Mathematics – Dynamical Systems
Scientific paper
2008-06-24
Mathematics
Dynamical Systems
17 pages, 3 figures
Scientific paper
Simple methods permit to generalize the concepts of iteration and of recursive processes. We shall give some simple definitions and, then, see briefly on several examples what these methods generate. In additive sequences Algebra, we shall encounter not only the golden or the silver ratio, but a dense set of ratio limits that corresponds to an infinity of conceivable recursive additive rules. We shall show that some of these limits have nice properties. Identities involving Fibonacci and Lucas sequences will be viewed as special cases of more general identities. We shall show that some properties of the Pascal Triangle belong also to other similar objects. We introduce p-Tribonacci and p-Lucas-Tribonacci sequences. In Dynamical Systems and Chaos Theory we shall encounter weird orbits, whose order is higher than the number of its distinct elements and, beyond the chaos point, a rather unexpected belated convergence to 0, after a pseudo chaotic behaviour during as many terms as one may wish. Time and again, we shall find the Feigenbaum constant. In Linguistics we shall see that recursive rules applied to concatenation are sometimes equivalent to formal grammars although generally more restrictive, therefore stronger.
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