# Strange asymptotic behavior on the Kozai-Lidov cycles

So, in Naoz et al. (2013) is described the Kozai mechanism on three-body systems. I've tried to solve the equations (B1) - (B17) for the example described in the Figure 3 and I got something like this:

Note that I got the exact same result until $$\sim$$ 33 Myr (black arrow), then the cycles change radically. Is this usual?. Do Kodai cycles tend to change over time like this or they actually remain with the same "shape" as time goes by? If the latter, isn't a way to predict the behavior of Kodai cycles as t tends to infinity?

• Wait, Figure 3 ends at 25 Myr, so you did not get "the exact same result until ~33 Myr". Perhaps they cut it off at 25 for just this reason? – uhoh Mar 21 '20 at 2:13
• If you want to explore any possible numerical errors you can also ask about that in Scientific Computing SE. I've had excellent luck receiving great answers there, see for example this and this. The first one reminds me; did you use a symplectic integrator? Have you tried adjusting any optional step size and/or precision or accuracy requirements specified to the integration routine to see what effect it has? What integrator did you use? – uhoh Mar 21 '20 at 2:15
• I use a Mathematica package call TIDES. I don't think there's a mistake, nor in the step size(I chose 5*10^6 points), but I may check out for mistakes. The thing is, do Kozai cycles maintain over time or can change like that? If they maintain is clearly a numerical error... – Carlos Vázquez Monzón Mar 21 '20 at 2:31
• @CarlosVázquezMonzón please try a larger and smaller step size and compare all three results. That's the first thing one does when running a numerical integrator, especially on what looks like it may be a stiff problem. You should include a description of what happens in your question because it's such a fundamental test when numerical integration produces unexpected results. Thanks! – uhoh Mar 21 '20 at 2:36
• Okay so "The thing is, do Kozai cycles maintain over time or can change like that?" is really a much better question! Why not adjust your title and intro and explain that instead - I think it is more likely to get an answer than "Why does my program do this?" Also, it's best to include "I use a Mathematica package call TIDES" in your question as well. – uhoh Mar 21 '20 at 2:38