On Chaos Controlling Mechanism for Ishikawa Iteration and Its Traffic Flow Model in Discrete Dynamical Systems

Abstract

The fact that very small changes cause unpredictable big changes in nature has led to the chaos theory. Many researchers focusing on chaos have improved this theory in several directions. One of the prominent areas in chaos theory is controlled chaos. Due to the usefulness of controlled chaos, it has been used in many working areas as the iteration methods of fixed point theory. In this study, we have developed a chaos controlling mechanism using the Ishikawa iteration method. Firstly, Ishikawa fixed point type controlling mechanism is defined and the theorems determining the control parameters are proved. Afterward, the results of Ishikawa fixed point type controlling mechanism were obtained by taking the logistic map as an example. In addition, by using the MATLAB software program, the convergence, periodical behavior and chaotic regions occurring outside the range of the parameters forming the controlling mechanism have been investigated and numerical simulations studied. Subsequently, the effectiveness of the stability intervals obtained for the fixed point and periodic fixed points of the Ishikawa fixed point controlling mechanism by applying the Lyapunov exponent is demonstrated. Finally, we have given an application as control chaos of Ishikawa traffic model.