Studies of one-dimensional unimodal maps in the chaotic regime
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For one-dimensional uninmodal maps hÎ»(x) a binary tree which includes all the periodic windows in the chaotic regime is constructed. By associating each element in the tree with the superstable parameter value of the corresponding periodic interval we define a different unimodal map. After applying a certain renormalization procedure to this new unimodal map, we find the period doubling fixed point g(x) which depends on the details of the map hÎ»(x) and the scaling constant Î±.
The thermodynamics and the scaling function of the resulting dynamical system are also discussed. In addition, the total measure of the periodic windows is calculated with results in basic agreement with those obtained previously by Farmer. Up to 13 levels of the tree have been included, and the convergence of the partial sums of the measure is shown explicitly. It is conjectured that the asymptotic behavior of the partial sum of the measure as the number of levels goes to 00 is universal for the class of maps that have the same order of maximum. A new scaling law has been observed, i.e., the product of the length of a periodic interval characterized by sequence Q and the scaling constant of Q is found to be approximately 1.
We also study two three-dimensional volume-preserving quadratic maps. There is no period doubling bifurcation in either case.
We have also developed an algorithm to construct the symbolic alphabet for some given superstable symbolic sequences for one-dimensional unimodal maps. Using this symbolic alphabet and the approach of cycle expansion the topological entropy can be easily computed. Furthermore, the scaling properties of the measure of constant topological entropy are studied. Our results support the conjectures that for the maps with the same order of maximum, the asymptotic behavior of the partial sum of the measure as the level of the binary goes to infinity is universal and the corresponding 'fatness' exponent is universal. Numerical computations and analysis are also carried out for the clipped Bernoulli shift.
- Doctoral Dissertations