Wayne Adkins Chaos and the Logistic Map February 2001

Chaos Conclusions

In my investigations of this problem and chaos in general I have come to a few conclusions that I will state here.

(1)   A chaotic system has something determining its behavior. In our system the variable a determines the convergence or oscillation of the sequence in [0,1].

(2)   A chaotic sysytem is very sensitive to the initial conditions. We see this in our system with seeds x₁ in (0,1) we can wildly different results with x₁ and x'₁ differing only minutely. This makes predictability very difficult because of the need for an exact intitial state.

(3)   Within the chaotic system there appears to be order and patterns. In our system there are periods of order where the system converges briefly befoperiods of order where the system converges briefly before returning to a chaotic state.

Thus with these observations I would have to say that Chaos Theory can allow us to understand previously unpredictable events such as the weather, evolutionary processes and even artificial intelligence with the understanding of how the initial conditions cause outcomes that vary wildly. In a dynamic system we come closer to understanding the system itself.

Now as computers become increasingly powerful we will be able to compute more systems that require multitudes of calculations andbe able to study the results of these dynamic systems and perhaps gain an understanding of the long term evolution of the system.