We present a method of detecting bifurcations by locating zeros of a signed version of the smallest singular value of the Jacobian. This enables the use of quadratically convergent root-bracketing techniques or Chebyshev interpolation to locate bifurcation points. Only positive singular values have to be computed, though the method relies on the existence of an analytic or smooth singular value decomposition (SVD). The sign of the determinant of the Jacobian, computed as part of the bidiagonal reduction in the SVD algorithm, eliminates slope discontinuities at the zeros of the smallest singular value. We use the method to search for spatially quasi-periodic traveling water waves that bifurcate from large-amplitude periodic waves. The water wave equations are formulated in a conformal mapping framework to facilitate the computation of the quasi-periodic Dirichlet-Neumann operator. We find examples of pure gravity waves with zero surface tension and overhanging gravity-capillary waves. In both cases, the waves have two spatial quasi-periods whose ratio is irrational. We follow the secondary branches via numerical continuation beyond the realm of linearization about solutions on the primary branch to obtain traveling water waves that extend over the real line with no two crests or troughs of exactly the same shape. The pure gravity wave problem is of relevance to ocean waves, where capillary effects can be neglected. Such waves can only exist through secondary bifurcation as they do not persist to zero amplitude. The gravity-capillary wave problem demonstrates the effectiveness of using the signed smallest singular value as a test function for multi-parameter bifurcation problems. This test function becomes mesh independent once the mesh is fine enough.
翻译:我们展示了一种方法来检测双曲线。 其方法是, 定位经签名的双曲线的零, 其值是雅各雅各雅各雅各雅各雅各雅各雅各雅各的最小单值的最小值。 这使得能够使用二次趋同的根基- 根曲裂技术或 Chebyshev 内插法来定位双曲线点。 只需要计算正单数值, 尽管该方法依赖于存在一个分析或平滑的单数值分解( SVD ) 。 雅各雅各的决定因素是SVD 算法的边际斜坡缩减法的一部分, 消除最小单数值零的零点的斜坡不适应性。 我们使用这一方法来寻找空间半周期间半流动的水流波, 从大增度周期周期的周期波分错位。 水波方方方程式的精确度方程式形成一个统一的绘图框架, 用于计算半周期间断面- Neurit- Neumann 。 我们发现纯重心波的纯度波点波点波点波流的纯度波比重度波比重。 我们从直处一直到直行到直径直向直行, 。 直向直向直向直向直向直行, 直行, 直向直向直向直行, 直行至直向直向直向直向直行, 直行, 直行至直向直向直向的轨道的轨道的轨道的轨道的轨道的波形的波形的波形波形直行至至至至至至直行至直行至直行至直行至直行至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直行至直至直至直至直至直至直至直至直至直至直至直行, 直行, 直至直至直行至直行至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直行, 直至直至直至直至直至直直