扰动sine-Gordon方程的混沌动力学

为了研究具有阻尼和强迫激励的截断sine-Gordon方程的全局分叉和混沌动力学,使用广义渐近惯性流形方法获得了二模态截断sine-Gordon方程.然后利用多尺度法获得了主共振和1:1内共振情况下截断sine-Gordon方程的平均方程.利用规范形理论和Maple符号程序,获得了一对双零特征值和一对纯虚特征值情况下平均方程的规范形.使用全局摄动法分析了二模态截断sine-Gordon方程的全局分叉和混沌动力学,数值模拟同样发现在二模态截断sine-Gordon方程中存在着混沌运动.

Study on Chaotic Dynamics of Perturbed sine-Gordon Equation

In order to study the global bifurcations and chaotic dynamics of the truncated sine-Gordon equation with damping and forced exciting, the two-mode truncated sine-Gordon equation is obtained by using the generalized asymptotic inertial manifold. Then, the method of multiple scales is used to find the averaged equations of the two-mode sine-Gordon equation in the case of the primary resonance and 1: 1 internal resonance. With the aid of theory of normal form, the explicit expression of normal form for the averaged equations associated with a pair of double-zero eigenvalues and a pair of pure imaginary eigenvalues is given by using Maple program. Finally, the global perturbation techniques are applied to this normal form to analyze the global bifurcations and chaotic dynamics. The chaotic motion of the two-mode sine-Gordon equation is also found by the numerical simulation.