Narrow Results

In this paper, a novel four-dimensional fractional-order chaotic system is proposed, and its dynamic characteristics are analyzed by equilibrium points, phase diagram, Lyapunov exponent spectrum and bifurcation diagram. Then, the electronic circuit of the chaotic system is designed and implemented by PSpice. The results of numerical simulation are in good agreement with the analog circuit simulation. In addition, the four-dimensional fractional-order system is implemented with field programmable gate array (FPGA). Finally, the synchronization of two four-dimensional chaotic systems with different initial values is realized by finite time control method, and it is also realized by FPGA. The design of fractional-order system based on hardware concerned in this paper will provide a certain theoretical basis for its application in the fields of secure communication, image encryption and so on.

In this paper, we present a concise four-dimensional (4D) conservative chaotic system with a wide parameter range. Since there are no terms higher than first order, the circuit does not contain multipliers, resulting in a simple circuit implementation. The nonlinear dynamic characteristics, such as phase diagrams, equilibrium points, divergence, Poincaré cross-sections, Lyapunov exponents, bifurcation diagrams, and Lyapunov dimension, are analyzed in detail, which illustrates the conservativity. Besides, the system exhibits different offset boosting behaviors. Through offset boosting, the system can propagate along a line, convert signal polarity, control variable amplitude, generate coexisting attractors, and even induce changes in its state. Specially, we realize the transition from a single-vortex attractor to a multivortex one by some changes in the initial values. Furthermore, the Pearson correlation coefficient is used to demonstrate the higher initial value sensitivity of the system. Finally, the system is implemented through Multisim simulation and analog circuit separately, and their consistency validates the system effectively.

This study employs the Caputo–Fabrizio and Atangana–Baleanu fractional derivatives to determine the impedance and admittance model of fractional capacitor and inductor. The analog implementation circuits are proposed aiming at fractional-order electric elements based on these two derivatives, which can be widely used in a variety of electrical systems using new fractional operators. Constant phase capacitor and inductor are approximated by the Oustaloup algorithm and the recursive net-grid-type analog circuit, respectively. Based on that, approximation circuits of fractional electric components under Caputo–Fabrizio and Atangana–Baleanu definitions are given. For the purpose of judging whether the implementation topology of fractional-order capacitor and inductor is accurate, taking fractional RC and RL circuit defined by Caputo, Caputo–Fabrizio and Atangana–Baleanu derivatives as examples, the comparison of numerical and circuit simulations is carried out. The correctness of the analog implementation circuits using the Caputo–Fabrizio and Atangana–Baleanu definitions is verified. Fractional-order RC charging circuit experiments based on Caputo, Caputo–Fabrizio and Atangana–Baleanu derivatives are taken as examples. Several experiments with different fractional-order and circuit parameters are carried out. The validity of the implementation methods is ulteriorly proved with experiment data.

Many electrical systems can be characterized more authentically by fractional-order dynamic systems. The Atangana–Baleanu and the Caputo–Fabrizio fractional derivatives have solved the singularity problem in Caputo derivative. This work uses Atangana–Baleanu and Caputo–Fabrizio fractional derivatives to model the fractional-order mutual inductance in the frequency domain. To use the fractional mutual inductance in circuit design, the T-model equivalent circuits are presented with different fractional derivatives. The fractional impedance matching networks based on proposed fractional mutual coupling circuits are simulated as an application. The impedance characteristics of networks with different fractional orders are analyzed. The results indicate that the proposed fractional mutual coupling circuits based on Atangana–Baleanu and Caputo–Fabrizio fractional derivatives can be applied to the complex electrical systems to increase the design degree of freedom, which provides more choices for describing the nonlinear characteristics of the system.