A Posteriori Verification of Invariant Objects of Evolution Equations: Periodic Orbits in the Kuramoto-Sivashinsky PDE
A Posteriori Verification of Invariant Objects of Evolution Equations: Periodic Orbits in the Kuramoto-Sivashinsky PDE
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演化方程不变对象的后验验证：Kuramoto-Sivashinsky PDE 中的周期轨道
DOI：
10.1137/16m1073789
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
J. Lessard
中科院分区：
文献类型：
--
作者：
Marcio Gameiro;J. Lessard
本文提出了一种用严格数值计算Kuramoto- Sivashinsky PDE周期轨道的方法。这是[J.-L.]中介绍的理论方法的应用和实现。Figueras， M. Gameiro, j . p .。Lessard和R. de la Llave，“演化方程不变对象的数值计算框架和后验验证”，SIAM J. appll。Dyn系统（出现）。利用牛顿—kantorovich型论证（半径多项式方法），在加权$\ell^\infty$的傅里叶系数的Banach空间中得到了解的存在性。一旦证明了周期轨道，就求解了相关的特征值问题，并严格计算了Floquet指数，从而证明了一些周期轨道是不稳定的。最后，提出了一种预测校正延拓方法来严格计算周期轨道的全局光滑分支。一种替代方法和独立实现[J.-L.]Figueras， M. Gameiro, j . p .。更少……
In this paper, a method for computing periodic orbits of the Kuramoto--Sivashinsky PDE via rigorous numerics is presented. This is an application and an implementation of the theoretical method introduced in [J.-L. Figueras, M. Gameiro, J.-P. Lessard, and R. de la Llave, “A framework for the numerical computation and a posteriori verification of invariant objects of evolution equations,” SIAM J. Appl. Dyn. Syst., to appear]. Using a Newton--Kantorovich-type argument (the radii polynomial approach), existence of solutions is obtained in a weighted $\ell^\infty$ Banach space of Fourier coefficients. Once a proof of a periodic orbit is done, an associated eigenvalue problem is solved and Floquet exponents are rigorously computed, yielding proofs that some periodic orbits are unstable. Finally, a predictor-corrector continuation method is introduced to rigorously compute global smooth branches of periodic orbits. An alternative approach and independent implementation of [J.-L. Figueras, M. Gameiro, J.-P. Less...