Analysis and circuit implementation for the fractional-order Lorenz system

Abstract

Transfer function approximation in frequency domain is not only one of common numerical analysis methods studying portraits of fractional-order chaotic systems, but also a main method to design their chaotic circuits. According to it, in this paper we first investigate the chaotic characteristics of the fractional-order Lorenz system, find some more complex dynamics by analyzing Lyapunov exponents diagrams, bifurcation diagrams and phase portraits, that is, we display the chaotic characteristics as well as periodic characteristics of the system when changing fractional-order from 0.7 to 0.9 in steps of 0.1, and show that the chaotic motion exists in the a lower-dimensional fractional-order Lorenz system. Then, according to transfer function approximation and the approach to designing integer-order chaotic circuits, we also design an analog circuit to implement the fractional-order system. The resistors and capacitors in the circuit are selected according to the system parameters and transfer function approximation in frequency domain. Some phase portraits including chaotic attractors and periodic attractors are observed by oscilloscope, which are coincident well with numerical simulations, and the chaotic characteristics of the fractional-order Lorenz system are further proved by the physical implementation.