Abstract:In this article, we will discuss some spectacularly beautiful images known as Fractals such as Sierpiński Triangle, Koch Curve, Dragon Curve, Koch Island, H Fractal, The Levy Curve Fractal, Box Fractal etc. We will investigate and calculate the area, perimeter and self-similar dimension of fractals. Observing the results we see some similarities about the said properties for some fractals those are generated by particular method. Our attention is restricted to find the mathematical behavior of Fractals so that we can establish mathematical formulas concerning the fractals.

In this article, we have discussed basic concepts of one-dimensional maps like Cubic map, Sine map and analyzed their chaotic behaviors in several senses in the unit interval. We have mainly focused on Orbit Analysis, Time Series Analysis, Lyapunov Exponent Analysis, Sensitivity to Initial Conditions, Bifurcation Diagram, Cobweb Diagram, Histogram, Mathematical Analysis by Newton's Iteration, Trajectories and Sensitivity to Numerical Inaccuracies of the said maps. We have tried to make decision about these mentioned maps whether chaotic or not on a unique interval of parameter value. We have performed numerical calculations and graphical representations for all parameter values on that interval and have tried to find if there is any single value of parameter for which those maps are chaotic. In our calculations we have found there are many values for which those maps are chaotic. We have showed numerical calculations and graphical representations for single value of the parameter only in this paper which gives a clear visualization of chaotic dynamics. We performed all graphical activities by using Mathematica and MATLAB.

This paper aim to describe a number of simplifications that can be made to the Lorenz system that preserve its dynamics as well as a number of chaotic systems. The butterfly effect was proven. The solution of differential equation and Lorenz Attractor were investigated. This study compares the dynamical behaviors of the Lorenz system with complex variables to that of the standard Lorenz system involving real variables. Different methodologies, including the Lyapunov exponents spectrum, the bifurcation diagram, the first return map to the Poincare section, topological entropy, periodic and quasi-periodic phase portraits, and chaotic behavior of the resulting system were discussed in Matlab.

In this article, we studied that no homeomorphism on unit interval into itself is chaotic in the sense of R.L. Devaney. We also studied the behavior of orbits of points in the dynamical system defined by homeomorphism on the unit interval.

In this paper we are concerned with a general form of the Henon map as a retarded functional equation. The existence of a unique solution is proved. The continuous dependence of the solution and the local stability of fixed points are investigated. Dynamics of periodic points, Chaos, bifurcation, and topological conjugacy of the resulting system are discussed in Matlab.