非线性动力系统在分岔点处的临界慢化行为  

Critical-slowing-down of nonlinear systems at bifurcation points

作　　者：张杰[1] 杜瑶 李晴 于一真 李海红[2] 王新刚 ZHANG Jie;DU Yao;LI Qing;YU Yi-zhen;LI Hai-hong;WANG Xin-gang(School of Physics and Information Technology,Shaanxi Normal University,Xian,Shaanxi 716001,China;School of Science,Beijing University of Posts and Telecommunications,Beijing 100876,China)

机构地区：[1]陕西师范大学物理学与信息技术学院,陕西西安710100 [2]北京邮电大学理学院,北京100876

出　　处：《大学物理》2023年第5期9-17,65,共10页College Physics

基　　金：陕西师范大学课堂教学模式创新研究项目(21KT-JG02)资助;高校教改阵线课程教学研究项目(JZW-21-JW-02)资助。

摘　　要：Lyapunov指数是定量描述非线性动力系统轨道稳定性的主要方法之一,同时也是分析系统分岔行为的常用手段.实际应用中,人们通常只关心Lyapunov指数的正负,并以此来判断系统轨道是否稳定,而对于Lyapunov指数为零,即动力学分岔点处系统的行为特征讨论甚少.本文以几类经典的非线性动力系统为例,针对系统在分岔点处的轨道稳定性进行理论和数值分析.研究发现,不同系统在分岔点处其微扰后的轨道均以幂律,而非指数的形式收敛,呈现出经典物理系统在相变临界点处的慢化行为.通过理论分析,我们解析得到分岔点处计算临界指数的一般公式,并通过数值模拟对理论公式的准确性进行了验证.临界慢化是物理系统在相变点处的普遍现象,文中关于非线性系统在分岔点处临界慢化行为的发现将加深人们对于动力学分岔本质的认识,同时也是对现有教材中关于Lyapunov指数相关知识的有益补充.As a key concept in nonlinear science,Lyapunov exponent has been extensively studied in literature.In general,to analyze the stability of a dynamical state,it is the sign of the Lyapunov exponent that is interested.Yet,this approach fails when the state is located exactly at the bifurcation point,where the Lyapunov exponent is equal to zero,suggesting that the perturbations will be keeping unchanged during the system evolution at this point.This prediction,however,is conflicting with the true results,as attractors also exist at the bifurcation points.Here we are able to argue mathematically and demonstrate numerically that at the bifurcation points the perturbation is damped by a power-law scaling to the equilibrium,with the scaling exponent determined by the nonlinear term of the variational equation of the perturbations.That is,the dynamics of the system shows the typical feature of critical slowing down observed at the critical point of many physical systems.The phenomenon of criticalslowing-down is demonstrated in different nonlinear models,and the theoretical predictions are in good agreement with the numerical results in all cases studied.

关 键 词：LYAPUNOV指数 分岔 临界慢化 非线性动力系统 混沌 

分 类 号：O4-1[理学—物理]