The construction of homoclinic and heteroclinic orbit in two-dimentsional nonlinear systems based on the quasi-Pade approximation
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摘要: 提出一种求解强非线性系统同异宿轨道解析解的改进类Pad\'{e}逼近方法, 该方法在分析带有扰动参数的系统时无需预先限定参数的取值范围. 首先研究了具有三次非线性项的系统,分析其产生同宿或异宿轨道时参数的取值范围, 分别提出直接体现参数的同宿及异宿解的设解通式, 据此获得了一类强非线性下的自治系统方程的同宿及异宿解. 其次, 对于非自治系统, 研究了具有三次非线性项系统的强迫振动, 直接考虑扰动参数对整个系统的影响, 得到了满足同(异)宿边界条件的周期解. 最后, 构造了两种不同形式的异宿解, 从而减少了保守系统异宿解的计算量. 借助数值模拟验证了该方法的有效性及精确性.
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关键词:
- 强非线性
Abstract: The conventional quasi-Pad\'{e} approximants aredeveloped to study the homoclinic and heteroclinic solutions in nonlineardynamic system, and in the solution process in which the disturbanceparameters of system don't be restricted in advance. Firstly, the systemwith cubic nonlinear oscillators is considered. The value ranges of itsparameters can be determined when the homoclinic and heteroclinic orbits areoccurred. Respectively suppose the general formulations homoclinic andheteroclinic solutions which reflect the parameters of the systemdirectly.Meanwhile, the homoclinic and heteroclinic solutions ofthe strongly nonlinear autonomous system are derived successfully. Secondly,the periodic solutions of the non-autonomous system are derived under thedirect consideration of the disturbance parameters, which are satisfied theconditions of the homoclinic and heteroclinic solutions. Finally, twoheteroclinic solutions functions are constructed in order to reduce thecomputational complexity. The validity and accuracy of the quasi-Pad\'{e}approximants are proved by comparing with the numerical computation.-
Key words:
- strongly nonlinear oscillation
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