基于Melnikov方法的分数阶Duffing振子混沌阈值解析研究

Analytical study on the chaos threshold of a Duffing oscillator with a fractional-order derivative term by the Melnikov method

研究了含有分数阶微分项的Duffing振子的分岔与混沌行为,利用等效刚度和等效阻尼的概念对分数阶微分项进行处理,将分数阶微分项等效成三角函数与指数函数的形式,用Melnikov方法分析了分数阶Duffing振子产生分岔与混沌的必要条件,得到了其解析结果。进行了解析解和数值解的比较,证明了解析结果的精确度,并通过仿真计算研究了分数阶的阶次和系数对系统产生混沌必要条件的影响。在数值模拟过程中,还发现分数阶Duffing振子中存在双稳态特性,从两个稳态解出发,随着外激励参数的变化都能通过倍周期分岔到达混沌的状态。通过分析系统的动力学响应验证了这一现象。

The bifurcations and chaos of a Duffing oscillator with a fractional-order derivative term was studied.The equivalent stiffness and equivalent damping were used to deal with the fractional-order derivative,where the derivative term was made equivalent to a term in the form of trigonometric function and exponential function.Then,the Melnikov method was used to analyze the necessary conditions for the bifurcation and chaos generation of the fractional-order Duffing oscillator.The approximate analytical solution of the fractional-order Duffing oscillator was obtained.Finally,the comparison between the analytical solution and the numerical solution was investigated,and the accuracy of the analytical result was proved.The influences of fractional order and coefficient of the fractional-order derivative on the necessary condition of chaos were studied by simulation.Additionally,it is found that there is a bistability characteristic in the fractional-order Duffing oscillator.Starting from the two steady-state solutions,the system can reach the chaos state through the period-doubling bifurcation with the change of external excitation parameter f,which was then confirmed by analyzing its dynamic response.