Dynamic Geomag: Chaos Theory explained
^ Umm yeah somthing i found..its a simple pendulum demonstrating the Chaos theory. The pendulum ends in a south magnetic pole, attracted by the four coloured north poles below...
Mathematical theory
Sarkovskii's theorem is the basis of the Li and Yorke (1975) proof that any one-dimensional system which exhibits a regular cycle of period three will also display regular cycles of every other length as well as completely chaotic orbits.
Mathematicians have devised many additional ways to make quantitative statements about chaotic systems. These include: fractal dimension of the attractor, Lyapunov exponents, recurrence plots, Poincaré maps, bifurcation diagrams, and transfer operator.
Sharkovskii's Theorem
In mathematics, Sharkovskii's theorem is a result about discrete dynamical systems. One of the implications of the theorem is that if a continuous discrete dynamical system on the real line has a periodic point of period 3, then it must have periodic points of every other period.
Suppose
f : R → R
is a continuous function. We say that the number x is a periodic point of period m if f m(x) = x (where f m denotes the composition of m copies of f) and having least period m if furthermore f k(x) ≠ x for all 0 < k < m. We are interested in the possible periods of periodic points of f. Consider the following ordering of the positive integers:
3, 5, 7, 9, ... ,2·3, 2·5, 2·7, ... , 22·3, 22·5, ..... , 24, 23, 22, 2, 1.
We start, that is, with the odd numbers in increasing order, then 2 times the odds, 4 times the odds, etc., and at the end we put the powers of two in decreasing order. Sarkovskii's theorem states that if f has a periodic point of period m and m ≤ n in the above ordering, then f has also a periodic point of period n.
As a consequence, we see that if f has only finitely many periodic points, then they must all have periods which are powers of two. Furthermore, if there is a periodic point of period three, then there are periodic points of all other periods.
Sharkovskii's theorem does not state that there are stable cycles of those periods, just that there are cycles of those periods. For systems such as the logistic map, the bifurcation diagram shows a range of parameter values for which apparently the only cycle has period 3. In fact, there must be cycles of all periods there, but they are not stable and therefore not visible on the computer generated picture.
Interestingly, the above "Sharkovskii ordering" of the positive integers also occurs in a slightly different context in connection with the logistic map: the stable cycles appear in this order in the bifurcation diagram, starting with 1 and ending with 3, as the parameter is increased. (Here we ignore a stable cycle if a stable cycle of the same order has occurred earlier.)
Veiws
I think.....really.....i dont know what to think...Theres scarse evidence to support the theorys, even though models and demo's have been constructed...I would like to think that maybe if this process is out there then maybe we could use it for our advantage...though as its been shown with the pendulum its terribly hard to predict...
Linear Mode
