Circuit Implementation Synchronization between Two Modified Fractional-Order Lorenz Chaotic Systems via a Linear Resistor and Fractional-Order Capacitor in Parallel Coupling

In this study, a modified fractional-order Lorenz chaotic system is proposed, and the chaotic attractors are obtained. Meanwhile, we construct one electronic circuit to realize the modified fractional-order Lorenz chaotic system. Most importantly, using a linear resistor and a fractional-order capacitor in parallel coupling, we suggested one chaos synchronization scheme for this modified fractional-order Lorenz chaotic system. The electronic circuit of chaos synchronization for modified fractional-order Lorenz chaotic has been given. The simulation results verify that synchronization scheme is viable.

Parameters Identification of Nonlinear Lorenz Chaotic System and its Application to High Precision Model Reference Synchronization

The Lorenz chaotic system synchronization has been a popular research topic in the last two decades. Most of the studies focus on the design of model reference adaptive control (MRAC) synchronization schemes. In the existing MRAC schemes, adaptive laws are designed to estimate the system parameters online. However, due to the system parameters being unknown, arbitrary selection results in a longer estimation period. Although applying large values of adaptive gains can increase the estimation convergence speed, it usually induces obvious estimation oscillations and large control efforts. On the contrary, small adaptive gains result in smooth but sluggish transient estimations. None of the studies addressed on the parameters estimation and its contribution to precise synchronization. To address this issue, two system identification schemes are presented. The first applied scheme is called observer/Kalman filter identification (OKID). The second one is called bilinear transform discretization (BTD). The related detail derivations for the Lorenz chaotic system parameter identification will be presented in this paper. Results show that the proposed BTD identification algorithm is relatively simple and computationally efficient. Moreover, highly precise parameter estimations can be achieved as well. Nevertheless, due to the complex nonlinearity of the chaotic system, it will be illustrated that even extremely small parameter deviations could lead to dramatic mismatch for the chaotic system model output prediction. To further solve this issue, a MRAC is further designed in which the initial guess of the system parameters is obtained through the proposed BTD identification algorithm. Since the identified parameters are already very close to the true value, smaller values of adaptive gains can be used. With the aid of the precise parameter identification, the transient dynamics and the convergence performance of the MARC are both improved significantly. Simulations demonstrate the effectiveness of the proposed scheme.

A image encryption algorithm based on chaotic Lorenz system and novel primitive polynomial S-boxes

We have proposed a robust, secure and efficient image encryption algorithm based on chaotic maps and algebraic structure. Nowadays, the chaotic cryptosystems gained more attention due to their efficiency, the assurance of robustness and high sensitivity corresponding to initial conditions. In literature, there are many encryption algorithms that can simply guarantees security while the schemes based on chaotic systems only promises the uncertainty, both of them can not encounter the needs of current scenario. To tackle this issue, this article proposed an image encryption algorithm based on Lorenz chaotic system and primitive irreducible polynomial substitution box. First, we have proposed 16 different S-boxes based on projective general linear group and 16 primitive irreducible polynomials of Galois field of order 256, and then utilized these S-boxes with combination of chaotic map in image encryption scheme. Three chaotic sequences can be produced by the disturbed of Lorenz chaotic system corresponding to variables x, y and z. We have constructed a new pseudo random chaotic sequence ki based on x, y and z. The plain image is encrypted by the use of chaotic sequence ki and XOR operation to get a ciphered image. To show the strength of presented image encryption, some renowned analyses are performed.

On the Caputo-Fabrizio fractal fractional representation for the Lorenz chaotic system

<abstract><p>The widespread application of chaotic dynamical systems in different fields of science and engineering has attracted the attention of many researchers. Hence, understanding and capturing the complexities and the dynamical behavior of these chaotic systems is essential. The newly proposed fractal-fractional derivative and integral operators have been used in literature to predict the chaotic behavior of some of the attractors. It is argued that putting together the concept of fractional and fractal derivatives can help us understand the existing complexities better since fractional derivatives capture a limited number of problems and on the other side fractal derivatives also capture different kinds of complexities. In this study, we use the newly proposed Caputo-Fabrizio fractal-fractional derivatives and integral operators to capture and predict the behavior of the Lorenz chaotic system for different values of the fractional dimension $ q $ and the fractal dimension $ k $. We will look at the well-posedness of the solution. For the effect of the Caputo-Fabrizio fractal-fractional derivatives operator on the behavior, we present the numerical scheme to study the graphical numerical solution for different values of $ q $ and $ k $.</p></abstract>