Fractional-order Chua's System Research Articles

Hidden attractors in a new fractional-order Chua system with arctan nonlinearity and its DSP implementation

This paper reports a new fractional-order Chua’s system with arctan function and an algorithm for determining the initial value of fractional-order hidden attractors. Firstly, dynamics of the system is investigated by employing the stability analysis, dissipativeness, phase diagrams, 0-1 test, bifurcation diagram and SampEn complexity measure. It shows rich dynamics in the proposed system. Secondly, using the proposed algorithm, the initial values of the new fractional-order Chua’s system are calculated. And according to two sets of initial conditions, hidden attractors are found and verified by the numerical simulation. Meanwhile, the basin attraction is analyzed to confirm that the hidden attractor is indeed not near the fixed point. Thirdly, the new fractional-order Chua’s system is implemented on the DSP platform, and the arctan function’s processing algorithm is proposed. The DSP experimental results show that the system can produce a pair of coexisting fractional-order hidden attractors, and the experimental results are consistent with the numerical simulation results. It verifies the effectiveness of the presented methods and shows the potential engineering application value of the proposed Chua’s system.

Generating Chaotic Series via Encryption Method in Fractional-Order Chua Systems

Fractional-order Chua systems have drawn wide attention in view of rich dynamic characteristics, e.g., chaos and hyperchaos. How to generate chaotic series via encryption method in fractional-order Chua systems is a difficult point all the while. In this paper, we derive several encryption methods in fractional-order Chua systems to generate chaotic series. As one of the most direct application, image encryption is also discussed by the presented methods.

Tracking the chaotic behaviour of fractional-order Chua’s system by Mexican hat wavelet-based artificial neural network

In this paper, a process is devised systematically to scrutinize the scrolling chaotic behaviour of fractional-order Chua's system. The process is composed of fractional Laplace transformation, artificial neural network with Mexican hat wavelet as an activation function and simulated annealing. Sequentially, the parametric expansion of fractional Laplace transform is employed to convert the governing fractional system into an ordinary differential system. Next, artificial neural network and simulated annealing approximate and optimize the attained system and produce accurate solutions. The predictability and elaboration of double scrolling chaotic structures of fractional-order Chua's system are also studied using the Lyapunov exponent and fifth–fourth Runge–Kutta method. Moreover, the mean absolute error and root mean square error are measured for the convergence analysis of the proposed scheme. On the whole, the accurate approximate solutions, the phase plots of Lyapunov exponent spectrum and bifurcation maps of the dynamical evolution of fractional Chua's system are a triumph of this endeavour.

Chaos in Fractional Order Cubic Chua System and Synchronization

This paper deals with fractional order Chua system with cubic nonlinearity (cubic Chua system), which is a modification of Chua system. The aim is, first, to study the chaotic behavior in fractional order cubic Chua system. We found that chaos indeed exists in the fractional version of this system. The necessary condition for exhibiting chaotic attractors similar to its integer order counterpart is presented. This condition is used to distinguish for which parameters and orders the system generates double scroll chaotic attractors. The synchronization problem between two coupled chaotic fractional order cubic Chua systems is then addressed. An adaptive feedback control scheme for the synchronization with suitable feedback nonlinear controller applied to the response system is presented. Numerical simulations are performed to verify the theoretical analysis.

Non-inductive modular circuit of dislocated synchronization of fractional-order Chua's system and its application

In this paper, from the stability theory of fractional-order chaotic system, a kind of dislocated projective synchronization for fractional-order Chua's system is successfully completed through a nonlinear controller. Meanwhile, the fractional-order unit circuit is designed, according to the series-parallel structure of resistor-capacitor and the approximate linear transfer function expression for the complex frequency domain. Thus, non-inductive modular circuit of dislocated projective synchronization of fractional-order Chua's system is realized. The circuit simulation results prove the feasibility of the scheme. Furthermore, the method can be applied in secure communication through the improved chaotic masking. The information signal can be concealed and recovered. Numerical simulation results show the effectiveness of the proposed method.

Chaos and synchronization of the fractional-order Chua’s system

Chaotic synchronization of fractional-order Chua’s system is further studied. An algorithm for numerical solution of fractional-order differential equations is presented; the chaos in a fractional-order Chua system with some parameters is discussed. The scheme of synchronization system consist of fractional-order Chua’s system is constructed. The synchronization conditions are investigated theoretically. And the synchronization thresholds are discussed by utilizing bifurcation graphs.

Chaos in a fractional order Chua's system

This brief studies the effects of fractional dynamics in chaotic systems. In particular, Chua's system is modified to include fractional order elements. By varying the total system order incrementally from 3.6 to 3.7, it is demonstrated that systems of "order" less than three can exhibit chaos as well as other nonlinear behavior. This effectively forces a clarification of the definition of order which can no longer be considered only by the total number of differentiations or by the highest power of the Laplace variable. >