On bifurcation and chaos in a discrete dynamical system

The aim of the present paper is to define a class of nonlinear functions on the set of non-negative real numbers and study the dynamics and periodic points of these functions. We show that this class of functions contains chaotic functions and exhibits the phenomenon of bifurcation. We also apply the results on dynamics of these functions to obtain general tests of divisibility of numbers and conditions of divisibility of polynomials. We not only obtain generalizations of the well-known remainder theorem and the factor theorem of algebra but also compute an explicit value of the quotient.