A Method of Following the Unstable Path Between Two Saddle-Node Bifurcation Points in Nonlinear Dynamic System
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摘要: 采用将伪弧长延拓法与Poincaré映射法相结合的方法,确定非自治动力系统中两鞍-结分岔点间非稳定曲线,并对采用一般延拓法时出现的奇异性进行了证明。该方法引入了一正则化方程,避免了在求解过程中出现的奇异问题,并给出了相应的迭代格式。在曲线的延拓过程中,由于存在两个延拓方向,为保证将曲线延拓出来,给出了一种确定切线方向的方法,该方法在分析非线性振动系统中的双稳态现象等问题是很有效的。Abstract: A computation algorithm based on the Poincar Mapping in combination with Pseudo-Arc Length Continuation Method is presented for calculating the unstable response with saddle-node bifurcation,and the singularity,which occurs using the general continuation method combined with Poincar Mapping to follow the path,is also proved. A normalization equation can be introduced to avoid the singularity in the process of iteration, and a new iteration algorithm will be presented too.There will be two directions in which the path can be continued at each point,but only one can be used,the method of determining the direction will be presented in the paper.It can be concluded that this method is effective in analysis of non-linear dynamic system with saddle-node bifurcations.
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Key words:
- nonlinear dynamic system /
- bifurcation /
- stability
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