一种非线性动力学系统混沌现象的深入研究

In-depth study of chaos in a nonlinear dynamical system

本文在实验教学中引入一种非线性混沌摆系统,通过调节混沌摆的驱动力周期演示了该非线性动力学系统出现混沌现象的过程,从而让学生了解混沌现象的参数敏感性、相图特点、频谱特性等基本特性.为了进一步了解该混沌摆的特性,本文建立了该非线性摆系统的简化动力学方程,在数值上对其进行了研究.基于动力学方程的数值模拟,克服了实验上相关参数定量改变困难、摆动稳定性不易控制、实验时间周期长等问题.在数值模拟中,通过改变不同参数得到了相图、频谱图以及分岔图,比较深入详细地对这种混沌摆的相关特性进行了描述,也有利于学生加深对混沌摆的理解.

A kind of nonlinear chaotic pendulum system is introduced into the experiment teaching.By adjusting the period of the driving force of the chaotic pendulum,the process of chaotic phenomena appearing in the nonlinear dynamic system is demonstrated so that students can understand the basic characteristics of chaotic phenomena,such as parameter sensitivity,phase diagram characteristics,spectrum characteristics and so on.In order to further understand the characteristics of the chaotic pendulum,a simplified dynamic equation of the nonlinear pendulum system is established and studied numerically.The numerical simulation based on dynamic equation overcame the difficulties of quantitative change of experimental parameters,difficult control of swing stability and long experimental period.In the numerical simulation,the related characteristics of the chaotic pendulum is described in detail by changing the phase diagram,spectrum diagram and bifurcation diagram obtained by different parameters,which is also helpful for students to deepen their understanding of the chaotic pendulum.