Fractal Trap for The Chaotic Behavior Of Real Cubic Polynomials

Abstract

A real quadratic family functions f(x) = k x ( 1 - x ), k  R, has indicated that even the simplest looking functions may have the complicated dynamics. This logistic map exhibits the properties like topological transitivity, sensitivity dependence on initial conditions and density of periodic points. The cubic family functions f : R → R, f(x) = x 3 + x,   R, are no exceptions. The dynamics of this family is 'controlled' for an interval of values of , but it becomes more complicated as the value of  decreases from -1.5. At λ = -3, f(x) is chaotic on the interval (-2,2). In this paper, we shall see that the chaotic behaviour of f(x) = x 3 + x is even