LogisticMap.html

For a Study of the Logistic Map

We will first use the un-renormalized equation, which says that

n[t+1] = r*n[t]*(1-n[t]/N) , where N is the total carrying capacity of the system.

We then choose parameters as in the text.

(1)

(2)

(3)

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At this point we get tired, and do a lot of them at once.

(6)

Then we should plot the results, but it's not very complicated, so we don't bother for now.

Below is the value that it would have had if we simply use the exponential map:

(7)

Now let's try a much larger value of r

(8)

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Having seen how it works, now let's introduce x = n/N, which then can only vary from 0 to 1:

the equation is x[t+1] = r*x[t]*(1-x[t]).

(10)

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Plot_2d

We see that this is indeed a period-two situation, as one thought as well from the numbers.

Let's make r back to its arbitrary value, and look at our underlying functions for a moment.

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To find the fixed points we look for the intersections of x and f(x):

(15)

To determine their stability we evaluate the derivative of f(x) at those points:

(16)

As the derivative must have absolute value less than or equal to 1, for stability, we see that

the first fixed point is stable up to r=1, when the second one then takes over, and remains stable until r=3.

Plot_2d

Now, let's think about double period situations, such as the one with r=3.3 above.

We begin by looking at the first iterate---often called second iterate---of our map

(17)

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We will ask when it intersects x

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As should have been expected, the fixed points are of course also fixed points for the iterate;

However, there are also two new ones, but only when r is greater than or equal to 3.

(21)

It is the last two that are of interest---noting that both of them give the same value to the derivative, and remembering that we only care when r is in [3,4]

(22)

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Of course only r=3 and r=1+sqrt(6) are relevant. These tell us that these two double points are stable fixed points, for g, when r is between 3 and 1+\sqrt{6) =

3.4495

The plot just below is for exactly 3, where all these points are the same, while the one below that isfor r=3.4, with the two double points