In this paper, analysis and control of the chaotic vibrations in bounce dynamic of vehicle have been studied according to the comparison of controller based on the nonlinear control and CHAOS controller on the basis of the chaotic system properties. After modeling the vehicle dynamic, the chaotic behavior of the uncontrolled system was determined using combination of the numerical analysis including bifurcation diagrams and max Lyapunov exponent. The system parameters values were then identified in the quasi-periodic and chaotic behavior system. In order to eliminate the chaotic vibrations, the control signal was first developed using a nonlinear fast-terminal sliding mode control algorithm that its control gains are estimated online by fuzzy logic which was designed for vehicle vertical dynamics. Then the delayed feedback control was designed based on the development of Pyragas algorithm to control the system based on the properties of the chaotic system and generation of a small control signal. Comparison of the feedback system depicts priority of the Fuzzy-Pyragas controller in less energy consumption and better behavior.

The sequence of events is always either regular or chaotic. Regularity is an ordered sequence of events. In a chaotic sequence, however, such a law does not appear. This is our judgment, but from a philosophical point of view, this problem can be approached realistically. The sequence of natural numbers, the series of objects that cause and effect each other, many natural phenomena, and the limits of the set of human forces follow each other with a certain regularity. The sequence of natural numbers follows one another according to the law "n + 1". As well as it is also possible to show the formulas for the regularity of these sets, i. e. regular systems. This applies to regular systems that have both finite and infinite number of limits. However, is it possible to collect them under a certain law, as in chaotic sequences? At first glance, it seems that, as its name suggests, if a sequence is "chaotic", then there is no question of "order". But in any case, as if in any plurality, if you look closely, you can see any order, taking into account certain "signs" of the limits in this or that arrangement. In this article, we will look for a mathematical answer to such a philosophical question and try to show that there is a certain order in any CHAOS. In other words, no set of events in the universe is irregular. We will explain this according to Lagrange's "interpolation" formula.

Uncertainty evaluation is an important task in almost all activities in the field of calibration and standard conformity assessment. Evaluation of uncertainty is a process in which the statistical characteristics (mainly the first and second moments,i. e. the average and the standard deviation) of the result of a measurement are estimated based on those of contributing factors. The result of this process is used to determine how appropriate a measurement is for a specific application. The measurement uncertainty influences the result of binary decisions made on conformity assessment, calibration and many other applications in the fields of jurisdiction, commerce, law enforcement or healthcare. Another important application of uncertainty evaluation, is to evaluate and validate mathematical or computer models simulating real world phenomena. For example, an uncertainty evaluation process can be utilized to quantify the error characteristics of a simulation software. Many techniques have been introduced and practiced for uncertainty evaluation. The applicability of each method relies upon the measurement model, possibility of performing repeated simulations and availability of statistical data. In this paper, we examine a celebrated uncertainty propagation method, namely,the Polynomial CHAOS Expansion in the uncertainty evaluation of measurements described by a rigorous mathematical model. These measurement processes have been conventionally studied using sensitivity coefficients which can be calculated via partial derivatives. The two approaches are compared in an illustrative example.

A CHAOS control method suggested by Erjaee has been reviewed. It has been shown that this technique can be applied in various evolutionary systems of 2-dimensional types. The method has been applied for cases of the Henon map, as well as Burger’s map. The limitations of the control technique have also been discussed by considering the Standard Map and the Gumowski-Mira map. The results obtained through numerical calculations are very interesting and significant. This technique has some advantages over many other techniques of CHAOS control in discrete systems.