00434 u32b79da7b4e1841dd92bd2bf819342
A Graphical Aid for Analyzing Autocorrelated Dynamical Systems 439
X,=r*X,_*( 1-*M) and X, e(0,l)
where Xt represents the fraction of the possible population at any time t.
For all values of r between 0 and about 3.0, the Verhulst model converges to a single value for each chosen r. In particular when r = 2.99 the iterates converge to 0.665552 approximately.
When the growth ratę r is set between 3 and 3.4495 approximately, the iterates eventually cycle between two values. In this case, the orbit of the iterates is said to have a period of two. In particular if r = 3.449 the iterates cycle between 0.849864 and 0.440075 approximately. If r = 3.50 the iterates cycle among 4 distinct values, and if r = 3.55 the iterates cycle among 8 distinct values.
Similarly, when the population growth parameter, r, is set to 3.569, the population eventually oscillates about 16 fixed values. However, when the population growth parameter is set at approximately 3.570, the modeled population can assume an infinite number of values seemingly at random.
We can readily find values of r between 3 and about 3.570 that produce iterates of period 2, 4, 8, 16, 32, and so on. Upon initial study, it appears that there is a phenomena called period doubling occurring. That is, as the setting of the value of r increases through this interval, we notę that the iterates appear to have periods that are double the immediately prior period. However upon morę careful examination of the iterates we can also find values of r for which there are orbits of period 3, 5, 7, 9 and so on. In addition there are orbits of iterates that exhibit no period at all -- these orbits have been shown to exhibit deterministic chaos (Falconer, 1990).
In the generał case for dynamical systems, the iterates of the orbits may appear to behave randomly and still remain close to some set, A, of points. That is, in these cases the orbits that come near to the set A tend to remain close to A. We cali this set of points A an attractor. In addition, if the set of points A is a fractal set, then the set A is called a strange attractor. The converse of an attractor is a repeller and a fractal repeller is called a strange repeller.
A completely deterministic dynamical system that contains either a fractal attractor or repeller may exhibit chaotic behavior. Falconer suggests that such systems commonly exhibit the following properties:
a) the orbit of a member of the attractor or repeller is dense in the attractor or repeller.
b) the periodic points of the iterates of the attractor or repeller are dense in the attractor or repeller.
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