This coursework relates to an investigation and description of the systems behavior adopted by 1-D attractors typified by the logistic equation (the version to use is given below, in its difference equation form). The logistic equation: PN+1 = PN + kPN(1-PN). The coursework comprises an appreciation of data representation and its visualization, plus an indication of the equations use in engineering. The way we have set out to do this coursework is by simulating the behavior of the equation by using an Excel spreadsheet model, and then using this to plot graphs.
We plotted 40 graphs for values of k within the range of 0. 5 – 3. 0, with the interval of 0. 0625. We believe this has given us a big enough range of graphs to show the varying behavior of the equation, with different values of k. We have then split the graphs into categories depending on the characteristics that they have shown. We also observed different steady states of the value PN for different values of k, and then plotted a graph with these results to show the overall system behavior. After having plotted all of these graphs, we observed that the system behavior shows bifurcation.
This is when the steady states that we have found multiply in a process called period doubling . The number of steady states multiply from 2 to 4, and then to 8 and so on. This occurs until the system reaches chaos, where no more steady states can be found. The process of bifurcation can be shown in a bifurcation diagram, where all the values for the steady states of PN are plotted against k. The values for the steady states are found by looking at the individual graphs for PN against k, and observing the values to which PN converge. These can then be put into an Excel spreadsheet to plot for the bifurcation diagram.
Plot a family of curves (use at least 20 values of k; you will probably need to use more) of PN against iteration number to illustrate your findings. Graphs of the logistic equation We had been given a starting value for PN at 0. 350, we then used this to create a table in an excel spreadsheet, which consisted of the different parts of the logistic equation, as shown below in Fig 1 and Fig 2. Having created a working spread sheet model for this equation, we set about presenting this data in graphs. Upon plotting the graphs, the first observation we made was that from 0.
5 – 1. 125 the graphs converged towards one without PN exceeding the value of one. Therefore, they all have a common steady state of one. Another observation made was that as the value of k tends towards 1. 125, the quicker the value of PN reaches its steady state value of one. This is shown in the first ten graphs we created. (See appendix Figures 3-13) We observed in our next set of graphs that the value of PN begins to oscillate around the value of one (see appendix Figures 14 – 26). As it oscillates it converges towards a steady state of one.
It shows that the behaviour of the logistic equation is now becoming more complex. We can observe the equations increasing complexity with the increase in the value of K. Figure 26 shows the oscillations of the equations in detail. The graph for which the value of k is 2 is a bit different from the others as it doesn't reach a steady state within the range of the graph (250 iterations). We decided to investigate further to see if it reached a steady state at a higher iteration; we found that after 100,000 iterations it still hadn't converged to a steady value.
After looking at this in detail, we found that the value of PN actually decreases exponentially, tending towards the value of one but never actually reaching one. This is a turning point in the pattern of graphs because the graphs now converge to two steady states, instead of tending towards the value of one. After some investigation we have found that this shows signs of bifurcation of the value of PN as a small change of K can result in a big change in the behaviour of the system. For the graphs with values of k which are between 2 and 2.
5 (in our range of graphs, but upon further investigation 2. 4495 is closer to the point on which the graph bifurcates), we have observed that the system now reaches 2 steady states (see appendix Figures 28-34). An interesting factor observation from the set of graphs is that the points do not oscillate about PN =1 which we would have expected from the last set of graphs mentioned previously. The fact that the graphs now reach two steady states instead of suggest that the system has bifurcated between these points. Our next value of K changes the behaviour of the equation again.
Now the equation has 4 steady states with 2 extreme steady states and 2 confined within them, shown in Figure 35. This occurs very suddenly with the graphs values having a very wide range. The increase in steady states shows that the graph has bifurcated again, and the steady states show period doubling as they multiply from 2 to 4. Figure 36 shows the final graph we found in our range which had steady states. We took time to try and find the point at which the equation flows into chaos; we found that it occurs around 2. 567. Which to find a better value for when the bifurcation occurs, onwards of 200000 iterations need to be calculated.
The next group of graphs reach chaos, as they never converge to steady states. There is no set pattern and the numbers randomly change although, every now and again it begins to look like the values are converging into a pattern when suddenly they can break down into a chaotically random state again. From this point on no further steady states are reached and the system stops bifurcating as it reaches chaotic behaviour. (See appendix Figures 37 – 43) Plot Psteady state against k using the values found in part a); what is the name given to this diagram?
Define the ? and ? values and their associated mathematical functions. Now that we have observed the logistic equation responding to different values of k, we were set the task of plotting the steady value of PN against the value of k. This was done simply by setting up a graph on Excel with k as the X axis and by plotting one of the steady state values against it. For the values of k which had more than one steady value we plotted these again on top of the original graph to build up a complete understanding of what happens in a bifurcation diagram.