So now we come to the next example.
This is more of an example from mathematics, and it's called a Lawrence system.
So the Lawrence problem or a Lawrence Attractor also Lawrence System became very famous due to its relation
to the so-called butterfly effect.
And it basically is a very simple mathematical system that consists out of three differential equations
that describe the change of the X, Y and Z coordinate of a particle, and it has chaotic solutions.
And this is a pretty interesting topic by itself because basically what we are saying when we deal with
classical physics a lot quantum physics, but classical physics is that everything is deterministic.
And so this means that if you know the starting conditions and you know the laws of physics, then you
also know the state of the system at any point in time.
But the special feature about chaotic systems is that when you even change the starting conditions by
a tiny bit, even if there is a butterfly in your physical system somewhere and you don't know precisely
where, then the outcome of the system will be totally different after some time.
So the small change in the starting conditions will have really tremendous and really gigantic effects
on the coordinates at a later point of time.
And this is like a cool mathematical system where you can investigate this chaotic behavior and also
the trajectories that we are going to calculate.
Here are butterflies.
So this is why it's such a beautiful example.
So it turns out that later on, there have also been some physical systems where this equation applies
to.
So this could, for example, be some atmospheric convection.
But for our purposes, we don't really want to talk here about the physics because it's really more
of a mathematical system.
All right.
So that's enough with the introductory talk.
Let's see how we can program this, and let's analyze the equations first.
So we have here a differential equation for X that depends on X, which is good.
This is what we had previously.
Then it could also depend on the time, but you see, nowhere is there any dependence on time.
So this is even better.
So it's pretty easy equation.
But you see here we have a dependence of X on Y.
You have a dependence on the change of Y, on X and Z.
And here we have a change of Z, the independence of X and Y.
So this means we have three differential equations and they are couples.
This makes it maybe a bit more difficult than our previous example of the harmonic oscillator, where
all of these equations were uncoupled.
So let's go ahead and see how we can solve this example.