基于分形的双涡卷系统特性分析与电路实现

Characteristic Analysis and Circuit Implementation of Double-Scroll System Based on Fractal

摘要

为获得形态复杂的涡卷混沌系统,将具有涡卷结构的Julia分形式应用到变形Liu系统中构建新系统,对其相图、平衡点、Lyapunov指数谱和分岔图等主要特性进行分析,在Multisim平台设计系统的乘方和分式单元电路,并用单片机完成对新系统的数字电路实现.结果表明:新系统产生形态更丰富的四翼双涡卷吸引子;引入二次项系数k未改变吸引子形态,仅对吸引子的尺寸产生影响;对应同一参数a,改变系数k,最大Lyapunov指数值均保持不变,即系数k不影响系统的混沌状态;采用模块化设计可使电路更清晰、参数调节更方便.电路仿真和数字电路实现结果均与MATLAB数值计算结果相符,从而验证了分形和混沌相结合的可行性.

Abstract

In order to obtain a complex scroll chaotic system, a new system was constructed by applying the Julia fractal with a scroll structure to the deformed Liu system. The main characteristics such as phase diagram, equilibrium point, Lyapunov exponent spectrum and bifurcation diagram were analyzed. The power and fractional unit circuits of the system were designed on the Multisim platform, and the digital circuit of the new system was implemented using a microcontroller. The results show that the new system produces a more diverse form of four-winged double-scroll attractor, and the introduction of the coefficient k of the quadratic term does not change the shape of the attractor, only affecting the size of the attractor. Corresponding to the same parameter a, changing the coefficient k, the value of the maximum Lyapunov exponent remains unchanged, the coefficient k does not affect chaotic state of the system. Adopting modular circuit design can make the circuit more clear and the parameter adjustment more convenient. The results of the circuit simulation and the digital circuit implementation are consistent with the numerical calculation results performed in MATLAB, thus validating the feasibility of the combination of fractal and chaos.

关键词

Julia分形 / 双涡卷混沌系统 / 双参数 / 电路仿真 / 单片机

Key words

Julia fractal / double-scroll chaotic system / two-parameter / circuit simulation / microcontroller

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基金资助

河北省自然科学基金面上项目(A2022106001)

河北省高等教育教学改革研究与实践项目(2022GJJG543)