(Student’s name withdrawn)
MAT200
Prof. Frank Wang
Nov 08, 2005
The Observation of Logistic Difference Equation
For the equation “y=λ*x*(1-x)”, I choseλ valuein the interval [0, 4]. Meanwhile, I chose two different lambdas in the interval (0, 1) with five different seeds, two different lambdas in the interval (1, 2) with three different seeds, each lambda with three different seeds in the interval (2, 3) and 4 lambdas in the interval (3, 4) with three different seeds and one lambda when it equals 4 with one seed.
The most interesting thing that I found out is whatever seed changes, as long as lambdas are in the interval (0, 1), the graphs are periodic, and all of them are declining to 0 and settle down there. In this interval, the greater lambda and seed are, the faster lines drop. It’s showed obviously in the graph that when lambda equals 0.75 and seed is 0.75, the orbit drops by 0.61 (0.75-0.14). From the following graph which is when lambda = 0.75 and seed = 0.75 we can see it clearly.
The trend that graphs are going to 0 could explain the growth of population in our real world. In a scientific article “Chaos in the Pond”, λ is used as growth rate and x is used as initial population so that the equation shows the growth of population. If the growth rate is less than 1, the whole population is going to 0 eventually. That’s the best explanation why so many small European countries which have low population growth rate encourage their people to give birth.
Also, when lambda is in the interval (1, 3), the graph keeps to be periodic. But when the interval is (1, 2), whenever the graph changes, there is no bobbling. The difference between the graphs in this interval is decided by the seed. If the seed is less than 0.5, the orbit rises quickly and keeps to be a constant number. The graph below is when lambda equals 1.75 and seed equals 0.25.
On the contrary, when seed equals 0.5, the orbit falls extremely without any bobbling but is to be constant when y is going to 0.328. However, when the seed is over 0.5, I chose 0.789, the orbit also drops fast till 0.32 then increases quickly to 0.48, after that it keeps constant. We can see it from the following graph. That’s the graph when lambda equals 1.92 and seed equals 0.789.
The bobbling appears in the graphs when the interval is (2, 3). Likewise, the orbit increases quickly when the seed is less than 0.5 and decreases extremely when the seed is over 0.5. Moreover, all graphs in the interval (2, 3) bobbles around a bit before they settle down to a constant number.
Furthermore, when lambda is over 3 but does not reach 3.5, all graphs almost have the same characters as graphs in the interval (2, 3) have, and the changing of orbit is periodic clearly. For example, when lambda equals 3.2 and 3.44, whatever seed is 0.25, 0.5 or 0.789, the orbits respectively approach to period-2 and -4 points. Graph 1 is period -2 and graph 2 is period -4:
Graph 1:
Graph 2
However, all graphs in this interval do not keep to be a constant number. All graphs keep bobbling regularly. It’s very interesting that although I chose 3 different seeds when lambda equals 3.2, there are no much difference between these graphs. All graphs seem have the same orbit and all of them bobbles in the interval (0.510, 0.80).
Nevertheless, when lambda is over 3.5, the bigger it is, the more chaotic the graph is, regardless seeds. The following graph is lambda equals 3.65 and seed equals 0.789.
Before lambda reaches 4, the graphs keep changing and bobbling. But they have one same thing that all graphs keep up and down by turns. When lambda reaches 4, the graph is not up and down regularly. The graph below is when lambda equals 4 and seed equals 0.789.
When lambda is over 4, there is no visible pattern.