Discrete Superior Hyperbolicity in Chaotic Maps

In the last few decades, the dynamics of one-dimensional chaotic maps have gained the tremendous attention of scientists and scholars due to their remarkable properties such as period-doubling, chaotic evolution, Lyapunov exponent, etc. The term hyperbolicity, another important property of chaotic maps is used to examine the regular and irregular behavior of the dynamical systems. In this article, we deal with the hyperbolicity and stabilization of fixed states using a superior two-step feedback system. Due to the superiority in the chaotic evolution of one-dimensional maps in the superior system we are encouraged to examine the hyperbolicity and stabilization in chaotic maps. The hyperbolic notion, hyperbolicity in periodic states of prime order, stabilization, and the hyperbolic set of the chaotic maps are studied. The numerical, as well as experimental simulations, are carried out, followed by theorems, examples, remarks, functional plots, and bifurcation diagrams.