Impending Role Of Period Three Points In Chaotic Functions Through Sturm Method And Matlab Programming

Chaos Theory and Dynamical Systems has been considered as one of the most significant breakthroughs in Mathematics in this century. Its applications in a wide range of subjects including Physics, Biology, Chemistry, Ecology, Fluid Mechanics, Engineering, Economics etc., have made the field very attractive and important for the researchers from various disciplines. The salient feature of the chaos theory is that even simplest looking maps illustrate virtually every important phenomena that occur in the Dynamical Systems. The logistic map f (x) = k x ( 1 - x ) is a popular example of such a behaviour. This family has been one of the simplest of the nonlinear maps, but it exhibits the various aspects of dynamical systems varying from stability to chaos. However this concept needs to be generalized. The quite unknown dynamics of a cubic family of functions defined by f ( x ) = x 3 + x shows some interesting similarities as well as differences with this family of functions. The advent of modern computers has quickened the pace of development in the field of Chaos. Matlab programming can also play a vital role to visualize the inner properties of chaotic functions, including one of the three essential properties of chaotic functions viz. existence of period three points. This paper explains the existence of period three points for cubic family of functions through the Sturm method for finding the solution of the higher degree polynomials and Matlab programming.