
Title:
08x05 Lorenz Peaks

Description:

For our second class problem in the final exam, we're going to talk a little bit more

about chaos theory especially in relation to the Lorenz Butterfly.

We have some system in a state defined by x, y, and z.

Now the solution that we might get for this system depending on how we define

different parameters within the equations could look something like this.

I'm not very good at drawing in 3D but this should look something like the classic butterfly image

that Jorn showed you in the unit.

Remember that our solution has two wings. One over here and one over here.

Both spiraling outward.

We can plot how z, this coordinate, varies as a function of time.

And when we do this, we see a series of peaks that ascend and then descend

and then reascend and so on and so forth.

This set of peaks of increasing height corresponds to spiraling outward along one wing.

And then as you jump down to a local minimum to stepping upward again over here

corresponding to the spiraling outward along this wing.

What we want to know is if we can predict the progression

of this local maxima along the z axis.

You can make a plot of the current local maximum of z

versus the previous local maximum and then take each of these maxima and plot them down here.

So here's a preview of what you should get as your solution to this problem.

Up here we can see how the peaks in z vary with time.

Clearly, these zigzags are a little bit more accurate than mine were.

And down here we see how the present local maximum depends on the previous local maximum.

You'll know that you've gotten the problem right if something like this appears.

Now in order to plot each local maximum we do in fact need to know

the exact z coordinate of that maximum.

One issue that we have when using approximation methods, however,

is that sometimes our time steps are not going to line up exactly with the locations of the peaks.

So if you look at this visual for example at step1 we have this value of z.

At step, we have this value.

And after one more time step, we've passed over the peak and downwards again.

We plotted this on its own completely ignoring where the actual peak is

and shows instead it's substantially lower.

In order to approximate better where this actual peak is,

we can pretend that there is a parabola connecting these three different points.

The vertex of the parabola will be our estimate for where the true location of the peak is.

To find out the t coordinate of the peak, you can just simply use the quadratic formula.

Just a couple of hints for you. You want only one z value to be equals the maximum.

So think about what that means in terms of our equations down here.

Also, think about how you could use the essential difference formula

when you're discussing this local maxima.

Remember that one way to calculate a, b, and c is to plug in information

about the points that you know.

Then you'll have these coefficients and be able to find the proper value of z and t.

Let's take a look at the code for a second.

Now the first thing that we want you to do for this problem is to use Heun's method, which you

remember from section 2.9, with a fixed step size and not a variable step size for the Lorenz system.

Remember that Heun's method depends on the forward Euler method

so we've included that for you right here.

So again just to be clear, use Heun's method with nonvariable step size to figure out how x, y, and z

are going to change with each time step.

Once you do this, you're going to use the parabola fitting method that I just discussed

to figure out better where the local maxima are.

Once you've done that, you're going to enter your estimate at the local maximum value.

This should be a fun look into chaos theory. Good luck.