The orbit of this asteroid is remarkable: plotting the distances to Mars and Earth gives the following plot over the first couple of thousand years .  
Although the ast is really bound to Mars it seems to spend more time closer to Earth than to Mars .  
In this time span the mean distance to Mars was much more than the distance to earth ...
I wonder how this ast may evolve ...

This is one facinating asteroid.
 
I added the animated GIF with a rotating period set to Earth's period to the first post in this thread.  It shows that 1999 ND43 is in a 25:47 mean motion resonance with Earth.  This resonance is responsible for the variability it the asteroid's speed in the frame with a rotating period set to Mars' period.
 
After the asteroid encounters Mars in about the year 5000, Mars pulls the asteroid into a slightly higher and slower orbit, breaking it from its 25:47 mean motion resonance with Earth.  But when it encounters Mars again from the other direction in about the year 7800, Mars pulls the asteriod into a slightly lower and faster orbit, and restores the asteroid's 25:47 mean motion resonance with Earth.

Pasting reverse and forward sim gives the following picture of the asteroid .  
There's a lot of dynamics in the orbit . Remarkable in the past is the time -5000 years where the asteroid slows down and becomes a co-orbital of Mars . In some way we are "lucky" today .  
Furthermore  the asteroids inclination has a mean period of about 16000 years .( Is this somehow related with some product of orbital periods of other planets  ? ) .  
Also in the past the asteroid spended more time near earth then to Mars ( related to the mean distance ) . It seems however that between "now" and 2500 years the influence of Earth is at biggest .
By the way : I have a series of screenshots in gif of the orbit of the Asteroid in which the precession of the perihelion or aphelion becomes clear . I put them together in .ppt format , but the file is rather big . Can this forum accept a .ppt or do I have to convert to animated gif ?

You're right ... After the first "peak" our results don't match .  
However the peakvalue of 1.002 matches .  
Perhaps some parameters were different . I used 14 bodies up to Saturn ( Nept, Uranus , Pluto : not ) . Moon and Earth were two bodies . Maybe the value of G was different . I used g = 6.67259E-11

After T=6500 my 16 second run starts to differ from my 32 second run.

Don't delete the Moon because the asteroid makes numerous close approaches to the Earth.  If you wanted to delete the Moon, it's not as easy as simply deleting it.  You would need to replace Earth with an accurate value for the mass, position and velocity of the Earth barycenter.  This is what I did with Jupiter, Saturn, Uranus, Neptune, and Pluto.  The asteroid makes no close passages to these objects, so it is justified for these planets.
 
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If you look at my animation, the asteroid travels counter-clockwise around Mars' orbit until it encounters Mars.  Then it travels clocwise away from Mars about 90 degrees before reversing direction and heading back to Mars.  After Mars pushes it away for the 2nd time, it travels completely around Mars' orbit in a clockwise direction.  My 16 second run shows it makes 2 attempts to get past the 90 degree point before travelling all the way around the orbit.

I ran the sim as mentionned above without the small minor planets . There is no significant change. The only difference a notice is that the program performed 15 days more to perform 300000 iterations . I think his is due to the fact that the program didn't have to slow down while there were less bodies. I think we have to check our initial conditions . Herunder the sim . By curiousity I added the velocity of the asteroid relative to Mars . I really am surprised to see that this relative velocity seems to follow the distance to Mars .