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So now we come to the next example.

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This is more of an example from mathematics, and it's called a Lawrence system.

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So the Lawrence problem or a Lawrence Attractor also Lawrence System became very famous due to its relation

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to the so-called butterfly effect.

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And it basically is a very simple mathematical system that consists out of three differential equations

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that describe the change of the X, Y and Z coordinate of a particle, and it has chaotic solutions.

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And this is a pretty interesting topic by itself because basically what we are saying when we deal with

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classical physics a lot quantum physics, but classical physics is that everything is deterministic.

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And so this means that if you know the starting conditions and you know the laws of physics, then you

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also know the state of the system at any point in time.

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But the special feature about chaotic systems is that when you even change the starting conditions by

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a tiny bit, even if there is a butterfly in your physical system somewhere and you don't know precisely

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where, then the outcome of the system will be totally different after some time.

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So the small change in the starting conditions will have really tremendous and really gigantic effects

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on the coordinates at a later point of time.

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And this is like a cool mathematical system where you can investigate this chaotic behavior and also

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the trajectories that we are going to calculate.

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Here are butterflies.

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So this is why it's such a beautiful example.

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So it turns out that later on, there have also been some physical systems where this equation applies

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to.

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So this could, for example, be some atmospheric convection.

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But for our purposes, we don't really want to talk here about the physics because it's really more

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of a mathematical system.

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All right.

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So that's enough with the introductory talk.

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Let's see how we can program this, and let's analyze the equations first.

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So we have here a differential equation for X that depends on X, which is good.

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This is what we had previously.

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Then it could also depend on the time, but you see, nowhere is there any dependence on time.

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So this is even better.

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So it's pretty easy equation.

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But you see here we have a dependence of X on Y.

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You have a dependence on the change of Y, on X and Z.

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And here we have a change of Z, the independence of X and Y.

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So this means we have three differential equations and they are couples.

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This makes it maybe a bit more difficult than our previous example of the harmonic oscillator, where

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all of these equations were uncoupled.

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So let's go ahead and see how we can solve this example.