分数阶Lorenz系统的分析及电路实现

Analysis and circuit implementation for the fractional-order Lorenz system

频域传递函数近似方法不仅是常用的分数阶混沌系统相轨迹的数值分析方法之一,而且也是设计分数阶混沌系统电路的主要方法.应用该方法首先研究了分数阶Lorenz系统的混沌特性,通过对Lyapunov指数图、分岔图和数值仿真分析,发现了其较为丰富的动态特性,即当分数阶次从0.7到0.9以步长0.1变化时,该分数阶Lorenz系统既存在混沌特性,又存在周期特性,从数值分析上说明了在更低维的Lorenz系统中存在着混沌现象.然后又基于该方法和整数阶混沌电路的设计方法,设计了一个模拟电路实现了该分数阶Lorenz系统,电路中的电阻和电容等数值是由系统参数和频域传递函数近似确定的.通过示波器观测到了该分数阶Lorenz系统的混沌吸引子和周期吸引子的相轨迹图,这些电路实验结果与数值仿真分析是一致的,进一步从物理实现上说明了其混沌特性.

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.