Chaos and bifurcations in chaotic maps with parameter q: Numerical and analytical studies

In this paper, a class of chaotic maps with parameter q are introduced and bifurcations and chaos in proposed maps are numerical and analytical studied. Euler method is employed to get the continuous systems corresponding to chaotic maps and the fractional styles in Caputo's definition. Based on that, we finally infer a class of chaotic maps with the Adams–Bashforth–Moulton predictor-corrector method. In the simulation and analysis, we discuss the Logistic map with q and Hénon map with q, observe the route from period to chaos and do tests to analyze properties of maps with parameter q

Symmetry in Chaotic Systems and Circuits

Symmetry can play an important role in the field of nonlinear systems and especially in the design of nonlinear circuits that produce chaos. Therefore, this Special Issue, titled “Symmetry in Chaotic Systems and Circuits”, presents the latest scientific advances in nonlinear chaotic systems and circuits that introduce various kinds of symmetries. Applications of chaotic systems and circuits with symmetries, or with a deliberate lack of symmetry, are also presented in this Special Issue. The volume contains 14 published papers from authors around the world. This reflects the high impact of this Special Issue

A New No Equilibrium Fractional Order Chaotic System, Dynamical Investigation, Synchronization and Its Digital Implementation

YesIn this paper, a new fractional order chaotic system without equilibrium is proposed, analyti-cally and numerically investigated, and numerically and experimentally tested. The analytical and numerical investigation were used to describe the system dynamical behaviors including, the system equilibria, the chaotic attractors, the bifurcation diagrams and the Lyapunov expo-nents. Based on the obtained dynamical behaviors, the system can excite hidden chaotic attrac-tors since it has no equilibrium. Then, a synchronization mechanism based on the adaptive con-trol theory has been developed between two identical new systems (master and slave). The adaptive control laws are derived based on synchronization error dynamics of the state varia-bles for the master and slave. Consequently, the update laws of the slave parameters are ob-tained, where the slave parameters are assumed to be uncertain and estimate corresponding to the master parameters by the synchronization process. Furthermore, Arduino Due boards were used to implement the proposed system in order to demonstrate its practicality in real-world applications. The simulation experimental results are obtained by MATLAB and the Arduino Due boards respectively, where a good consistent between the simulation results and the ex-perimental results. indicating that the new fractional order chaotic system is capable of being employed in real-world applications

A new three-dimensional chaotic system without equilibrium points, its dynamical analyses and electronic circuit application

U radu se predstavlja i analizira novi trodimenzionalni kaotični sustav bez točaka ekvilibrija. Osnovna dinamička analiza tog novog kaotičnog sustava bez točaka ekvilibrija izvodi se pomoću sustava ekvilibrija, faznih slika (portreta), osjetljivosti na početne uvjete, fraktalne dimenzije i kaotičnog ponašanja. Uz to je izvedena analiza spektra Lyapunovljevih eksponenata i bifurkacijska analiza predloženog kaotičnog sustava primjenom izabranih parametara. Kaotični sustav bez točaka ekvilibrija dobiven je detaljnom teorijskom analizom kao i simulacijama s dizajniranim elektroničkim krugom. Sustav kaosa bez točaka ekvilibrija također je poznat kao sustav kaosa sa skrivenim atraktorom i o tome postoji mali broj istraživanja. Budući da ne postoje homokliničke i heterokliničke orbite, Schilnikova metoda se ne može primijeniti kako bi se ustanovilo je li sustav kaotičan ili nije kaotičan. Stoga kaotični sustav bez točaka ekvilibrija može biti od koristi u mnogim tehničkim primjenama, naročito u kriptologiji i kodiranju zasnovanom na kaosu. Nadalje, ovaj predloženi kaotični sustav bez točaka ekvilibrija može se ponašati na mnogo dinamički nepoznatih načina. Takve vrste ponašanja nepoznatih kaotičnih atraktora zahtijevaju dodatna istraživanja.In this paper, a new three-dimensional chaotic system without equilibrium points is introduced and analysed. Basic dynamical analysis of this new chaotic system without equilibrium points is carried out by means of system equilibria, phase portraits, sensitivity to initial conditions, fractal dimension and chaotic behaviours. In addition, in this paper Lyapunov exponents spectrum and bifurcation analysis of the proposed chaotic system have been executed by means of selected parameters. The chaotic system without equilibrium points has been executed by detailed theoretical analysis as well as simulations with designed electronical circuit. A chaotic system without equilibrium points is also known as chaotic system with hidden attractor and there are very few researches in the literature. Since they cannot have homoclinic and heteroclinic orbits, Shilnikov method cannot be applied to find whether the system is chaotic or not. Therefore, it can be useful in many engineering applications, especially in chaos based cryptology and coding information. Furthermore, introduced chaotic system without equilibrium points in this paper can have many unknown dynamical behaviours. These behaviours of the strange chaotic attractors deserve further investigation

A Novel Encryption Method for Dorsal Hand Vein Images on a Microcomputer

In this paper, a Lorenz-like chaotic system was developed to encrypt the dorsal hand patterns on a microcomputer. First, the dorsal hand vein images were taken from the subjects via an infrared camera. These were subjected to two different processes called contrast enhancement and segmentation of vein regions. Second, the pre- and post-processed images were encrypted with a new encryption algorithm in the microcomputer environment. For the encryption process, random numbers were generated by the chaotic system. These random numbers were subjected to NIST-800-22 test which is the most widely accepted statistical test suite. The speeded up robust feature (SURF) matching algorithm was utilized in the initial condition sensitivity analysis of the encrypted images. The results of the analysis have shown that the proposed encryption algorithm can be used in identification and verification systems. The encrypted images were analyzed with histogram, correlation, entropy, pixel change rate (NPCR), initial condition sensitivity, data loss, and noise attacks which are frequently used for security analyses in the literature. In addition, the images were analyzed after noise attacks by means of peak signal-to-noise ratio (PSNR), mean square error (MSE), and the structural similarity index (SSIM) tests. It has been shown that the dorsal hand vein images can be used in identification systems safely with the help of the proposed method on microcomputers.This work was supported by the Qatar National-LibraryScopu

A Numerical Study of the Complex Lorenz System as a Dynamical Model

A nonlinear dynamical system is a mathematical model for a portion of the physical world where continuous components are interacting with each other. Such systems are complex, and it is difficult to predict how they will react towards changing the driving parameters and initial conditions. \\ This dissertation is concerned with the numerical aspect of controlling the Lorenz system that studies the production of the magnetic field in sunspots. The study aims to do this by applying different approaches to the system