19,340 research outputs found
On the Floquet Theory of Delay Differential Equations
We present an analytical approach to deal with nonlinear delay differential
equations close to instabilities of time periodic reference states. To this end
we start with approximately determining such reference states by extending the
Poincar'e Lindstedt and the Shohat expansions which were originally developed
for ordinary differential equations. Then we systematically elaborate a linear
stability analysis around a time periodic reference state. This allows to
approximately calculate the Floquet eigenvalues and their corresponding
eigensolutions by using matrix valued continued fractions
A New Algorithm to Approximate Bivariate Matrix Function via Newton-Thiele Type Formula
A new method for computing the approximation of bivariate matrix function is introduced. It uses the construction of bivariate Newton-Thiele type matrix rational interpolants on a rectangular grid. The rational interpolant is of the form motivated by Tan and Fang (2000), which is combined by Newton interpolant and branched continued fractions, with scalar denominator. The matrix quotients are based on the generalized inverse for a matrix which is introduced by C. Gu the author of this paper, and it is effective in continued fraction interpolation. The algorithm and some other important conclusions such as divisibility and characterization are given. In the end, two examples are also given to show the effectiveness of the algorithm. The numerical results of the second example show that the algorithm of this paper is better than the method of Thieletype matrix-valued rational interpolant in Gu (1997)
The Calogero-Fran\c{c}oise integrable system: algebraic geometry, Higgs fields, and the inverse problem
We review the Calogero-Fran\c{c}oise integrable system, which is a
generalization of the Camassa-Holm system. We express solutions as (twisted)
Higgs bundles, in the sense of Hitchin, over the projective line. We use this
point of view to (a) establish a general answer to the question of
linearization of isospectral flow and (b) demonstrate, in the case of two
particles, the dynamical meaning of the theta divisor of the spectral curve in
terms of mechanical collisions. Lastly, we outline the solution to the inverse
problem for CF flows using Stieltjes' continued fractions.Comment: 22 pages, 2 figure
Discrete integrable systems generated by Hermite-Pad\'e approximants
We consider Hermite-Pad\'e approximants in the framework of discrete
integrable systems defined on the lattice . We show that the
concept of multiple orthogonality is intimately related to the Lax
representations for the entries of the nearest neighbor recurrence relations
and it thus gives rise to a discrete integrable system. We show that the
converse statement is also true. More precisely, given the discrete integrable
system in question there exists a perfect system of two functions, i.e., a
system for which the entire table of Hermite-Pad\'e approximants exists. In
addition, we give a few algorithms to find solutions of the discrete system.Comment: 20 page
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