On the Complex and Chaotic Dynamics of Standard Logistic Sine Square Map

In this article, we set up a new nonlinear dynamical system which is derived by combining logistic map and sine square map in Mann orbit (a two step feedback process) for ameliorating the stability performance of chaotic system and name it Standard Logistic Sine Square Map (SLSSM). The purpose of this paper is to study the whole dynamical behavior of the proposed map (SLSSM) through various introduced aspects consisting fixed point and stability analysis, time series representation, bifurcation diagram and Lyapunov exponent. Moreover, we show that our map is significantly superior than existing other one dimensional maps. We investigate that the chaotic and complex behavior of SLSSM can be controlled by selecting control parameters carefully. Also, the range of convergence and stability can be made to increase drastically. This new system (SLSSM) might be used to achieve better results in cryptography and to study chaos synchronization.

Border collision bifurcations of chaotic attractors in one-dimensional maps with multiple discontinuities

In one-dimensional piecewise smooth maps with multiple borders, chaotic attractors may undergo border collision bifurcations, leading to a sudden change in their structure. We describe two types of such border collision bifurcations and explain the mechanisms causing the changes in the geometrical structure of the attractors, in particular, in the number of their bands (connected components).

On the generation of self-similar with long-range dependent traffic using piecewise affine chaotic one-dimensional maps (renewed version-2)

A qualitative and quantitative extension of the chaotic models used to generate self-similar traffic with long-range dependence (LRD) is presented by means of the formulation of a model that considers the use of piecewise affine one-dimensional maps. Based on the disaggregation of the temporal series generated, a valid explanation of the behavior of the values of Hurst exponent is proposed and the feasibility of their control from the parameters of the proposed model is shown.

On the generation of self-similar with long-range dependent traffic using piecewise affine chaotic one-dimensional maps (renewed version)

A qualitative and quantitative extension of the chaotic models used to generate self-similar traffic with long-range dependence (LRD) is presented by means of the formulation of a model that considers the use of piecewise affine one-dimensional maps. Based on the disaggregation of the temporal series generated, a valid explanation of the behavior of the values of Hurst exponent is proposed and the feasibility of their control from the parameters of the proposed model is shown.

On the generation of self-similar with long-range dependent traffic using piecewise affine chaotic one-dimensional maps (extended version)

A qualitative and quantitative extension of the chaotic models used to generate self-similar traffic with long-range dependence (LRD) is presented by means of the formulation of a model that considers the use of piecewise affine one-dimensional maps. Based on the disaggregation of the temporal series generated, a valid explanation of the behavior of the values of Hurst exponent is proposed and the feasibility of their control from the parameters of the proposed model is shown.

BEHAVIOR OF FIXED POINT CONGRUENT PERIODIC TRAJECTORIES OF NONLINEAR MAPS IN DYNAMICAL SYSTEMS THEORY

This paper considers problems that arise during number sequence generation based on non- linear dynamical systems. Complex systems can depend on many parameters analysis and examination of one-dimensional maps was per-formed since these maps are dynamical systems. Dependence of iterative fixed points for nonlinear maps on the properties of functions and function domain numbers was investigat- ed. Several approaches to randomness evaluation and, accordingly, methods for estimating the degree of randomness of a particular sequence were considered. The properties and internal structure of sequences obtained on the basis of nonlinear maps were also examined in accordance to their influence on the degree of randomness.