Precisely chaotic models survey with Qualitative Bifurcation Diagram

The most important method of recognizing behavior of recurrent maps is to plot Bifurcation Diagram. Conventional method for plotting is to generate a variety of time series for different values of model parameter and plotting these points with respect to it after transient state. Bifurcation Diagrams are destitute of analytical interpretations for determining the cause of chaotic and period doubling behaviors. In this study, a diagram is presented which can be called Qualitative Bifurcation Diagram (QBD). One of the special features of Qualitative Bifurcation Diagram is recognition of parameter values and initial conditions, in which, the map has periodic, Aperiodic or chaotic behavior and for recognizing these behaviors, no primary information with regards to the map is needed.
The implementation of QBD on the logistic map indicates high susceptibility of the algorithm in distinguishing high periods and periodic windows; it has not only reduced the calculation time, but also has made it possible to examine initial conditions and logistic parameter effects with high velocity simultaneously. In spite of existence arrangement in Bifurcation Diagram, the result showed that logistic is not a mosaic arrangement, but it has a dynamic arrangement.
This article dealt with the investigation of Lyapunov Exponent as one of the tools of chaos examination which is analyzed analytically.