Abstract :
The non-linear recurrence scheme gn + 1(x) = 4λ(1 − gn(x)) • gn(x) is studied where λ is the growth parameter 0 λ 1 and gn(x) is a function of −1 x 1 instead of a simple number as is the case in the classical logistic mapping scheme. This kind of a new functional logistic mapping is solved by the ansatz gn(x) = 0.5(1 − S2n(x)). It is shown that the generated novel polynomials, S2n(x) with n = 0, 1, 2, 3, …, possess the non-linear recurrence relation S2n + 1(x) = 2λS2n2(x) − (2λ − 1) and represent some kind of a set of generalized Chebyshev polynomials. Their properties are discussed thoroughly for various λ-values, i.e. in the ‘smooth’, the bifurcated and the chaotic regimes.
