So welcome back to this very short and small additional lecture.
So in previous lecture, we have solved D3 coupled differential equations and we have found a chaotic
solution where we had to use a value for bead that is larger than 25.
So we used 50 and we have seen that we have such a chaotic solution where we have here just a tiny difference
in the starting conditions.
But then after some time, there is really a large difference in the end, coordinates of the two solutions
blue and orange.
So now I want to just show you an example what happens if the value of B is smaller than this critical
value?
So I just go to this point and copy it.
So this is nice thing we can know.
Just change these parameters here without having to redefine the function because we have added them
here and then we have added or the arts optional argument.
So I copied this whole cell and then I copy also the plots, which is this one.
So when I now change the value of B to a value smaller than 24, like 20, for example, I get such
a solution.
And let's see what happens if I add another solution here with a slightly different starting position.
One point one.
Sorry again, the same typo.
1.1 times X0.
And then I plot here both of them.
Hmm.
Sorry, I forgot to run it.
Yeah.
Now you see, we have here on top of each other both of the solutions and maybe just to explain it to
you a bit better.
Let me export the plots.
To
and open it.
OK, so now you see here we have this small difference in the starting positions of X equal to one and
one point one.
And you see they follow exactly or almost exactly the same trajectory and then they end up in the same
point.
This is because by changing the value of B, we have turned the two repulsive points in space to two
attractive points.
And so in this case, the system does not behave chaotic anymore, and it will move towards the same
solution almost.
Or it will always move to the two same solution.
But even it will move on the same trajectory if the starting conditions are just slightly different.
So this is really the difference of chaotic and deterministic behavior.
And so even though this is more of a mathematical model, it teaches us quite a lot about physics as
well.
And I told you in the very beginning that there are some physical applications of this model.
But yeah, I think I said already enough about the background and we see that from such a nice mathematical
model, we can learn quite a lot about physics.
And of course, we can use it as a nice model for our function.
Integrate lots, solve underscore IVP, which had absolutely no problem handling such a system.