This bifurcation diagram shows how the behaviour of the logistic equation can suddenly change at certain points. After doing research on the logistic equation we noticed that a strong relation to the logistic equation is the bifurcation diagram. A more detailed example of this can be seen below in Figure 45 . In Figure we can see how our diagram matches the same diagram for a slightly different logistic equation. They both break onto chaos very quickly after the first bifurcation, and the distance between the following bifurcations decreases dramatically each time.
In Weisstein's diagram Figure 45 he has plotted after the diagram reaches chaos, therefore we went back and plotted a few more of the values that we acquired for the values of k after 2. 5625 in order to see if we could see if a trend was occurring. This diagram also shows where the bifurcations occur. Figure 46 is a version of our bifurcation diagram, with sample values from the graphs which didn't converge to any steady states. It shows how in our equation, chaos takes over very rapidly. If we had plotted more values for the chaotic values of k, we would have seen a closer relationship to the graph above it.
While looking into bifurcation how the distance between each bifurcation brought up Feigenbaum constant, a good description of this was illustrated on Wolfram Alpha they described it as "The Feigenbaum constant ? is a universal constant for functions approaching chaos via period doubling" . Therefore, this implies that there is a ratio that governs the distance between each bifurcation. It also suggests that this value can be used for any dynamic system as long as it shows period doubling, and due to the fact that our system does this we can apply this to our results.
So there exists an equation to solve for ?. The other Feigenbaum constant ? is "defined as the separation of adjacent elements of period doubled attractors from one double to the next", just like ? it's a ratio for the bifurcation diagram. When the diagram's steady states multiply, it describes the distance between the values. Using appropriate cited evidence from the literature, describe TWO engineering applications where the Logistic equation has been used to model or simulate real world phenomena.
Application 1- Chaotic noise MOS generator based on logistic map Introduction: what it is simulating? One of the engineering applications where the Logistic equation has been used to simulate real world phenomena is with an analog noise generator. This noise generator simulates the behavior of an electronic circuit which is iterated using a current amplifier with the gain i?? , in the same way that our equation is iterated with the gain k. In this case the logistic equation is; f(x(n)) = x(n+1) = i?? x(n) (1-x(n))
It is simulating an analogue noise MOS generator circuit using different values of i?? to find the steady state of the current. The logistic equation and a bifurcation diagram are used to study the behaviour of the system to help in the design of the circuit. Certain conditions are needed for the circuit to run smoothly, and to find what these specific conditions are, a logistic equation is used to study the effects of using different values for the gain in current. Different graphs are plotted of transient analysis, for different values of to analyse the behaviour at intervals, shown in fig 50. The steady states are then taken from these graphs to plot on the bifurcation diagram, shown in fig 51.
How the logistic equation is used In this application the logistic equation is used in circuit design as a model to mathematically simulate the behaviour of the circuit in order to find the stable values for the current. The logistic equation works alongside Lyapunov's experiment and Birkhoff's Ergodic Theorem in the design of the circuit, but the model is mainly governed by the logistic equation.
In transferring the logistic equation into practical use in the analogue noise MOS generator circuit, the equation simulates an amplifier of current with a gain of. The differing values of affect the transfer function in the circuit. The bifurcation diagram and Lyapunov's experiment work together to indentify the steady states due to the logistic equation, and this is used in the design of the circuit, as these steady states are to be avoided. The Logistic Equation has chaotic behaviour as there is no set pattern and the numbers randomly change, which we found by plotting the graphs.