Critical Point Identification In 3D Velocity Fields

Classification of flow fields involving strong vortices such as those from bluff body wakes and animal locomotion can provide important insight to their hydrodynamic behavior. Previous work has successfully classified 2D flow fields based on critical points of the velocity field and the structure of an associated weighted graph using the critical points as vertices. The present work focuses on extension of this approach to 3D flows. To this end, we have used the Gauss-Bonnet theorem to find critical points and their indices in the 3D velocity vector field, which functions similarly to the Poincare-Bendixon theorem in 2D flow fields. The approach utilizes an initial search for critical points based on local flow field direction, and the Gauss-Bonnet theorem is used to refine the location of critical points by dividing the volume integral form of the Gauss-Bonnet theorem into smaller regions. The developed method is cable of locating critical points at sub-grid level precision, which is a key factor for locating critical points and determining their associated eigenvalues on coarse grids. To verify this approach, we have applied this method on sample flow fields generated from potential flow theory and numerical methods

Locating the Eigenvalues of Matrix Polynomials

Some known results for locating the roots of polynomials are extended to the case of matrix polynomials. In particular, a theorem by Pellet [Bull. Sci. Math. (2), 5 (1881), pp. 393--395], some results from Bini [Numer. Algorithms, 13 (1996), pp. 179--200] based on the Newton polygon technique, and recent results from Gaubert and Sharify (see, in particular, [Tropical scaling of polynomial matrices, Lecture Notes in Control and Inform. Sci. 389, Springer, Berlin, 2009, pp. 291--303] and [Sharify, Scaling Algorithms and Tropical Methods in Numerical Matrix Analysis: Application to the Optimal Assignment Problem and to the Accurate Computation of Eigenvalues, Ph.D. thesis, École Polytechnique, Paris, 2011]). These extensions are applied to determine effective initial approximations for the numerical computation of the eigenvalues of matrix polynomials by means of simultaneous iterations, like the Ehrlich--Aberth method. Numerical experiments that show the computational advantage of these results are presented

Efficient detection of periodic orbits in chaotic systems by stabilising transformations

An algorithm for detecting periodic orbits in chaotic systems [Phys. Rev. E, 60 (1999), pp.~6172--6175], which combines the set of stabilising transformations proposed by Schmelcher and Diakonos [Phys. Rev. Lett., 78 (1997), pp.~4733--4736] with a modified semi-implicit Euler iterative scheme and seeding with periodic orbits of neighbouring periods, has been shown to be highly efficient when applied to low-dimensional systems. The difficulty in applying the algorithm to higher-dimensional systems is mainly due to the fact that the number of the stabilising transformations grows extremely fast with increasing system dimension. Here we analyse the properties of stabilising transformations and propose an alternative approach for constructing a smaller set of transformations. The performance of the new approach is illustrated on the four-dimentional kicked double rotor map and the six-dimensional system of three coupled Henon maps