Computation of Limit Cycles and Their Isochrons: Fast Algorithms and Their Convergence
Computation of Limit Cycles and Their Isochrons: Fast Algorithms and Their Convergence
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极限环及其等时线的计算：快速算法及其收敛性
DOI：
10.1137/120901210
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
R. Llave
中科院分区：
文献类型：
--
作者：
G. Huguet;R. Llave
我们提出了有效的算法来计算极限环及其等时线（即，具有相同渐近相位的点的集合）。我们制定了一个功能方程的参数化不变的周期和等时线，我们表明，它可以通过牛顿法来解决。利用正确的变换，我们可以有效地求解牛顿步方程。算法是有效的，在这个意义上，如果我们离散的功能使用$N$点，牛顿步骤需要$O（N）$存储和$O（N\log N）$操作傅立叶离散或$O（N）$操作在其他离散。我们证明了算法的收敛性，并提出了一个验证定理的后验格式。也就是说，我们证明了，如果存在一个近似解的不变性方程，满足一些温和的非退化条件，那么有一个真正的解决方案附近。因此，我们的主要定理可以用来验证数值计算的解决方案。这个定理...
We present efficient algorithms to compute limit cycles and their isochrons (i.e., the sets of points with the same asymptotic phase) for planar vector fields. We formulate a functional equation for the parameterization of the invariant cycle and its isochrons, and we show that it can be solved by means of a Newton method. Using the right transformations, we can solve the equation of the Newton step efficiently. The algorithms are efficient in the sense that if we discretize the functions using $N$ points, a Newton step requires $O(N)$ storage and $O(N\log N)$ operations in Fourier discretization or $O(N)$ operations in other discretizations. We prove convergence of the algorithms and present a validation theorem in an a posteriori format. That is, we show that if there is an approximate solution of the invariance equation that satisfies some some mild nondegeneracy conditions, then there is a true solution nearby. Thus, our main theorem can be used to validate numerically computed solutions. The theorem ...