两类受迫Duffing方程混沌解的图形化比较

Compare and Analysis of the Chaotic Solutions of the Two Types of Duffings Equations

Duffing方程是常微分方程理论研究中比较有代表性的非线性系统之一。笔者通过运用数值计算可视化和解析分析方法,对两类受迫Duffing方程的混沌解进行图形化研究,揭示了Duffing系统在不同强弱激励下混沌表现的不同:强激励下系统的混沌吸引子在Poincaré映射图上具有明显自相似的分形结构,而弱激励下则呈现出模糊状态。并进一步对混沌吸引子结构形态的差异进行了比较和分析,利用耗散结构理论对产生差异的原因进行了合理解释。

The Duffings equations are the representative non-linear systems in the study of the ordinary differential equations. In this paper, the chaotic solutions of two types of the equations are studied using visualization and analytic method. The differences, which is obvious in the Poincaré mappings, between the two chaotic solutions are presented: the attractors in systems forced by high exciters are self-similar and fractal, but the ones in systems forced by weak exciters are blurry and non-structural. The analysis and explanation about the difference are given in accordance to the theory of dissipation structure.