slows the program down to about 0.5 second for 5 days , meaning 2 years per minute of calculation .  
The bigger the distances between bodies the quicker a simulation is performed . Multiple star systems can easily be simulated for millions of years .  
What's your experience with the Solsyin program ?  
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Re: Lyapunov Time
Reply #3 - 12/21/07 at 05:26:49
 
Hello Tony and Frank, thanks by the aswers, and sorry by my delay.  
 
Solsyin can simulate a graph of the dispersion of the mean anomaly of a asteroid against a hundred of virtual images of the asteroid in a time ( they use a covariance data catalog ). For this graph the aproximate Lyapounov Time can be determinate.
 
For example, this graph can permit calculate the aproximate Lyapounov Time of the asteroid 1950DA**:
 
 
 
font: http://math.ubbcluj.ro/~sberinde/thesis/impact.html
 
The calculate Lyapounov time for asteroid by this graph is aproximate 50 years. The chaotic behavior is after 3000 years ( after the close encounter with Earth in 2880).
 
Gravity Simulator can create virtual images of a asteroid. Can you imagine some way to compute the Lyapounov time of Output data of a simulation of some asteroid against many virtual images ?
 
----///----
 
Frank,
 
Im very curious by your Picard integrator, I hope you release it soon !
 
I use Solsyin often, i like it.  It provides great analysis of data and your "auto-control" timestep prevents many time dispended in "trial and error" process to determinate a more exact simulation.  
 
I use too the Mercury integrator but I prefer Solsyin but really like the visual results of Gravity Simulator for classrom educational purposes. But I test Gravity Simulator only recently, Im really newbie !  Wink
 
 
 
 
** Someone here already held a simulation of 1950DA ?  
 
 
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Re: Lyapunov Time
Reply #4 - 12/22/07 at 01:27:49
 
Quote from APODman on 12/21/07 at 05:26:49:


Frank,

Im very curious by your Picard integrator, I hope you release it soon !

I use Solsyin often, i like it. It provides great analysis of data and your "auto-control" timestep prevents many time dispended in "trial and error" process to determinate a more exact simulation.

I use too the Mercury integrator but I prefer Solsyin but really like the visual results of Gravity Simulator for classrom educational purposes. But I test Gravity Simulator only recently, Im really newbie ! Wink


** Someone here already held a simulation of 1950DA ?


[ ]s

 
Apodman , concerning the Picardintegrator : I have running it at home as an executable which I can transmit if you want to try . It works well as a stand-alone program , but it hasn't far-off the nice interactive features as GravitySimulator has . Integrating the integror with GravitySim slows the program down we noticed , some work might to be done i guess.  
 
By curiousity I wanted to download the Solsyin and the Mercury package . Is it possible that Solsyin works under MS-Dos or has it a windows version ? I found the Mercury files , but I don't get out of this files any executable . Any idea?  
 
Concerning 1950da : I simulated this one some time ago but didn't get an impact in the year 2880. I'll try to run it again .  
 
Concerning the idea of Lyapunov : what algorithm is needed to calculate this ? Can you give me some hint ? I wasn't familiar with this .  
 
You use the GravSim programm for classroom presentations ?  Smiley
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Re: Lyapunov Time
Reply #5 - 12/27/07 at 04:50:38
 
Quote:
Apodman , concerning the Picardintegrator : I have running it at home as an executable which I can transmit if you want to try . It works well as a stand-alone program , but it hasn't far-off the nice interactive features as GravitySimulator has . Integrating the integror with GravitySim slows the program down we noticed , some work might to be done i guess.
 
 
There is no problem, I'm used to command lines.
 
I sincerely would like to test Picard if you allows. Your integrador seems to present interesting analysis levels and I'm interested in explore them!
 
I don't know the size but I believe that could be sent to my email: apodman@hotmail.com or carlos.apodman@gmail.com
 
thank in advance !
 
Quote:
By curiousity I wanted to download the Solsyin and the Mercury package . Is it possible that Solsyin works under MS-Dos or has it a windows version ? I found the Mercury files , but I don't get out of this files any executable . Any idea?

 
Both work through MS-DOS only.
 
The executable of the Mercury need to be compiled.  
 
I put both of my versions of the software packages in the internet. You can download it here:
 
- http://www.4shared.com/dir/5092376/4d582275/sharing.html
 
The executable of Mercury files is also compiled and I included the configuration files already formatted in this way you can easily understand how configure it for comparison. The Solsyin included the great thesis of the autor of the integrator and the Gnuplot software needed to make the graphics.  
 
Quote:
Concerning 1950da : I simulated this one some time ago but didn't get an impact in the year 2880. I'll try to run it again .

 
I didn't also obtain any impact.
 
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Re: Lyapunov Time
Reply #6 - 12/27/07 at 04:53:37
 
-->
 
Quote:
Concerning the idea of Lyapunov : what algorithm is needed to calculate this ? Can you give me some hint ? I wasn't familiar with this .

 
The algorithm for Lyapounov Team's calculation I obtained from the theory of doctorate of the own solemnity of SOLSYIN (a long abstract in English tb is available. He is practically a complete article!  Details can be obtained of the complete version (immense) of the tesis although she is in Rumanian. But Reading the theory first in English is not so difficult to understand the text in Rumanian. )
 
A importance of the Lyapounov Time is that ...
 
"The phenomenon of chaos appears even in one of the simplest problems of celestial
mechanics: the restricted three-body problem. Close encounters with the perturbatrice
planet always induce such a chaotic behaviour for an asteroidal orbit. But
the phenomenon of chaos is more subtle, since it appears in motions totaly free of close
encounters. For the planar, circular, restricted three-body problem the Poincare
surface of section is a tool to distinguish between chaotic and regular motions. For the
general case, we have the Lyapounov exponents method.
...
In summary, the exponential divergence in time of the specific three types of errors (error in initial
conditions, approximation error and round-off error) limits the deterministic nature of
the final numerical solution."

 
reference: http://math.ubbcluj.ro/~sberinde/thesis/abstract.pdf
 
Simulating some dozens or hundreds of virtual images of an asteroid and calculating the medium difference among their longitudes is possible to obtain Lyapounov Time ( the length of time for a dynamical system to become chaotic. The Lyapunov time reflects the limits of the predictability of the system. By convention, it is measured as the time for nearby trajectories of the system to diverge by e. )
 
Starting from the generated graph it is possible if to obtain the approximate data of the variables of the algorithm
 
The algorithm is:   L=1/ ((ln(Di)-ln(Df))/(Tf-Ti))
 
In the graph below Di = initial distance, in this case is 1e-04. The final distance (Df) is 100. The difference of time so that the chaotic state is reached is the time to the graph reach the point that the distance is 100 (Tf), in the case 2300, less the initial time of the simulation (Ti) that is 2000 in the graph.
This gives a time of Lyapounov 21,7 years old, in other words, this is the time that not modeled forces and mistakes of data act on the previsibility of  the orbit. In the case of the graph we can foresee with safety the evolution of the orbit of the asteroid to approximately 21,7 years.
 
For Excel to use the following format:
 
1 / ((LN (B1) - LN (A1)) / (C1-D1) )
 
 (A1, B1, C1 and D1 is the cells with the data.)
 
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Re: Lyapunov Time
Reply #7 - 12/27/07 at 04:56:55
 
Quote:
You use the GravSim programm for classroom presentations ?

 
Yes,
 
Here in Brazil the scientific divugation is precarious. So I work with an organization that seeks to take scientific popularization of quality the public schools waking up the students' interest for the astronomy, a field that without a doubt active the students' interest for the sciences. Possesses a telescope or simulators as Gravity Simulator in hands is possible to observe the natural forces in action, something that can be deeply discouraging when just work in school books.
 
I liked Gravity Simulator for supplying a fort visual aspect of the solutions of the orbital problems and for the practicity of use. This sharpens the students' curiosity that want to observe the alterations that the gravitational relationships supplies.
 
It's a success!
 
After some time we trained the teachers in the use of the integrator so that they can continue supplying the classes.
 
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Re: Lyapunov Time
Reply #8 - 12/27/07 at 07:57:30
 
Apodman , I will send you the Picard executable per mail and also a dataset of data .  
I works like GravSim in this sense that there is a program which reads a datafile .  
Outpunt is done on screen and on a datafile . Standard I have data outputted of each body : position , velocity , inclination , eccentricity and semi-major axis .  
These datafiles can be imported in Excel for further analysis .
Contrary to GravSim the program is not as user-friendly in this sense that once the program runs one can not adapt something . It's just running , calculating and generating data .
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Re: Lyapunov Time
Reply #9 - 12/28/07 at 10:10:15
 
Thanks Frank !
 
I will test it.
 
 
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Re: Lyapunov Time
Reply #10 - 12/28/07 at 10:45:56
 
I've been thinking about putting together a lab manual for educators where they can use Gravity Simulator to teach things about Astronomy.  If you have any ideas or suggestions, let me know.  I'll be in Austraila and New Zealand in January on vacation, and may work on that during some of my down time.
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Re: Lyapunov Time
Reply #11 - 12/28/07 at 10:54:42
 
Quote:
I've been thinking about putting together a lab manual for educators where they can use Gravity Simulator to teach things about Astronomy. If you have any ideas or suggestions, let me know. I'll be in Austraila and New Zealand in January on vacation, and may work on that during some of my down time.

 
Good Tony !
 
I will organize some ideas but initially ( by my experience )  you can adress the basic of Kepler Laws. The teachers of fundamental cicle can use easily the Gravity Simulator to teach basic astronomy !
 
 
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