<p>So, in <a href="https://arxiv.org/pdf/1107.2414.pdf" rel="nofollow noreferrer">Naoz et al. (2013)</a> 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:</p>
<p><a href="https://i.sstatic.net/BwQj6.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/BwQj6.png" alt="enter image description here" /></a></p>
<p><a href="https://i.sstatic.net/fT0Ux.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/fT0Ux.png" alt="enter image description here" /></a></p>
<p>Note that I got the exact same result until <span class="math-container">$\sim$</span> 33 Myr (black arrow), then the cycles change radically. Is this usual? Do Kozai cycles tend to change over time like this, or do they actually remain with the same &quot;shape&quot; as time goes by? If the latter, isn't there a way to predict the behavior of Kozai cycles as time tends to infinity?</p>

CC BY-SA 4.0
2020-03-21T02:13:05.173Z
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
CC BY-SA 4.0
2020-03-21T02:15:07.433Z
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](https://scicomp.stackexchange.com/q/29149/17869) and [this](https://scicomp.stackexchange.com/q/26537/17869). 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
CC BY-SA 4.0
2020-03-21T02:31:55.560Z
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
CC BY-SA 4.0
2020-03-21T02:36:24.607Z
@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](https://en.wikipedia.org/wiki/Stiff_equation). 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
CC BY-SA 4.0
2020-03-21T02:38:15.177Z
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
CC BY-SA 4.0
2020-03-21T02:41:48.980Z
because as I found in the answer to [“Pythagorean Three Body Problem” - need some points from an accurate solution for comparison](https://space.stackexchange.com/q/15364/12102) (and is in hindsight obvious), Mathematica may do a better job than some random off-the-shelf integrator for stiff problems.
uhoh
<celestial-mechanics>
Strange asymptotic behavior on the Kozai-Lidov cycles
Carlos Vázquez Monzón
# Strange asymptotic behavior on the Kozai-Lidov cycles

(Asked by: username_0 on 2020-03-20)

So, in [Naoz et al. (2013)](https://arxiv.org/pdf/1107.2414.pdf) 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:

[![enter image description here](https://i.sstatic.net/BwQj6.png)](https://i.sstatic.net/BwQj6.png)

[![enter image description here](https://i.sstatic.net/fT0Ux.png)](https://i.sstatic.net/fT0Ux.png)

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

**Tags:** celestial-mechanics

> **Comments:**
> **username_1**: 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?  
> **username_1**: 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](https://scicomp.stackexchange.com/q/29149/17869) and [this](https://scicomp.stackexchange.com/q/26537/17869). 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?  
> **username_0**: 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...  
> **username_1**: @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](https://en.wikipedia.org/wiki/Stiff_equation). 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!  
> **username_1**: 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.  
> **username_1**: because as I found in the answer to [“Pythagorean Three Body Problem” - need some points from an accurate solution for comparison](https://space.stackexchange.com/q/15364/12102) (and is in hindsight obvious), Mathematica may do a better job than some random off-the-shelf integrator for stiff problems.  

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## 0 answers

