小波变换在浑沌研究中的应用

Application of wavelet transform to chaos study

利用D.E.Newland提出的谐波小波变换来识别浑沌.鉴于任何非线性振动系统,其解最多有3种不同形式,即特种形式的周期响应、拟周期响应和浑沌响应,将小波变换和Poincare映射结合起来,用Poincare映射来确定周期及周期数,用小波变换来区分拟周期响应和浑沌响应,从而对系统运动的特种形式进行准确判断;此外,用这种方法分析了参数空间中对应于特种形式解的存在域,揭示了非线性振动系统的响应特性.该方法可用于对初值空间及吸引域进行分析.

The response of a nonlinear vibration system may be of three types, namely,periodic, quasiperiodic or chaotic, when the parameters of the system are changed. The periodic motions can be identified by Poincare map, and harmonic wavelet transform (HWT) can distinguish quasiperiod from chaos, so the existing domains of different types of motions of the system can be revealed in the parametric space with the method of HWT joining with Poincare map. In this paper, by using the HWT suggested by D.E.Newland to identify chaotic motion, and considering Poincare map at the same time, an effective method is introduced to analyze the existing domains of different types of motions in the parametric space of a nonlinear system.