So you reckon it's all down to the timestep? That'd be annoying if it is, it means we can't do test runs that can last tens or hundreds of mlilions of years without leaving them running for weeks! How low does the timestep have to be to notice this effect?
 
I guess I should stop the run I'm doing now now anyway... at 36 million years the planet inclination is at about 95° and eccentricity is still wobbling around 0.0017. It's just silly to carry on with this one now, it's clearly not going anywhere.

The time step definately ruined this one.  Using formula 2, I computed its Kozai period at 61000 years.  And the first half-cycle does appear to take about 30000 years.  But with its ecc approaching 1 (not quite this time since I used inc=80 instead of 90), its perihelion was too close to the Sun for a 32K time step.  This introduced error which is obvious from this point onward.

 
I'll play with this some more.  We'll get to the bottom of this  8)

OK, starting again with a 16k timestep we're seeing something interesting - the planet eccentricity is now increasing over time! It's still varying by +/-0.0002, but the value around which it's varying has increased from 0 to about 0.0025 in 3 million years, which isn't what happened before.  
 
Inclination of the planet is still increasing monotonically over time though, it's at 8 degrees after 3 million years.  
 
The other interesting thing is that the argument of pericentre of the planet seems to be converging to around 90 degrees - that didn't happen with the 65k timestep (the arg. of peri. for the companion is 39 degrees though).

That's interesting, I wouldn't have thought time step would be that critical in this case.  But that's still a very small ecc gain compared to the maxEcc it needs to achieve halfway through the cycle at 12M years.
 
Frank, it should be outputting as a .txt file.  Send me an e-mail and attach what you have.  Are you running 2.0 or the beta version?

I can't find the file now to make an inc graph.  I remember it fluxuated, but not like ecc.  And I discovered a error in the inclination output.  Under some conditions, its off by 180 degrees.  It's easy to spot though,  If your inclination all of a sudden instantly jumps 180 degrees, that a bug.  I'll try to find the file later and post an inc graph.

This is identical to your sim #1, except the secondary is at 200 AU instead of 750 AU.  All other values are the same.
 
The formulas give:
Pkoz=486265 years
emax = .94
 

 
Again, the simulation is right on the money.  But I tried your sim at secondary sma=750 AU and got the same results you got... Insignificant delta eccentricity, and a linear inclination advancment.  It almost seems as if the Kozai Mechanism turns off at a certain distance.  Maybe that makes sense.  It turns off completely at inclinations <~35 degrees.
 
I ran this at time step 16K, which was too fast once the planet's ecc approached 0.94, so I don't trust the graph beyond what's shown.
 
So at what point between 200 AU and 750 AU does the Kozai Mechanism simply stop working as advertised?
 
Quote from Mal   on 10/26/06 at 18:15:42:

Me too.  Despite all the forumlas in all the papers, I think there's still stuff to discover about the Kozai Mechanism.
 
I'll try a 400 AU run and see what happens.

Hi Tony , Mal , out of my simulation of the 5 earths I got the following result : the time to get a noticible influence decreases rapidly with distance ... so the sim with 750 might run longer .  
Also , as distance increases the frequency of the mechanism decreases with time .  
And further a noticed that the maximum eccentricity decreases with distance . F.i. in the 3AU earth simulation the moon didn't get to an ecc of 1.  
This makes sense to me , intuitionally ...  
In one of the articles a also read that time step should be reduced if eccentrity gets high . I guess especcially in the points were the planet is close to the mother .

Here's a graph of the state of affairs for the planet at 8.493 million years.
 

 
Blue is Arg of Pericentre, you can see how that's converging to about 96 degrees. That uses the left y-axis. I have absolutely no idea why this is converging to 96 - the only thing that could possibly be related is the Longitude of Ascending Node of the companion, which is about 179 degrees, and 96 is almost half of that. But it's a really tenuous link.  
 
Red is inclination. This uses the left y-axis too, and it's at 22 degrees so far.  
 
Pink is eccentricity. This uses the right y-axis. You can see that its rate of increase is accelerating with time. It may yet increase very steeply - I guess we'll know for sure if it's going to be like the graph in the paper at around the 15 to 20 million year mark (which will be tomorrow evening, I guess).
 
(1000 on the x-axis is about 1 million years)