计算非线性参数系统分岔的自适应同伦方法
An adaptive homotopy method for computing bifurcations of nonlinear parametric systems
摘要：
本文提出了一种计算非线性系统歧点的自适应步长同伦跟踪方法。该方法包括4个部分：1）在歧点附近发展自适应跟踪技术; 2）在自适应跟踪失败时，采用膨胀技术; 3）利用Pu I-x级数插值计算歧点; 4）对歧点的切锥结构进行数值逼近，以计算不同分支上的解.各种数值例子的非线性系统，以说明这种新的方法的效率。这种新的自适应同伦跟踪方法也适用于一个系统的非线性偏微分方程和大规模非线性离散系统的鲁棒性和效率。
英文摘要：
In this paper, we present an adaptive step-size homotopy tracking method for computing bifurcation points of nonlinear systems. There are four components in this new method: 1) an adaptive tracking technique is developed near bifurcation points; 2) an inflation technique is backed up when the adaptive tracking fails; 3) Puiseux series interpolation is used to compute bifurcation points; and 4)the tangent cone structure of the bifurcation point is approximated numerically to compute solutions on different branches. Various numerical examples of nonlinear systems are given to illustrate the efficiency of this new approach. This new adaptive homotopy tracking method is also applied to a system of nonlinear PDEs and shows robustness and efficiency for large-scale nonlinear discretized systems.
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