On Direct Bifurcations into Chaos and Order for a Simple Family of Interval Maps
نویسنده
چکیده
In this note, we present a simple one-parameter family of interval maps which has a direct bifurcation from order to chaos and then another direct bifurcation from chaos back to order. (See also [4, 5].) In fact, for this family of interval maps, the creation of the first non-fixed periodic point is more complicated than we expect. It is the limit point of a series of bifurcations of period 2n (n ^ 3 odd) points. Consequently, the creation of the first non-fixed periodic point is a bifurcation of period 12 points. After the bifurcation into chaos, this family undergoes a series of bifurcations of period 2n points with n (^ 3 odd) in decreasing order. After the period 6 points are created and live for a while, then, all of a sudden, all chaotic phenomena cease to exist and we have order again. To be more precise, we shall prove the following two results. THEOREM 1. Let 6 be a fixed number in [3/8, 1/2). For 0 ^ c < 6, let
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