For the following project I'm supposed to complete, I am not fully understanding what it is exactly asking for. Am I supposed to run a code using the first equation or the second? What is the difference between the first and second equation?

Logistic map is given as:

$$x_{n+1} = r\cdot x_n(1 - x_n) ,\qquad x_n \in [0,1] \tag{1}$$ This recursive equation has different behavior for different values of $r$. The purpose of this project is to experience the effect of $r$ on this dynamic. To do so, let’s consider the member of the following set for values of $r$:

$$R = \left\{r_i \mid r_i = \frac{10}{i}, 1≤i≤1000\right\} \tag2$$

Now, we have the recursive equation, and we can calculate this sequence for any given initial values. In this step, I am interested in $x_{10000}$ for 100 random initial values $x_0$. In other words, you have to choose 100 random initial values. Run your code to find $x_{10000}$ for each of these initial values.

up vote 1 down vote accepted

You are supposed to implement both sets. Here is the idea. If you choose a single value for $r$, say $r=1$, then you have what is called a dynamical system that takes a value $x_0$ and maps it to a new value $x_1$, which it maps to $x_2$, and so on. If you pick a starting value $x_0$ then you can follow the trajectory. Say let's pick $x_0 = 0.5$:

$$0.5 \to 0.25\to 0.1875 \to 0.1523\to 0.1291\to 0.1125\to\cdots\to 0.00009989\to\cdots$$

each value of $x$ is followed by $r\cdot x\cdot (1-x)$, where remember we selected $r=1$. It's pretty clear from looking at this that the $x$ values are heading toward zero. Do they do that for every possible starting value $x_0$? Your instructor suggests that you pick a hundred random $x_0$'s and look at the trajectory of each one: for each one calculate the corresponding $x_{10000}$ that you get by applying the transformation $x\to r\cdot x\cdot (1-x)$ 10,000 times, and see where you get. Maybe make a scatter plot comparing $x_0$ with $x_{10000}$ in each case.

Then pick a different $r$, say $r=4$, and do the same experiment. The behavior of the system is extremely different! Most values of $x_0$ do not head toward 0. The scatter plot of 100 different $x_0$ values and their corresponding $x_{10000}$'s will look totally different.

So each experiment involves choosing one value for $r$, and looking at the behavior of the $x\to r\cdot x\cdot (1-x)$ map for various $x_0$'s, and seeing what you can notice about it. When you understand that map, try one with a different $r$.

OK?

  • Thanks, this clears it up quite a bit. I'm going to try this out. Thanks for editing my bad formatting btw. – Mike Jun 5 '13 at 18:11

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