ON THE NUMERICAL CONTINUATION OF ISOLAS OF EQUILIBRIA
ON THE NUMERICAL CONTINUATION OF ISOLAS OF EQUILIBRIA
复制标题
DOI：
10.1142/s021812741250277x
复制
发表时间：
2012-11-01
影响因子：
2.2
通讯作者：
Rodrigues, S.
中科院分区：
文献类型：
--
作者：
Avitabile, D.;Desroches, M.;Rodrigues, S.
我们提出了一种计算微分方程平衡解的单参数孤立族的数值策略。孤立体是封闭在参数空间中的解分支。由于单个孤立体至少包含一个不稳定段，因此需要进行数值延拓。我们展示了如何使用伪弧长预测校正方案，以便在参数空间中跟踪整个孤立体，作为一个单独的对象，通过提出一个合适的代数问题。我们继续在二维动力系统中，即所谓的连续搅拌槽反应器模型，以及在与等离子体物理相关的三维模型中进行平衡。然后，我们构建了一个玩具模型，并跟随一组孤立体经过折叠，并说明如何在靠近形成中心的地方启动计算，使用受范德波尔方程启发的模型中的近似椭圆。我们还展示了如何在孤立体的离散化中引入节点自适应，以便集中于曲率较高区域的节点。最后，我们评论了将所提出的数值策略推广到周期轨道孤立体的情况。
We present a numerical strategy to compute one-parameter families of isolas of equilibrium solutions in ODEs. Isolas are solution branches closed in parameter space. Numerical continuation is required to compute one single isola since it contains at least one unstable segment. We show how to use pseudo-arclength predictor-corrector schemes in order to follow an entire isola in parameter space, as an individual object, by posing a suitable algebraic problem. We continue isolas of equilibria in a two-dimensional dynamical system, the so-called continuous stirred tank reactor model, and also in a three-dimensional model related to plasma physics. We then construct a toy model and follow a family of isolas past a fold and illustrate how to initiate the computation close to a formation center, using approximate ellipses in a model inspired by the van der Pol equation. We also show how to introduce node adaptivity in the discretization of the isola, so as to concentrate on nodes in region with higher curvature. We conclude by commenting on the extension of the proposed numerical strategy to the case of isolas of periodic orbits.