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So let's begin and let's implement these equations.

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And of course, first of all, we must implement all three of those.

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And we write death and then we can use the same method as before.

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So you see, this is pretty, pretty powerful and universal methods.

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And we just write everybody.

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And then we could have just roads T and R.

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And what I write then is just for simplicity.

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X y z is equal to ALM.

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So this means that three individual components will be stored in the variables called X, Y and Z.

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And the first derivative of X I call X one is equal to eight times Y means X and then Y one is equal

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to X times in brackets B minus C minus Y and z one is equal to x times y minus z c times z.

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And then we return everything and we return x1, y1, Z1.

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So you see, it's the same logic or the same methods or methodology behind it, where we return three

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individual values, which are the right hand side of these equations.

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And all of this is just to make it look nicer and to name these things.

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But we could have just wrote here, for example, here we could have written eight times our brackets,

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one minus our brackets zero.

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So we could have implemented this thing here in just a single line of code.

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And now we have this function.

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Sorry, first, I must run this and then this, and now we can look at some solutions.

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And as I told you, the system has chaotic and non chaotic solutions.

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And it turns out you can investigate this in more detail.

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It turns out that this constant be here is determining if it is chaotic or non chaotic.

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We will start with a chaotic solution.

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So let's continue and let's define the parameters for the function so we can right here.

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For example, a is equal to then a typical set of parameters would be 10.

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And then something larger than 24 Dot seven four.

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So, for example, 50 and see, I just used a typical value that is used everywhere in literature that

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gives a nice butterfly, which would be eight over three.

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But you can play it with the parameters here and now I want to show you a nice trick.

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So you see, we want to later on change these parameters.

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So this means later on we would have to change these values and then redefine the function.

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But we can also do it in a different way.

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So I paste them here and then we must right here a b c as parameters or as our input values here for

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the function itself.

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And then we just have to change one thing when we call the the command from this Typekit module dot

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integrate.

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So let me show you how this works.

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So we right solution is equal to and like before the right integrate thought, so underscore IVP.

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Then first the function, then the times which would be t start come off t end.

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So this would be t start equal to zero T and equal to 50, for example.

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Then we continue to rewrite x zero y zero zero.

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So this would be the starting positions.

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So let's just use some values here.

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And they made a typo here.

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Yup.

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So now it looks good.

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And the methods we want to use once again, the cutoff 45 methods, then we want to give it the time

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values.

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So of course, this is not really necessary to do, but to generate nice plots and to know everything

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about the intermediate time between zero and 50.

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It's always helpful to have this trial in space from teh start to T and then the number of steps, for

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example, like this.

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And now comes the difference.

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So if it would run this now, it would not work because in our function, we have to find several inputs

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here.

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And the integrated solve underscore IVP only knows how to handle functions with the arguments T.

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And then here are our can be in the right.

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But then it doesn't know what is ABC, so we must specify that ABC are parametres that do not change.

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So we write.

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This arcs is equal to in brackets, a comma, become a C.

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No, it uses these values that we have to find here and treats them as arguments in the function f o

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d e.

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And of course, they have to be called exactly like here.

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So now if I run this and if I run this, oh no, I get a problem here because I made a typo and T and

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not a typo.

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Yes.

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No, it works.

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Sorry for the two typos, and then we can continue and plot the trajectory.

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So the right plot trajectory, this will be the name of our plot peel dot axes.

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And first of all, since it's three dimensional system, we must activate the 3D option and then we

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can go ahead and write plot trajectory

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is equal to.

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No, that's not correct.

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Plot trajectory dot plot three and then we must adhere the X, Y and Z coordinates.

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So this will be a solution.

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Thought why X component will be the first component with the index zero one two, and then we can change

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into line with to make this plot look a bit nicer.

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For example, like this.

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So you see here is our butterfly.

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We start from this particular point, which I think corresponds to one zero zero, and then it moves

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around these two points in space.

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And I've written already before here, that's too repulsive.

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Points exist.

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And so these are here, these centers.

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So the trajectory of the particle will move on a trajectory around these around these repulsive points.

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And now we want to check if this is really a chaotic solution.

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So let me add here.

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Another calculation was I just copied this line of code and call it solution two, and I would just

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briefly change the starting conditions, so just multiply the X zero by 1.1

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and oops, sorry, 1.1 times x zero.

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And now we can add here and other plots so we can just copy and write solution to solution two, solution

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two.

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And now we see here two of these trajectories on top of each other.

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And you see at the starting position you see really hit blue and an orange line almost perfectly on

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top of each other.

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And then at a later point of time, you see here these two lines, they don't agree with each other

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anymore, and the position at the end of time is then totally different, of course.

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So we can compare just by writing solution dots y and then all of the coordinates at the last point

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of time that we have calculated would be this one and would be this one.

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So you see quite different, even though in the beginning they were just positioned at one zero zero

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and 1.1 zero zero.

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And this is what is meant when people say that this is a chaotic system and that a butterfly effect

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exists where if you don't exactly and precisely know the starting conditions, you don't anymore know

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about the end positions, even though the system is of course, deterministic and it's not a quantum

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system, it's really a classical, deterministic system, but it behaves chaotic.

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So now I just want to analyze this in a bit more detail.

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The first thing is that we have here such a bad resolution, so it's really hard to see anything.

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So what we can do is we can save it.

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So let's safe fig and then we give it a name.

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For example, Lauren's trying to read thoughts PMG and to increase the resolution.

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Oops.

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Sorry.

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You're right.

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DPI dots per inch and then quite a high number, for example, three hundreds.

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And now I can open this.

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So file open.

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And here it is.

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So you see here, you can really much better see, you know, the trajectory, and at the beginning,

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the two lines orange and blue are really on top of each other.

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And then the keep on top of each other.

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But then it's, of course, a bit hard to see here because you cannot follow all of the trajectory.

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But, you know, at some point, the two trajectories will be independent of each other.

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And then the really just the chaotic behavior winds over the over the deterministic behavior.

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All right.

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So

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we have also figured this out by looking at the coordinates that are here totally different, especially

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X and Y.

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And this is really the indication of this chaotic behavior.

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The last thing that we can do is before I will zoom in on the last thing that we can do is analyzing

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the coordinates.

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And for this, we can just write Peel T dot plot and then we plot the solution dots T versus just one

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of the coordinates, for example, the X coordinate like this.

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And so you see how there are so many sign changes here.

135
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So the X coordinate over time changes from negative to positive over and over again.

136
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And this is whenever a particle is closer to this one or to this one.

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Of course, we could also analyze hit y coordinate, and it's the same thing, basically.

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And for the Z coordinate, it's different because you see both of the wings of this butterfly, they

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are at positive values.

140
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So it really makes more sense to use here X or Y as the coordinates that we want to analyze.

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All right.

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So we have now figured out what this chaotic behavior means, and we have seen that we can really propagate

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this differential equation and that we can solve such differential equations in three dimensions that

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depend on each other.

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So here the coordinates X, Y and Z are not independent of each other as before.

146
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But here really, we have three couples differential equations.

147
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But still, it was no problem we could have.

148
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We could use the same methods, which is to find a function, return the three right hand sides of the

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equations.

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And then we just call, integrate, dot, solve, underscore IVP and then plot the solution and analyze

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it.